![Page 1: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/1.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
![Page 2: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/2.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation
![Page 3: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/3.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
![Page 4: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/4.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
measure
![Page 5: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/5.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
interpolation integrable
measure
![Page 6: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/6.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Interpolation of spaces of integrable functions with respect to a vector measure
Antonio Fernández (Universidad de Sevilla)
interpolation integrable
measure
![Page 7: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/7.jpg)
The team (FQM–133).
![Page 8: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/8.jpg)
The Bartle–Dunford–Schwartz integral.
![Page 9: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/9.jpg)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
![Page 10: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/10.jpg)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
![Page 11: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/11.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
f : ⌦ �! K
![Page 12: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/12.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
![Page 13: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/13.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! KRloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
![Page 14: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/14.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! K
h⌫, x0i (A) := h⌫(A), x0i , A 2 RR ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
![Page 15: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/15.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
f : ⌦ �! K
h⌫, x0i (A) := h⌫(A), x0i , A 2 R
k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]
R ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}
![Page 16: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/16.jpg)
The Bartle–Dunford–Schwartz integral.
Measurable. (sigma-algebra)
Null set.
A set is said to be null if
f : ⌦ �! K
A 2 Rloc
h⌫, x0i (A) := h⌫(A), x0i , A 2 R
k⌫k : A 2 Rloc �! k⌫k(A) 2 [0,1]
k⌫k(A) = 0.
R ⌫�! X
x
0�! K, x
0 2 X
0
Rloc := {A ⇢ ⌦ : A \B 2 R, 8B 2 R}
k⌫k(A) := sup {|h⌫, x0i| (A) : kx0k 1}
![Page 17: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/17.jpg)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
![Page 18: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/18.jpg)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
is said to be integrable if:
I) for each x
0 2 X
0,
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
f 2 L
1 (|h⌫, x0i|)
![Page 19: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/19.jpg)
The Bartle–Dunford–Schwartz integral.
The ingredients: set, sigma-algebra
( delta-ring), Banach space and
( ) vector measure.
The definition. A measurable function
is said to be integrable if:
I) for each and
II) For every there exists
such that
x
0 2 X
0,
A 2 Rloc
⌦ ⌃
R X
m : ⌃ ! X ⌫ : R ! X
f : ⌦ �! K
f 2 L
1 (|h⌫, x0i|) Z
Afd⌫ 2 X
⌧Z
Afd⌫, x
0�
=
Z
Afd h⌫, x0i , x
0 2 X
0.
![Page 20: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/20.jpg)
The function spaces.
![Page 21: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/21.jpg)
The function spaces.
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
![Page 22: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/22.jpg)
The function spaces.
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
![Page 23: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/23.jpg)
The function spaces.
For
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
1 p < 1
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
Lpw(⌫) :=
�f : ⌦ �! K : |f |p 2 L1
w(⌫) ,
Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)
.
![Page 24: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/24.jpg)
The function spaces.
For
kfk1 := sup
⇢Z
⌦|f | d |h⌫, x0i| : kx0k 1
�.
kfkp := sup
(✓Z
⌦|f |p d |h⌫, x0i|
◆ 1p
: kx0k 1
).
1 p < 1
L1w(⌫) := {f : ⌦ �! K : I)} ,
L1(⌫) := {f : ⌦ �! K : I)+ II)} .
Lpw(⌫) :=
�f : ⌦ �! K : |f |p 2 L1
w(⌫) ,
Lp(⌫) :=�f : ⌦ �! K : |f |p 2 L1(⌫)
.
![Page 25: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/25.jpg)
Basic properties.
![Page 26: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/26.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
Lp(⌫)
![Page 27: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/27.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
![Page 28: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/28.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
![Page 29: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/29.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
![Page 30: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/30.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
3) If then
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).
![Page 31: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/31.jpg)
Basic properties.
1) is a Banach lattice with o.c.n:
2) is a Banach lattice with the Fatou
property:
3) If then
In general,
Lp(⌫)
0 fn " f 2 Lp(⌫) ) kfn � fkp ! 0.
