Research ArticleChoi-Davis-Jensen Inequalities in Semifinitevon Neumann Algebras

Turdebek N Bekjan1 Kordan N Ospanov2 and Asilbek Zulkhazhav2

1College of Mathematics and Systems Science Xinjiang University Urumqi 830046 China2LN Gumilyov Eurasian National University Astana 010008 Kazakhstan

Correspondence should be addressed to Turdebek N Bekjan bekjantyahoocom

Received 4 July 2015 Accepted 19 August 2015

Academic Editor Quanhua Xu

Copyright copy 2015 Turdebek N Bekjan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We prove the Choi-Davis-Jensen type submajorization inequalities on semifinite von Neumann algebras for concave functions andconvex functions

1 Introduction

Let A and B be 119862lowast-algebras and let Φ be a linear map

between A and B It is said to be positive if for all positiveoperators 119860 isin A we have Φ(119860) ge 0 If for all strictly positiveoperators 119860 isin A (119860 gt 0) it follows that Φ(119860) is strictly pos-itive thenΦ is said to be strictly positiveΦ is called unital ifΦ(1) = 1 where 1 denotes the unities of the algebras

Davis [1] and Choi [2] showed that ifΦ is a unital positivelinearmap on119861(H) and if119891 is an operator convex function onan interval 119868 then the so-called Choi-Davis-Jensen inequality

119891 (Φ (119860)) le Φ (119891 (119860)) (1)

holds for every self-adjoint operator119860 onH whose spectrumis contained in 119868 where 119861(H) is the119862lowast-algebra of all boundedlinear operators onHilbert spaceH Khosravi et al [3] provedthat (1) still holds for positive linear map Φ A rarr 119861(H)

with 0 lt Φ(1) le 119868 Antezana et al [4] obtained the followingtype of Choi-Davis-Jensen inequality LetAB be unital119862lowast-algebrasΦ 119860 rarr 119861 a positive unital linear map 119891 a convexfunction defined on an open interval 119868 and 119860 isin A such that119860 is self-adjoint and 120590(119886) sub 119868 IfB is a von Neumann algebraand 119891 is monotone then 119891(Φ(119860)) ⪯ Φ(119891(119860)) (spectralpreorder) One can find some related results to these topicsin [5ndash8]

In [3] the authors proved the following refinement ofthe Choi-Davis-Jensen inequality let Φ

1 Φ

119899be strictly

positive linear maps from a unital 119862lowast-algebraA into a unital119862lowast-algebra B and let Φ = sum

119899

119895=1Φ119895be unital If 119891 is an

operator convex function on an interval 119868 then for every self-adjoint operator 119860 isin A with spectrum contained in 119868

119891 (Φ (119860)) le

119899

sum

119895=1

Φ119895(1)12

sdot 119891 (Φ119895(1)minus12

Φ119895(119860)Φ

119895(1)minus12

)Φ119895(1)12

le Φ (119891 (119860))

(2)

The purpose of this paper is to extend (2) for measurableoperators and for convex functions Let N M be semifinitevonNeumann algebrasΦ

1 Φ

119899positive linear continuous

maps from 1198710(N) into 119871

0(M) such that Φ

1|N Φ

119899|N are

positive linearmaps fromN intoM andΦ = sum119899

119895=1Φ119895unital

and let 119909119896isin 1198710(N)+ (119896 = 1 2 119899) We will prove

119899

sum

119896=1

Φ119896(119891 (119909119896)) ≼ 119891(

119899

sum

119896=1

Φ119896(119909119896)) (3)

for any concave function 119891 [0infin) rarr [0infin) and

119892(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

Φ119896(119892 (119909119896)) (4)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 208923 5 pageshttpdxdoiorg1011552015208923

2 Journal of Function Spaces

for any convex function 119892 [0infin) rarr [0infin) with 119892(0) = 0where ldquo≼rdquo mains submajorization

This paper is organized as follows Section 2 containssome preliminary definitions In Section 3 we prove themainresult and related results

