big bases ben mathesbig bases ben mathes overview kalisch one dimensional example two dimensional...
TRANSCRIPT
![Page 1: Big Bases Ben MathesBig Bases Ben Mathes Overview Kalisch One dimensional example Two dimensional example Sarason - Waterman Sarason Waterman Their result Strictly cyclic algebras](https://reader035.vdocuments.net/reader035/viewer/2022071211/6022c9fabbace9445f03de60/html5/thumbnails/1.jpg)
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.1
——–Big Basesand large diagonal operators
Big Bases May 2008
Ben MathesColby College
![Page 2: Big Bases Ben MathesBig Bases Ben Mathes Overview Kalisch One dimensional example Two dimensional example Sarason - Waterman Sarason Waterman Their result Strictly cyclic algebras](https://reader035.vdocuments.net/reader035/viewer/2022071211/6022c9fabbace9445f03de60/html5/thumbnails/2.jpg)
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.2
Big Bases and Large Diagonal Operators
2666666666666666666666666666664
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18
3777777777777777777777777777775
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.3
Overview
1 KalischOne dimensional example: Mx − VTwo dimensional example: Mx − V + i(Ny −W )
2 Sarason - WatermanInvariant subspaces of Mx + VInvariant subspaces of Mx − VSpectral synthesis!
3 Strictly cyclic algebrasSarason’s algebraTensor productsNew Examples
4 Substrictly cyclic algebrasSarason, Erdos, ...Idealsa substrict algebra
5 End
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.4
• Theorem
(Kalisch) Given any compact subset of the plane, there existsan operator whose spectrum equals that compact set andconsists entirely of simple point spectrum.
• We say that α is in the point spectrum of T when
Tv = αv
for some v 6= 0, and it is simple point spectra if thecorresponding eigenspace is one dimensional.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.5
First Kalisch paper......Mx − V
1
1t
A big basis......
{χ[t,1] : t ∈ [o,1)
}
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.6
apply Mx (red) and −V (blue), then add ......
1
1t
A continuum of eigenvectors for Mx − V ......
χ[t,1] 7→ tχ[t,1]
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.7
T = Mx − V
Theorem
(Kalisch) Take any closed subset E of (0,1), and letMEdenote the closed linear span of the correspondingeigenvectors. Then the restriction of T to this invariantsubspace has spectrum E and consists entirely of simple pointspectra.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.8
T = Mx − V + i(Ny −W )
• To accommodate sets with planar interior, move to L2(I)with I the unit square.
• Use the operator T = Mx − V + i(Ny −W ) whosespectrum is the closed unit square.
• Show that every α in the interior of I is simple pointspectra.
• Theorem
(Kalisch) Take any closed subset E contained in the interior ofI, and letME denote the closed linear span of thecorresponding eigenvectors. Then the restriction of T to thisinvariant subspace has spectrum E and consists entirely ofsimple point spectra.
• Technique of proof: here’s an operator, let’s roll up oursleeves and compute the spectrum!
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.9
Sarason and T = Mx + V
• Use V to map L2[0,1] bijectively onto the set A ofabsolutely continuous functions that vanish at the origin.
• Put a norm on A so that V becomes a unitary.• Observe that Mx + V is then unitarily equivalent to
multiplication by x on A• Since A is an algebra, find the closed ideals to
characterize the invariant subspaces.• Technique of proof: Banach algebras - function spaces
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.10
The relation of Sarason’s operator to Kalisch......
1
1t
Eigenvectors for T ∗......
{χ[0,t] : t ∈ (0,1]
}χ[0,t] 7→ tχ[0,t]
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.11
Like Kalisch, Waterman works with T = Mx − V(Waterman was a student of Kalisch)
• Characterize the algebra generated by T , the algebra of“large diagonal operators"
• The mappingχ[t,1] 7→ h(t)χ[t,1]
extends to a bounded operator when h is absolutelycontinuous on [0,1) with extra technical conditions aboutwhat happens at 1
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.12
A very nice algebra!
Theorem
(Sarason-Waterman) These operators admit spectralsynthesis. From Sarason’s Banach algebra perspective, thismeans every closed ideal is an intersection of maximal ideals.From Waterman’s perspective, every invariant subspace isspanned by eigenvectors.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.13
Definition (Hilbert Ring)
A Hilbert Ring is a Hilbert space that has a boundedmultiplication defined on it.
Definition (Strictly cyclic algebra)
A commutative strictly cyclic algebra is the set of multipliers{Mx : x ∈ H } where H is a unital commutative Hilbert ring.
Definition (Strictly cyclic operator)
A strictly cyclic operator is a multiplier corresponding to asingly generated unital Hilbert ring.
Definition (Substrictly cyclic operator)
A substrictly cyclic operator is a multiplier corresponding to asingly generated Hilbert ring.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.14
A cool thing...