Lpw(⌫)
0 fn " f 2 L0(⌫),
supn kfnkp < 1
))
8<
:
f 2 Lpw(⌫),
limn
kfnkp = kfkp .
⌫ = µ, Lp(⌫) = Lpw(⌫) = Lp(µ).
Lp(⌫) Lpw(⌫).
![Page 32: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/32.jpg)
Basic properties.
4) and can be non reflexive, even if
Lp(⌫) Lpw(⌫)
p > 1.
![Page 33: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/33.jpg)
Basic properties.
4) and can be non reflexive, even if
5) If then
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
![Page 34: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/34.jpg)
Basic properties.
4) and can be non reflexive, even if
5) If then
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
![Page 35: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/35.jpg)
Basic properties.
4) and can be non reflexive, even if
5) If then
6) In general,
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
p > 1.
1 p1 < p2 < 1,
L1(⌫) 6✓ L1(⌫).
![Page 36: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/36.jpg)
Basic properties.
4) and can be non reflexive, even if
5) If then
6) In general,
7) and
Lp2w (m) ⇢ Lp1
w (m) ⇢ L1w(m) ⇢ L0(m)
[ [ [L1(m) ⇢ Lp2(m) ⇢ Lp1(m) ⇢ L1(m)
Lp(⌫) Lpw(⌫)
Lp(m) Lp[0, 1] Lp(⌫) Lp (R)
p > 1.
1 p1 < p2 < 1,
L1(⌫) 6✓ L1(⌫).
![Page 37: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/37.jpg)
Interpolation methods.
![Page 38: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/38.jpg)
Interpolation methods.
Let be a (compatible) couple of Banach
spaces.
(X0, X1)
X0 \X1,! F(X0, X1) ,!X0 +X1
![Page 39: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/39.jpg)
Interpolation methods.
Let be a (compatible) couple of Banach
spaces. An interpolation method
with the interpolation property:
X0 \X1 ,! F(X0, X1) ,! X0 +X1
F
(X0, X1)
![Page 40: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/40.jpg)
Interpolation methods.
Let be a (compatible) couple of Banach
spaces. An interpolation method
with the interpolation property:
X0T�! Y0
X1T�! Y1
9>=
>;) F(X0, X1)
T�! F(Y0, Y1)
X0 \X1 ,! F(X0, X1) ,! X0 +X1
F
(X0, X1)
![Page 41: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/41.jpg)
Compatible couple of Banach spaces.
![Page 42: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/42.jpg)
Compatible couple of Banach spaces.
If and sigma-finite
1 p0 6= p1 < 1 ⌫ : R ! X
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
![Page 43: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/43.jpg)
Compatible couple of Banach spaces.
If and sigma-finite
we can consider the following (compatible) couples
of Banach spaces:
1 p0 6= p1 < 1 ⌫ : R ! X
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1
w (⌫))
(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1
w (⌫))
![Page 44: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/44.jpg)
Compatible couple of Banach spaces.
If and sigma-finite
we can consider the following (compatible) couples
of Banach spaces:
The ambient space:
1 p0 6= p1 < 1 ⌫ : R ! X
L0(⌫).
⌦ =[
n�1
⌦n [N, (⌦n)n ✓ R, k⌫k(N) = 0,
(Lp0(⌫), Lp1(⌫)) (Lp0w (⌫), Lp1
w (⌫))
(Lp0w (⌫), Lp1(⌫)) (Lp0(⌫), Lp1
w (⌫))
![Page 45: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/45.jpg)
Complex methods. [Calderón (1964)]
![Page 46: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/46.jpg)
Complex methods. [Calderón (1964)]
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
![Page 47: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/47.jpg)
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product: X1�✓0 X✓
1 .