2 Preliminaries

We use standard notions from theory of noncommutative119871119901-spaces Our main references are [9 10] (see also [9] for

more historical references) Throughout the paper let Mbe a semifinite von Neumann algebra acting on a Hilbertspace H with a normal semifinite faithful trace 120591 Let 119871

0(M)

denote the topological lowast-algebra of measurable operatorswith respect to (M 120591) The topology of 119871

0(M) is determined

by the convergence in measure The trace 120591 can be extendedto the positive cone 119871+

0(M) of 119871

0(M)

120591 (119909) = int

infin

0

120582 119889120591 (119890120582(119909)) (5)

where 119909 = intinfin

0

120582 119889119890120582(119909) is the spectral decomposition of 119909

For 119909 isin 1198710(M) we define

120582119904(119909) = 120591 (119890

perp

119904(|119909|)) (119904 gt 0)

120583119905(119909) = inf 119904 gt 0 120582

119904(119909) le 119905 (119905 gt 0)

(6)

where 119890perp

119904(|119909|) = 119890

(119904infin)(|119909|) is the spectral projection of

|119909| associated with the interval (119904infin) The function 119904 997891rarr

120582119904(119909) is called the distribution function of 119909 and 120583

119905(119909) is the

generalized 119904-number of 119909 We will denote simply by 120582(119909) and120583(119909) the functions 119904 997891rarr 120582

119904(119909) and 119905 997891rarr 120583

119905(119909) respectively

It is easy to check that both are decreasing and continuousfrom the right on (0infin) For further informationwe refer thereader to [11]

If 119909 119910 isin 1198710(M) then we say that 119910 is submajorised by 119909

(in the sense ofHardy Littlewood and Polya) andwrite119910 ≼ 119909

if and only if

int

119905

0

120583119904(119910) 119889119904 le int

119905

0

120583119904(119909) 119889119904 forall119905 gt 0 (7)

We remark that if M = M119899and 120591 is the standard trace

then

120583119905(119909) = 119904

119895(119909) 119905 isin [119895 minus 1 119895) 119895 = 1 2 (8)

and if 119909 119910 isin M then 119910 ≼ 119909 is equivalent to

119896

sum

119895=1

119904119895(119910) le

119896

sum

119895=1

119904119895(119909) 1 le 119896 le 119899 (9)

For further information we refer the reader to [11ndash13]Let 119909 119910 be self-adjoint elements ofM we say that 119909 spec-

trally dominates 119910 denoted by 119910 ⪯ 119909 if 119890(120572infin)

(119910) is equiva-lent in the sense ofMurray-vonNeumann to a subprojectionof 119890(120572infin)

(119909) for every real number 120572 (see [6]) It is clear thatif 119909 spectrally dominates 119910 then 119910 is submajorised by 119909

3 Main Results

Lemma 1 LetN andM be semifinite von Neumann algebrasLet Φ be a positive linear continuous map from 119871

0(N) into

1198710(M) such that the restriction of Φ on N is a unital positive

linear map fromN intoM

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

Φ(119891 (119909)) ≼ 119891 (Φ (119909)) forall119909 isin 1198710(N)+

(10)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (Φ (119909)) ≼ Φ (119892 (119909)) forall119909 isin 1198710(N)+

(11)

Proof (i) We may assume 119891(0) = 0 It implies 119891 is nonde-creasing First assume that 119909 isin N+ We use same methodas in the proof of Theorem 2 [4] (see Remark 32 in [4])Let 119886 gt 0 If 120585 isin 119890

119905119891(119905)le119886(Φ(119909))(H) cap 119890

(119886infin)(Φ(119891(119909)))(H)

with 120585 = 1 then 119891(⟨Φ(119909)120585 120585⟩) le 119886 and ⟨Φ(119891(119909))120585 120585⟩ gt