Being “selfdual", the maximal ideal space of a Hilbert ring livesinside the Hilbert space.
Examples
1 The algebra A of absolutely continuous functions, normedas Sarason did, is a unital Hilbert ring.
2 We can move the multiplicative structure of Sarason’salgebra to L2[0,1] obtaining the multiplication
f ? g = Vf g + f Vg
defined on L2[0,1]
3 Our big basis{χ[0,t] : t ∈ (0,1]
}is then seen to be the
maximal ideal space.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.15
Adjoints of multipliers...
Assume A is a commutative Banach algebra, a ∈ A, and Mathe multiplier on A:
Ma(b) = ab.
1 Every multiplicative functional is an eigenvector for M∗a .2 The eigenspaces are one-dimensional when a is a
generator.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.16
To use quotients...
Definition
A commutative Banach algebra A is Shilov regular when everyclosed subset of the maximal ideal space can be separatedfrom points not in it using elements of A:
< a,e >= 0 for e ∈ E but < a, f >6= 0
1 This is exactly what one needs to say that, for each closedE in the maximal ideal space, the maximal ideal space ofA/E⊥ is E .
2 This is a property lacking in many of the traditionalexamples of strictly cyclic algebras, those arising fromweighted shifts
3 Sarason’s algebra has this property, which is why Kalisch’smethod of restricting his operator to subspaces yielded anoperator with pure point spectrum.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.17
Use tensor products to fatten...
1 If H is Sarason’s Hilbert ring, then its spectrum is [0,1]
2 The Hilbert tensor product is also a Hilbert ring (that canbe identified with L2(I)) whose spectrum is the unit square.
3 Shilov regularity persists
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.18
Recapturing Kalisch...
1 Let T = Mx + V , a generator of Sarason’s algebra withspectrum [0,1]
2 The operator A = I ⊗ T + i(T ⊗ I) has spectrum equal tothe unit square.
3 Given a desired compact set, scale it and translate to fitinside the square, call the result E
4 The image of A in the quotient has spectrum E (regularityis used here).
5 The adjoint of this image is (unitarily equivalent to)Kalisch’s restriction!
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.19
Many new examples of strictly cyclic algebras andoperators...
Theorem
Given any compact subset of the plane, there exists a rationallystrictly cyclic operator whose spectrum equals that compactset.
Theorem
Given any polynomially convex compact subset of the plane,there exists a strictly cyclic operator whose spectrum equalsthat compact set.
Theorem
Given any compact subset of Euclidean space, there exists acommutative semisimple strictly cyclic algebra whose spectrumequals that compact set.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.20
Concept of substrictly cyclic operator...
Examples
1 Mx + V is a generator relative to Sarason’s multiplication
f ? g = Vf g + f Vg
2 Any Hilbert-Schmidt diagonal operator with distinct entries,the multiplier corresponding to a generator for pointwisemultiplication on `2
(ai)(bi) = (aibi)
3 The Volterra operator is also an example, with convolutionmultipication
f ◦ g(x) =
∫ x
0f (s)g(x − s)ds
Every substrictly cyclic operator is the restriction of a strictlycyclic operator to a maximal ideal.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.21
Can recapture another Theorem of Sarason:
Theorem
1 The strongly closed algebra generated by the Volterraoperator is maximal abelian.
2 A Kaplansky density result holds: the operators in the unitball of the strongly closed algebra generated by theVolterra operator are strong limits of operators in the unitball of multipliers.
3 The identity element is in the strongly closed algebragenerated by just the Volterra operator.
The ultra simple proof: there is an approximate identity inL2[0,1] for convolution, and the corresponding multipliers arecontractions.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.22
Can use this theory to characterize the strongly closedideals in the Volterra algebra
Theorem
1 The strongly closed ideals form a continuous chain It witht ∈ (0,1).
2 The annihilator of It is I1−t .3 These ideals consist entirely of nilpotents: the ideal I1/2
consists of square zero nilpotents.4 The ideal I1/n consists of nilpotents of order n.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.23
Examples
• The multiplication is on `2 via
(ai)(bi) = (aibi)
• The strictly cyclic algebra is
αI +
0 0 0 0 0
x1 x1 0 0 0x2 0 x2 0 0x3 0 0 x3 0
... 0 0. . . 0
• For the substrictly cyclic algebra, the multipliers are the
diagonal Hilbert-Schmidt operators, and the substrictlycyclic algebra is the algebra of all bounded diagonaloperators.
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Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.24
![Page 25: Big Bases Ben MathesBig Bases Ben Mathes Overview Kalisch One dimensional example Two dimensional example Sarason - Waterman Sarason Waterman Their result Strictly cyclic algebras](https://reader035.vdocuments.net/reader035/viewer/2022071211/6022c9fabbace9445f03de60/html5/thumbnails/25.jpg)
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.25
Dedicated to Heydar Radjavi