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
![Page 48: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/48.jpg)
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
![Page 49: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/49.jpg)
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
3) If and have the Fatou property:
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
[X0, X1][✓] ,! X1�✓0 X✓
1 = [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
X0 X1
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
![Page 50: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/50.jpg)
Complex methods. [Calderón (1964)]
1) Calderón-Lozanowskiĭ product:
2) If or has order continuous norm:
3) If and have the Fatou property:
4)
[Lp0(µ), Lp1(µ)][✓]=(Lp0(µ))1�✓ (Lp1(µ))✓
=[Lp0(µ), Lp1(µ)][✓]= Lp(µ).
1 p0, p1 1
1
p:=
1� ✓
p0+
✓
p1
[X0, X1][✓] = X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
[X0, X1][✓] ,! X1�✓0 X✓
1 = [X0, X1][✓] , 0 < ✓ < 1.
X0 X1
X1�✓0 X✓
1 .
X0 X1
[X0, X1][✓] ,! X1�✓0 X✓
1 ,! [X0, X1][✓] , 0 < ✓ < 1.
![Page 51: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/51.jpg)
Complex methods: the problem.
For sigma-finite, and
describe
0 < ✓ < 1 p0 6= p1 1⌫ : R ! X
[Lp0(⌫), Lp1(⌫)][✓] , [Lp0(⌫), Lp1(⌫)][✓] ,
[Lp0(⌫), Lp1w (⌫)][✓] , [Lp0(⌫), Lp1
w (⌫)][✓] ,
[Lp0w (⌫), Lp1(⌫)][✓] , [Lp0
w (⌫), Lp1(⌫)][✓] ,
[Lp0w (⌫), Lp1
w (⌫)][✓] , [Lp0w (⌫), Lp1
w (⌫)][✓] .
![Page 52: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/52.jpg)
Complex method [.,.]θ (sigma-algebra).
![Page 53: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/53.jpg)
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
![Page 54: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/54.jpg)
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1
1
p:=
1� ✓
p0+
✓
p1
m : ⌃ ! X
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
![Page 55: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/55.jpg)
Complex method [.,.]θ (sigma-algebra).
![Page 56: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/56.jpg)
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
![Page 57: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/57.jpg)
Complex method [.,.]θ (sigma-algebra).
[Mayoral, Naranjo, Sánchez-Pérez & AF, 2010]
If and
then
1
p:=
1� ✓
p0+
✓
p1
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)
0 < ✓ < 1 p0 6= p1 1m : ⌃ ! X
![Page 58: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/58.jpg)
Complex methods (delta-ring).
![Page 59: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/59.jpg)
Complex methods (delta-ring).
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)(R)
(R)(R)
(R)(⌃)
(⌃)(⌃)
(⌃)
![Page 60: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/60.jpg)
Complex methods (delta-ring).
[·, ·]✓ Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
Lp(m)
Lp(m) Lp(m)
Lp(m)
Lp0(m) Lp0w (m)
Lp1w (m)
Lp1(m)
[·, ·]✓
Lpw(m)Lp
w(m)
Lpw(m) Lp
w(m)
(R)
(R)(⌃)
(⌃)(⌃)
(⌃)
![Page 61: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/61.jpg)
Complex methods (delta-ring). Example.
![Page 62: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/62.jpg)
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
![Page 63: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/63.jpg)
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
It is not difficult to show that
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.
![Page 64: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/64.jpg)
Complex methods (delta-ring). Example.
Let be the delta-ring of finite subsets of
natural numbers, and consider the measure
It is not difficult to show that
Then, for 1 p0, p1 < 1,
Pf (N)
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0.
Lp(⌫) = c0 ✓ Lpw(⌫) = `1, 1 p < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = [`1, `1][✓] = `1 = Lpw(⌫),
[Lp0(⌫), Lp1(⌫)][✓] = [c0, c0][✓] = c0 = Lp(⌫).
![Page 65: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/65.jpg)
Intermediate spaces (sigma-algebra).
![Page 66: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/66.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
![Page 67: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/67.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
![Page 68: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/68.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
![Page 69: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/69.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
![Page 70: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/70.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Lp0(m) + Lp1(m)
Lp0w (m) + Lp1(m)
Lp0(m) + Lp1w (m)
Lp0w (m) + Lp1
w (m)
![Page 71: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/71.jpg)
Intermediate spaces (sigma-algebra).