119886 On the other hand using Jensenrsquos inequality for thestate ⟨Φ(sdot)120585 120585⟩ and nondecreasing concave function 119891we get ⟨Φ(119891(119909))120585 120585⟩ le 119891(⟨Φ(119909)120585 120585⟩) le 119886 Therefore119890119905119891(119905)le119886

(Φ(119909))(H) cap 119890(119886infin)

(Φ(119891(119909)))(H) = 0 Thus119890119905119891(119905)le119886

(Φ(119909)) and 119890(119886infin)

(Φ(119891(119909))) = 0 and

119890(119886infin)

(Φ (119891 (119909))) = 119890(119886infin)

(Φ (119891 (119909)))

minus 119890119905119891(119905)le119886

(Φ (119909))

and 119890(119886infin)

(Φ (119891 (119909)))

sim 119890119905119891(119905)le119886

(Φ (119909))

or 119890(119886infin)

(Φ (119891 (119909)))

minus 119890119905119891(119905)le119886

(Φ (119909))

le 1 minus 119890119905119891(119905)le119886

(Φ (119909))

= 119890119905119891(119905)gt119886

(Φ (119909))

= 119890(119886+infin)

(119891 (Φ (119909)))

(12)

that is Φ(119891(119909)) ⪯ 119891(Φ(119909)) Hence (10) holdsNow let 119909 isin 119871

0(N)+ For each 119898 = 1 2 observe that

119909 and 1198981 isin M and so using the first case it follows that

Φ(119891 (119909 and 1198981)) ≼ 119891 (Φ (119909 and 1198981)) (13)

Using the functional calculus and Corollary 12 in [13]observe that

119909 and 1198981uarr119898119909

119891 (119909 and 1198981)uarr119898119891 (119909)

(14)

and so by continuousness ofΦ it follows that

Φ (119909 and 1198981)uarr119898Φ (119909)

Φ (119891 (119909 and 1198981))uarr119898Φ(119891 (119909))

(15)

Journal of Function Spaces 3

Using (vi) of Lemma 25 in [11] we obtain that

int

119905

0

120583119904(119891 (Φ (119909))) 119889119904 = int

119905

0

119891 (120583119904(Φ (119909))) 119889119904

= lim119898rarrinfin

int

119905

0

119891 (120583119904(Φ (119909 and 1198981))) 119889119904

= lim119898rarrinfin

int

119905

0

120583119904(119891 (Φ (119909 and 1198981))) 119889119904

ge lim119898rarrinfin

int

119905

0

120583119904(Φ (119891 (119909 and 1198981))) 119889119904

= int

119905

0

120583119904(Φ (119891 (119909))) 119889119904 forall119905 gt 0

(16)

that is (10) holds(ii) The proof is similar to the proof of (i)

Theorem 2 Let N M be semifinite von Neumann algebrasΦ1 Φ

119899positive linear continuous maps from 119871

0(N) into

1198710(M) such that the restriction of Φ

119896onN is a positive linear

map from N into M (119896 = 1 2 119899) and Φ = sum119899

119895=1Φ119895

unital(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

for 119909119896isin 1198710(N)+

(119896 = 1 2 119899)119899

sum

119896=1

Φ119896(119891 (119909119896)) ⪯ 119891(

119899

sum

119896=1

Φ119896(119909119896)) (17)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin 1198710(N)+

(119896 = 1 2 119899)

119892(

119899

sum

119896=1

Φ119896(119909119896)) ⪯

119899

sum

119896=1

Φ119896(119892 (119909119896)) (18)

Proof LetN119899be the von Neumann algebra

N119899=

(

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

) 119909119896isinN 119896

= 1 2 119899

(19)

on Hilbert spaceH oplus sdot sdot sdot oplusH Define Φ N119899rarr M by

Φ((

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

)) =

119899

sum

119896=1

Φ119896(119909119896) (20)

then Φ is a unital positive linear map from N119899into M By

Lemma 1 we obtain desired result

Using Theorem 53 in [14] and Theorem 2 we obtain thefollowing

Proposition 3 Let N119872 be semifinite von Neumann alge-bras Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map fromN intoM (119896 = 1 2 119899) andΦ = sum119899