If and where
then
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
1 p0 6= p1 1r := min{p0, p1} < s := max{p0, p1},
r < p < s,
Ls(m) ✓ Lsw(m) ✓ Lp(m) ✓ Lp
w(m) ✓ Lr(m) ✓ Lrw(m).
Lp0(m) \ Lp1(m)
Lp0w (m) \ Lp1(m)
Lp0(m) \ Lp1w (m)
Lp0w (m) \ Lp1
w (m)
Lp0(m) + Lp1(m)
Lp0w (m) + Lp1(m)
Lp0(m) + Lp1w (m)
Lp0w (m) + Lp1
w (m)
![Page 72: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/72.jpg)
Gain of integrability (sigma-algebra).
![Page 73: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/73.jpg)
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
r < s, Lsw(m) ✓ Lr(m).
![Page 74: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/74.jpg)
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f 2 Lsw(m) ✓ Lr
w(m)f
⌦
r < s, Lsw(m) ✓ Lr(m).
![Page 75: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/75.jpg)
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f�[fn] 2 Lr(m)
f 2 Lsw(m) ✓ Lr
w(m)f
⌦
n f�[fn]
r < s, Lsw(m) ✓ Lr(m).
![Page 76: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/76.jpg)
Gain of integrability (sigma-algebra).
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then
f�[fn] 2 Lr(m)
f 2 Lsw(m) ✓ Lr
w(m)
��f � f�[fn]
��Lr
w(m)! 0
f
⌦
n f�[fn]
r < s, Lsw(m) ✓ Lr(m).
![Page 77: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/77.jpg)
Intermediate spaces (delta-ring).
![Page 78: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/78.jpg)
Intermediate spaces (delta-ring).
If then
1 p0 < p < p1 < 1,
(i) Lp0w (⌫) \ Lp1
w (⌫) ✓ Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫).
(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).
(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1
w (⌫).
(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0
w (⌫) + Lp1(⌫).
![Page 79: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/79.jpg)
Intermediate spaces (delta-ring).
If then
Nevertheless,
1 p0 < p < p1 < 1,
(i) Lp0w (⌫) \ Lp1
w (⌫) ✓ Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫).
(ii) Lp0(⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1(⌫).
(iii) Lp0(⌫) \ Lp1w (⌫) ✓ Lp(⌫) ✓ Lp0(⌫) + Lp1
w (⌫).
(iv) Lp0w (⌫) \ Lp1(⌫) ✓ Lp(⌫) ✓ Lp0
w (⌫) + Lp1(⌫).
Lp0w (⌫) \ Lp1
w (⌫) 6✓ Lp(⌫),
Lpw(⌫) 6✓ Lp0(⌫) + Lp1(⌫).
![Page 80: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/80.jpg)
Locally strongly additive measure.
![Page 81: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/81.jpg)
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
![Page 82: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/82.jpg)
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
![Page 83: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/83.jpg)
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
2) is not
locally strongly additive.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0
![Page 84: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/84.jpg)
Locally strongly additive measure.
is said to be locally strongly additive if
for each disjoint sequence
such that
1) The Lebesgue measure is l.s.a.
2) is not
locally strongly additive.
3) Every is (locally) strongly additive.
⌫ : R ! X
limn!1
k⌫(An)kX = 0
(An)n ✓ R k⌫k⇣S
n�1 An
⌘< 1.
m : ⌃ ! X
k⌫k(A) = 1,? 6= A 2 Pf (N), k⌫k(?) = 0
⌫ : A 2 Pf (N) �! ⌫(A) := �A 2 c0
![Page 85: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/85.jpg)
Intermediate spaces (delta-ring).
![Page 86: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/86.jpg)
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1 p0 < p1 < 1.
![Page 87: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/87.jpg)
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
⌫
1 p0 < p1 < 1.
![Page 88: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/88.jpg)
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
⌫
1 p0 < p1 < 1.
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
![Page 89: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/89.jpg)
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
2) for every (some)
⌫
1 p0 < p1 < 1.
p0 < p < p1.
Lp0w (⌫) \ Lp1
w (⌫) ✓ Lp(⌫)
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
![Page 90: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/90.jpg)
Intermediate spaces (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following assertions are
equivalent:
1) is locally strongly additive.
2) for every (some)
3) for every (some)
⌫
1 p0 < p1 < 1.
p0 < p < p1.
p0 < p < p1.
Lp0w (⌫) \ Lp1
w (⌫) ✓ Lp(⌫)
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫)
�B 2 Rloc, �B 2 L1
w(⌫) ) �B 2 L1(⌫)�
![Page 91: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/91.jpg)
Gain of integrability (delta-ring).
![Page 92: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/92.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
![Page 93: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/93.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 94: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/94.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
f�[fn]
n
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 95: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/95.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 96: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/96.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 97: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/97.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
��f�[f>n]
��L
p0w (⌫)
! 0
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 98: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/98.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
⌫ p0 < p < p1,
f
⌦
n
1
n
���f�[f< 1n ]
���L
p1w (⌫)
! 0
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 99: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/99.jpg)
Gain of integrability (delta-ring).
[del Campo, Mayoral, Naranjo & AF, 2012]
If is l.s.a. and then
f
⌦
n
1
n
���f�[f< 1n ]
���L
p1w (⌫)
! 0
f�[ 1nfn] 2 Lp0(⌫) + Lp1(⌫)
⌫ p0 < p < p1,
Lpw(⌫) ✓ Lp0(⌫) + Lp1(⌫).
��f�[f>n]
��L
p0w (⌫)
! 0
Lpw(⌫) ✓ Lp0
w (⌫) + Lp1w (⌫)
![Page 100: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/100.jpg)
Complex methods: equalities (R).
![Page 101: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/101.jpg)
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
0 < ✓ < 1 p0 < p1 < 1.
![Page 102: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/102.jpg)
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
⌫
0 < ✓ < 1 p0 < p1 < 1.
![Page 103: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/103.jpg)
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
2)
⌫
0 < ✓ < 1 p0 < p1 < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = Lp(⌫).
1
p:=
1� ✓
p0+
✓
p1
![Page 104: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/104.jpg)
Complex methods: equalities (R).
[del Campo, Mayoral, Naranjo & AF, 2012]
Let The following
assertions are equivalent:
1) is locally strongly additive.
2)
3)
⌫
0 < ✓ < 1 p0 < p1 < 1.
[Lp0w (⌫), Lp1
w (⌫)][✓] = Lp(⌫).
[Lp0(⌫), Lp1(⌫)][✓] = [Lp0w (⌫), Lp1(⌫)][✓]
= [Lp0(⌫), Lp1w (⌫)][✓]
= Lpw(⌫).1
p:=
1� ✓
p0+
✓
p1
![Page 105: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/105.jpg)
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
![Page 106: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/106.jpg)
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
1
p:=
1� ✓
p0+
✓
p1
Lp(⌫) [Lp0w (⌫), Lp1
w (⌫)][✓] Lpw(⌫)
![Page 107: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/107.jpg)
Complex methods (delta-ring).
If is not locally strongly additive, and
⌫
0 < ✓ < 1 p0 < p1 < 1.
1
p:=
1� ✓
p0+
✓
p1
Lp(⌫) [Lp0(⌫), Lp1(⌫)][✓] Lpw(⌫)
Lp(⌫) [Lp0w (⌫), Lp1
w (⌫)][✓] Lpw(⌫)
![Page 108: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/108.jpg)
The real method (.,.)θ,q. [Lions & Peetre (1964)]
![Page 109: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/109.jpg)
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
(X0, X1)
0 < ✓ < 1 q 1,
![Page 110: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/110.jpg)
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
is the space of
those such that
is finite.