119895=1Φ119895

unital

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

Φ119896(119891 (119909119896)) ≼ 119891(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

119891 (Φ119896(119909119896)) (21)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

119892 (Φ119896(119909119896)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

Φ119896(119892 (119909119896)) (22)

Proposition4 LetNM be semifinite vonNeumann algebrasand Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map from N into M (119896 = 1 2 119899) Suppose Φ =

sum119899

119895=1Φ119895is unital trace-preserving positive linear map fromM

into M If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (Φ119896(119909)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909)) ≼

119899

sum

119896=1

Φ119896(119892 (119909))

≼ 119892 (119909) forall119909 isin M+

(23)

Proof ByCorollary 29 in [15] we have thatsum119899119895=1

Φ119895is a trace-

preserving positive contraction Using Theorem 53 in [14]Lemma 31 in [16] (it is also holds for the semifinite case) andTheorem 2 we obtain the desired result

Corollary 5 Let 119886119896isin M (119896 = 1 2 119899) andsum119899

119896=1119886lowast

119896119886119896= 1

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin 1198710(M)+

(119896 = 1 2 119899)

119899

sum

119896=1

119886lowast

119896119891 (119909119896) 119886119896≼ 119891(

119899

sum

119896=1

119886lowast

119896119909119896119886119896) ≼

119899

sum

119896=1

119891 (119886lowast

119896119909119896119886119896) (24)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (119886lowast

119896119909119886119896) ≼ 119892(

119899

sum

119896=1

119886lowast

119896119909119886119896) ≼

119899

sum

119896=1

119886lowast

119896119892 (119909) 119886

119896

≼ 119892 (119909) forall119909 isin 1198710(M)+

(25)

LetM119899be von Neumann algebra of 119899 times 119899 complex matri-

ces and let 1198751 1198752 119875

119903be a family of mutually orthogonal

4 Journal of Function Spaces

projections in C119899 such that oplus119903119895=1

119875119895= 119868 where 119868 is unit matrix

inM119899 Then the operation of taking 119860 toC(119860) = sum

119903

119895=1119875119895119860119875119895

is called a pinching of 119860 The pinching C M119899

rarr M119899is a

trace-preserving positive map (see [17 18])

Proposition 6 LetC M119899rarr M

119899be a pinching

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

C (119891 (119860)) ≼ 119891 (C (119860)) forall119860 isin M+

119899 (26)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (C (119860)) ≼ C (119892 (119860)) ≼ 119892 (119860) forall119860 isin M+

119899 (27)

Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M) Define

diag (119909119896) =

((

(

1199091

0 sdot sdot sdot 0 sdot sdot sdot

0 1199092

sdot sdot sdot 0 sdot sdot sdot

d

0 0 sdot sdot sdot 119909119896

sdot sdot sdot

d

))

)

(28)

Proposition 7 Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M)

(i) If 119891 [0 +infin) rarr [0 +infin) where 119891(radic119905) is a concavefunction then

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

le (sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

forall0 lt 119901 le 1

(29)

(ii) If 119892 [0 +infin) rarr [0 +infin) where 119892(radic119905) is a convexfunction with 119892(0) = 0 then

(sum

119896ge1

1003817100381710038171003817119892 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

119892((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

forall1 le 119901 lt infin

(30)

Proof (i) Since

(

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

0 sdot sdot sdot

01003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

)

= C((

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

1199091119909lowast

2sdot sdot sdot

1199092119909lowast

1

1003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

))

= C((

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

))

(31)

and 119891(radic119905)119901 is concave by Lemma 1 we get

C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

≼ (

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

)

(32)

Hence

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

)))

le 120591((

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

))

= sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

(33)

Using the same arguments we can prove (ii)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Turdebek N Bekjan is partially supported by NSFC Grantno 11371304 Kordan N Ospanov is partially supportedby Project 3606GF4 of Science Committee of Ministry ofEducation and Science of the Republic of Kazakhstan