f 2 X0 +X1
(X0, X1)
0 < ✓ < 1 q 1, (X0, X1)✓,q
kfk✓,q :=
8>><
>>:
Z 1
0
�t�✓K(t, f)
�q dt
t
� 1q
, q < 1
supt>0
t�✓K(t, f), q = 1
![Page 111: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/111.jpg)
The real method (.,.)θ,q. [Lions & Peetre (1964)]
Given: a couple of Banach spaces and
is the space of
those such that
is finite.
f 2 X0 +X1
(X0, X1)
0 < ✓ < 1 q 1, (X0, X1)✓,q
kfk✓,q :=
8>><
>>:
Z 1
0
�t�✓K(t, f)
�q dt
t
� 1q
, q < 1
supt>0
t�✓K(t, f), q = 1
K(t, f) := inf {kf0kX0 + tkf1kX1 :
f0 2 X0, f1 2 X1, f = f0 + f1} .
![Page 112: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/112.jpg)
The real method: problem/classical results.
![Page 113: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/113.jpg)
The real method: problem/classical results.
For we want
to describe with 0 < ✓ < 1 q 1.(X0, X1)✓,q ,
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
![Page 114: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/114.jpg)
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
![Page 115: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/115.jpg)
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
![Page 116: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/116.jpg)
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
![Page 117: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/117.jpg)
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
![Page 118: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/118.jpg)
The real method: problem/classical results.
For we want
to describe with
Let be a positive sigma-finite measure, and
Then:
0 < ✓ < 1 q 1.(X0, X1)✓,q ,
µ
0 < ✓ < 1 p0 6= p1 1.
1
p:=
1� ✓
p0+
✓
p1
Xk 2 {Lpk(m), Lpkw (m)} , k = 0, 1,
•�L1(µ), L1(µ)
�✓,q
= L1
1�✓ ,q(µ).
• (Lp0(µ), Lp1(µ))✓,q = Lp,q(µ).
• Lp,p(µ) = Lp(µ), 1 p < 1.
• K(t, f, L1(µ), L1(µ)) =R t0 f⇤(s)ds.
![Page 119: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/119.jpg)
Lorentz spaces of a vector measure.
![Page 120: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/120.jpg)
Lorentz spaces of a vector measure.
Let and consider
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
⌃m�! X
x
0�! R, x
0 2 X
0,
![Page 121: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/121.jpg)
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
⌃m�! X
x
0�! R, x
0 2 X
0,
![Page 122: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/122.jpg)
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f : ⌦ �! R
⌃m�! X
x
0�! R, x
0 2 X
0,
![Page 123: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/123.jpg)
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
⌃m�! X
x
0�! R, x
0 2 X
0,
![Page 124: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/124.jpg)
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
kfkp,q
:= supn
kfkL
p,q(|hm,x
0i|) : kx0k 1
o
.
⌃m�! X
x
0�! R, x
0 2 X
0,
![Page 125: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/125.jpg)
Lorentz spaces of a vector measure.
Let and consider
If we say that if:
i) measurable and
ii)
hm,x
0i (A) := hm(A), x0i , A 2 ⌃.
1 p, q 1, f 2 Lp,qw (m)
f 2 L
p,q (|hm,x
0i|) , x
0 2 X
0.
f : ⌦ �! R
kfkp,q
:= supn
kfkL
p,q(|hm,x
0i|) : kx0k 1
o
.
⌃m�! X
x
0�! R, x
0 2 X
0,
�L1w(m), L1(m)
�✓,q
S(⌃)Lp,q
w (m)✓ Lp,q
w (m).
![Page 126: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/126.jpg)
Distribution and decreasing rearrangement.
![Page 127: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/127.jpg)
Distribution and decreasing rearrangement.
Distribution function. measurable:
f : ⌦ �! Rkmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)
kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})
= supn
|hm,x
0i|f (t) : kx0k 1
o
.
![Page 128: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/128.jpg)
Distribution and decreasing rearrangement.
Distribution function. measurable:
Decreasing rearrangement function.
kmkf : t 2 [0,1) �! kmkf (t) 2 [0,1)
kmkf (t) := kmk ({w 2 ⌦ : |f(w)| > t})
= supn
|hm,x
0i|f (t) : kx0k 1
o
.
f
⇤(s) := inf {t � 0 : kmkf
(t) s}= �kmkf
(s)
= sup {f⇤x
0(s) : kx0k 1}
f : ⌦ �! R
![Page 129: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/129.jpg)
Lorentz spaces of the semivariation.
![Page 130: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/130.jpg)
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! RLp,q(kmk)
kfkLp,q(kmk) < 1,
![Page 131: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/131.jpg)
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! R
1 p, q < 1
Lp,q(kmk)
kfkLp,q(kmk) :=
Z 1
0
⇣s
1p f⇤(s)
⌘q ds
s
� 1q
,
kfkLp,q(kmk) < 1,
![Page 132: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/132.jpg)
Lorentz spaces of the semivariation.
For the Lorentz space
consists of all measurable functions
such that where
1 p, q 1,
f : ⌦ �! R
1 p, q < 1
1 p < 1, q = 1
Lp,q(kmk)
kfkLp,q(kmk) :=
Z 1
0
⇣s
1p f⇤(s)
⌘q ds
s
� 1q
,
kfkLp,1(kmk) := supn
s1p f⇤(s) : s > 0
o
.
kfkLp,q(kmk) < 1,
![Page 133: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/133.jpg)
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
Lp,q(kmk)kf + gkLp,q(kmk) C(p, q)
�kfkLp,q(kmk) + kgkLp,q(kmk)
�
![Page 134: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/134.jpg)
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
![Page 135: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/135.jpg)
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
![Page 136: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/136.jpg)
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
3) Inclusions.
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
![Page 137: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/137.jpg)
Lorentz spaces of the semivariation.
✓✓
✓✓
Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
...
1 p1 < p2 < 1
![Page 138: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/138.jpg)
Lorentz spaces of the semivariation.
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
![Page 139: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/139.jpg)
Lorentz spaces of the semivariation.
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
![Page 140: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/140.jpg)
Lorentz spaces of the semivariation.
L1(m)✓
✓ ✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
...
......
...
1 p1 < p2 < 1
![Page 141: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/141.jpg)
Lorentz spaces of the semivariation.
L1(m)✓
✓
✓✓
✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
L1,1(kmk)
L1,1(kmk)
...
...
......
...
1 p1 < p2 < 1
![Page 142: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/142.jpg)
Lorentz spaces of the semivariation.
L1(m)✓
✓
✓✓
✓✓
✓✓✓
✓✓
Lp2,1(kmk)
Lp2,p2(kmk)
Lp2,1(kmk) Lp1,1(kmk)
Lp1,p1(kmk)
Lp1,1(kmk)
L1,1(kmk)
L1,1(kmk)
...
...
......
... Lp(m)
Lpw(m)
1 p1 < p2 < 1
![Page 143: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/143.jpg)
Lorentz spaces of the semivariation.
1) is a quasi-Banach lattice.
2) has separating dual:
a) b) and
3) Inclusions.
4) For
Lp,q(kmk)
Lp,q(kmk)p = q = 1, p > 1 1 q 1.
kf + gkLp,q(kmk) C(p, q)�kfkLp,q(kmk) + kgkLp,q(kmk)
�
Lp,q(kmk) ✓ Lp,q(µ), 1 p, q 1
1 p < 1,
Lp,p(kmk) Lp(m) Lpw(m) Lp,1(kmk).
![Page 144: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/144.jpg)
Estimates for the K−functional.
![Page 145: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/145.jpg)
Estimates for the K−functional.
If and then
f 2 L1w(m) t > 0,
![Page 146: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/146.jpg)
Estimates for the K−functional.
If and then
a)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
![Page 147: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/147.jpg)
Estimates for the K−functional.
If and then
a)
b)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
K�t, f, L1
w(m), L1(m)�
Z t
0f⇤(s)ds.
![Page 148: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/148.jpg)
Estimates for the K−functional.
If and then
a)
b)
c)
f 2 L1w(m) t > 0,
tf⇤(t) K�t, f, L1
w(m), L1(m)�.