Journal of Function Spaces 5

References

[1] C Davis ldquoA Schwarz inequality for convex operator functionsrdquoProceedings of theAmericanMathematical Society vol 8 pp 42ndash44 1957

[2] M D Choi ldquoA Schwarz inequality for positive linear maps onClowast-algebrasrdquo Illinois Journal of Mathematics vol 18 pp 565ndash574 1974

[3] M Khosravi J S Aujla S S Dragomir and M S MoslehianldquoRefinements of Choi-Davis-Jensenrsquos inequalityrdquo Bulletin ofMathematical Analysis and Applications vol 3 no 2 pp 127ndash133 2011

[4] J Antezana P Massey and D Stojanoff ldquoJensenrsquos inequality forspectral order and submajorizationrdquo Journal of MathematicalAnalysis and Applications vol 331 no 1 pp 297ndash307 2007

[5] T Ando Topics on Operator Inequalities Hokkaido UniversitySapporo Japan 1978

[6] L G Brown and H Kosaki ldquoJensenrsquos inequality in semi-finitevon Neumann algebrasrdquo Journal of OperatorTheory vol 23 no1 pp 3ndash19 1990

[7] FHansen andGK Pedersen ldquoJensenrsquos operator inequalityrdquoTheBulletin of the London Mathematical Society vol 35 no 4 pp553ndash564 2003

[8] F Hansen and G K Pedersen ldquoJensenrsquos inequality for operatorsand Lownerrsquos theoremrdquoMathematische Annalen vol 258 no 3pp 229ndash241 1982

[9] G Pisier andQ Xu ldquoNoncommutative L119901-spacesrdquo inHandbook

of the Geometry of Banach Spaces vol 2 pp 1459ndash1517 2003[10] Q Xu Noncommutative L

119901-Spaces and Martingale Inequalities

2007[11] T Fack and H Kosaki ldquoGeneralized 119904-numbers of 120591-measure

operatorsrdquo Pacific Journal of Mathematics vol 123 no 2 pp269ndash300 1986

[12] P G Dodds T K Dodds and B de Pagter ldquoNoncommutativeBanach function spacesrdquoMathematische Zeitschrift vol 201 no4 pp 583ndash597 1989

[13] P G Dodds T K Dodds and B de Pager ldquoNoncommutativeKothe dualityrdquo Transactions of the American MathematicalSociety vol 339 pp 717ndash750 1993

[14] P G Dodds and F A Sukochev ldquoSubmajorisation inequalitiesfor convex and concave functions of sums of measurable opera-torsrdquo Positivity vol 13 no 1 pp 107ndash124 2009

[15] V Paulsen Completely Bounded Maps and Operator Algebrasvol 78 of Cambridge Studies in Advanced Mathematics Cam-bridge University Press 2002

[16] T N Bekjan and M Raikhan ldquoAn Hadamard-type inequalityrdquoLinear Algebra and its Applications vol 443 pp 228ndash234 2014

[17] R Bhatia M D Choi and C Davis ldquoComparing a matrix to itsoff-diagonal partrdquoOperatorTheory Advances and Applicationsvol 40 pp 151ndash163 1989

[18] J-C Bourin andMUchiyama ldquoAmatrix subadditivity inequal-ity for 119891(119860 + 119861) and 119891(119860) + 119891(119861)rdquo Linear Algebra and itsApplications vol 423 no 2-3 pp 512ndash518 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

for any convex function 119892 [0infin) rarr [0infin) with 119892(0) = 0where ldquo≼rdquo mains submajorization

This paper is organized as follows Section 2 containssome preliminary definitions In Section 3 we prove themainresult and related results

2 Preliminaries

We use standard notions from theory of noncommutative119871119901-spaces Our main references are [9 10] (see also [9] for

more historical references) Throughout the paper let Mbe a semifinite von Neumann algebra acting on a Hilbertspace H with a normal semifinite faithful trace 120591 Let 119871