K�t, f, L1
w(m), L1(m)�
Z t
0f⇤(s)ds.
f 2 L1(m) 6)Z t
0f⇤(s)ds < 1,
6)Z 1
0f⇤(s)ds < 1.
![Page 149: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/149.jpg)
The real method. [Mayoral, Naranjo & AF, 2011]
![Page 150: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/150.jpg)
The real method. [Mayoral, Naranjo & AF, 2011]
1) For
with equivalence of norm−quasinorm.
0 < ✓ < 1 q 1,�L1(m), L1(m)
�✓,q
=�L1w(m), L1(m)
�✓,q
= L1
1�✓ ,q(kmk),
![Page 151: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/151.jpg)
The real method. [Mayoral, Naranjo & AF, 2011]
1) For
with equivalence of norm−quasinorm.
2) For
0 < ✓ < 1 q 1,
1 p0 6= p1 1; 0 < ✓ < 1 q 1,
1
p=
1� ✓
p0+
✓
p1.
�L1(m), L1(m)
�✓,q
=�L1w(m), L1(m)
�✓,q
= L1
1�✓ ,q(kmk),
(Lp0(m), Lp1(m))✓,q = (Lp0w (m), Lp1
w (m))✓,q
= Lp,q(kmk),
![Page 152: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/152.jpg)
The real method: consequences.
![Page 153: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/153.jpg)
The real method: consequences.
Consequence 1. If the space is
normable.
Lp,q(kmk)p > 1,
![Page 154: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/154.jpg)
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)
![Page 155: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/155.jpg)
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
[Beauzamy 1978] If and the
inclusion is weakly compact,
then is reflexive.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)0 < ✓ < 1 < q < 1
X0 \X1 ✓ X0 +X1
(X0, X1)✓,q
![Page 156: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/156.jpg)
The real method: consequences.
Consequence 1. If the space is
normable.
Concequence 2. If the space
is reflexive.
[Beauzamy 1978] If and the
inclusion is weakly compact,
then is reflexive.
[Mayoral, Naranjo, Sáez, Sánchez-Pérez & AF 2006]
If then is
weakly compact.
Lp,q(kmk)p > 1,
1 < p, q < 1,
Lp,q(kmk)0 < ✓ < 1 < q < 1
X0 \X1 ✓ X0 +X1
(X0, X1)✓,q
1 p0 < p1 1, Lp1w (m) ✓ Lp0(m)
![Page 157: Workshop Operators and Banach lattices WELCOME …obl/AFernandez.pdf · Welcome! This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices](https://reader035.vdocuments.net/reader035/viewer/2022070902/5f59221561c5417be571187c/html5/thumbnails/157.jpg)
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Welcome!
This is a two-day workshop bringing together specialists working in Operator theory and Banach lattices. The meeting will beheld at the Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, on Thursday 25 and Friday 26 October 2012.
If you wish to attend, please register by sending an email to [email protected], indicating your name and institution, beforeOctober 21. Also, please let us know if you are interested in presenting a horizontal poster (paper, preprint...). There will be noregistration fee.
We look forward to seeing you in Madrid. The organizers:
Julio Flores (URJC), Francisco L. Hernández (UCM), Pedro Tradacete (UC3M).
Speakers
Félix Cabello, Universidad de Extremadura.
María J. Carro, Universidad de Barcelona.
Ben de Pagter, Delft University of Technology (The Netherlands).
Antonio Fernández-Carrión, Universidad de Sevilla.
Manuel González, Universidad de Cantabria.
Miguel Martín, Universidad de Granada.
Yves Raynaud, CNRS/Université Paris VI (France).
José Rodríguez, Universidad de Murcia.
Luis Rodríguez-Piazza, Universidad de Sevilla.
Workshop Operators and Banach lattices
Universidad Complutense de Madrid,Facultad de Ciencias Matemáticas,25-26 October 2012
WELCOME PROGRAMME LOCAL INFO PARTICIPANTS
Interpolation of spaces of integrable functions with respect to a vector measure
Antonio Fernández (Universidad de Sevilla)