0(M)

denote the topological lowast-algebra of measurable operatorswith respect to (M 120591) The topology of 119871

0(M) is determined

by the convergence in measure The trace 120591 can be extendedto the positive cone 119871+

0(M) of 119871

0(M)

120591 (119909) = int

infin

0

120582 119889120591 (119890120582(119909)) (5)

where 119909 = intinfin

0

120582 119889119890120582(119909) is the spectral decomposition of 119909

For 119909 isin 1198710(M) we define

120582119904(119909) = 120591 (119890

perp

119904(|119909|)) (119904 gt 0)

120583119905(119909) = inf 119904 gt 0 120582

119904(119909) le 119905 (119905 gt 0)

(6)

where 119890perp

119904(|119909|) = 119890

(119904infin)(|119909|) is the spectral projection of

|119909| associated with the interval (119904infin) The function 119904 997891rarr

120582119904(119909) is called the distribution function of 119909 and 120583

119905(119909) is the

generalized 119904-number of 119909 We will denote simply by 120582(119909) and120583(119909) the functions 119904 997891rarr 120582

119904(119909) and 119905 997891rarr 120583

119905(119909) respectively

It is easy to check that both are decreasing and continuousfrom the right on (0infin) For further informationwe refer thereader to [11]

If 119909 119910 isin 1198710(M) then we say that 119910 is submajorised by 119909

(in the sense ofHardy Littlewood and Polya) andwrite119910 ≼ 119909

if and only if

int

119905

0

120583119904(119910) 119889119904 le int

119905

0

120583119904(119909) 119889119904 forall119905 gt 0 (7)

We remark that if M = M119899and 120591 is the standard trace

then

120583119905(119909) = 119904

119895(119909) 119905 isin [119895 minus 1 119895) 119895 = 1 2 (8)

and if 119909 119910 isin M then 119910 ≼ 119909 is equivalent to

119896

sum

119895=1

119904119895(119910) le

119896

sum

119895=1

119904119895(119909) 1 le 119896 le 119899 (9)

For further information we refer the reader to [11ndash13]Let 119909 119910 be self-adjoint elements ofM we say that 119909 spec-

trally dominates 119910 denoted by 119910 ⪯ 119909 if 119890(120572infin)

(119910) is equiva-lent in the sense ofMurray-vonNeumann to a subprojectionof 119890(120572infin)

(119909) for every real number 120572 (see [6]) It is clear thatif 119909 spectrally dominates 119910 then 119910 is submajorised by 119909

3 Main Results

Lemma 1 LetN andM be semifinite von Neumann algebrasLet Φ be a positive linear continuous map from 119871

0(N) into

1198710(M) such that the restriction of Φ on N is a unital positive

linear map fromN intoM

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

Φ(119891 (119909)) ≼ 119891 (Φ (119909)) forall119909 isin 1198710(N)+

(10)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (Φ (119909)) ≼ Φ (119892 (119909)) forall119909 isin 1198710(N)+

(11)

Proof (i) We may assume 119891(0) = 0 It implies 119891 is nonde-creasing First assume that 119909 isin N+ We use same methodas in the proof of Theorem 2 [4] (see Remark 32 in [4])Let 119886 gt 0 If 120585 isin 119890

119905119891(119905)le119886(Φ(119909))(H) cap 119890

(119886infin)(Φ(119891(119909)))(H)

with 120585 = 1 then 119891(⟨Φ(119909)120585 120585⟩) le 119886 and ⟨Φ(119891(119909))120585 120585⟩ gt

119886 On the other hand using Jensenrsquos inequality for thestate ⟨Φ(sdot)120585 120585⟩ and nondecreasing concave function 119891we get ⟨Φ(119891(119909))120585 120585⟩ le 119891(⟨Φ(119909)120585 120585⟩) le 119886 Therefore119890119905119891(119905)le119886

(Φ(119909))(H) cap 119890(119886infin)

(Φ(119891(119909)))(H) = 0 Thus119890119905119891(119905)le119886

(Φ(119909)) and 119890(119886infin)

(Φ(119891(119909))) = 0 and

119890(119886infin)

(Φ (119891 (119909))) = 119890(119886infin)

(Φ (119891 (119909)))

minus 119890119905119891(119905)le119886

(Φ (119909))

and 119890(119886infin)

(Φ (119891 (119909)))

sim 119890119905119891(119905)le119886

(Φ (119909))

or 119890(119886infin)

(Φ (119891 (119909)))

minus 119890119905119891(119905)le119886

(Φ (119909))

le 1 minus 119890119905119891(119905)le119886

(Φ (119909))

= 119890119905119891(119905)gt119886

(Φ (119909))

= 119890(119886+infin)

(119891 (Φ (119909)))

(12)

that is Φ(119891(119909)) ⪯ 119891(Φ(119909)) Hence (10) holdsNow let 119909 isin 119871

0(N)+ For each 119898 = 1 2 observe that

119909 and 1198981 isin M and so using the first case it follows that

Φ(119891 (119909 and 1198981)) ≼ 119891 (Φ (119909 and 1198981)) (13)

Using the functional calculus and Corollary 12 in [13]observe that

119909 and 1198981uarr119898119909

119891 (119909 and 1198981)uarr119898119891 (119909)

(14)

and so by continuousness ofΦ it follows that

Φ (119909 and 1198981)uarr119898Φ (119909)

Φ (119891 (119909 and 1198981))uarr119898Φ(119891 (119909))

(15)

Journal of Function Spaces 3

Using (vi) of Lemma 25 in [11] we obtain that

int

119905

0

120583119904(119891 (Φ (119909))) 119889119904 = int

119905

0

119891 (120583119904(Φ (119909))) 119889119904

= lim119898rarrinfin

int

119905

0

119891 (120583119904(Φ (119909 and 1198981))) 119889119904

= lim119898rarrinfin

int

119905

0

120583119904(119891 (Φ (119909 and 1198981))) 119889119904

ge lim119898rarrinfin

int

119905

0

120583119904(Φ (119891 (119909 and 1198981))) 119889119904

= int

119905

0

120583119904(Φ (119891 (119909))) 119889119904 forall119905 gt 0

(16)

that is (10) holds(ii) The proof is similar to the proof of (i)

Theorem 2 Let N M be semifinite von Neumann algebrasΦ1 Φ

119899positive linear continuous maps from 119871

0(N) into

1198710(M) such that the restriction of Φ

119896onN is a positive linear

map from N into M (119896 = 1 2 119899) and Φ = sum119899

119895=1Φ119895

unital(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

for 119909119896isin 1198710(N)+

(119896 = 1 2 119899)119899

sum

119896=1

Φ119896(119891 (119909119896)) ⪯ 119891(

119899

sum

119896=1

Φ119896(119909119896)) (17)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin 1198710(N)+

(119896 = 1 2 119899)

119892(

119899

sum

119896=1

Φ119896(119909119896)) ⪯

119899

sum

119896=1

Φ119896(119892 (119909119896)) (18)

Proof LetN119899be the von Neumann algebra

N119899=

(

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

) 119909119896isinN 119896

= 1 2 119899

(19)

on Hilbert spaceH oplus sdot sdot sdot oplusH Define Φ N119899rarr M by

Φ((

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

)) =

119899

sum

119896=1

Φ119896(119909119896) (20)

then Φ is a unital positive linear map from N119899into M By

Lemma 1 we obtain desired result

Using Theorem 53 in [14] and Theorem 2 we obtain thefollowing

Proposition 3 Let N119872 be semifinite von Neumann alge-bras Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map fromN intoM (119896 = 1 2 119899) andΦ = sum119899

119895=1Φ119895

unital

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

Φ119896(119891 (119909119896)) ≼ 119891(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

119891 (Φ119896(119909119896)) (21)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

119892 (Φ119896(119909119896)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

Φ119896(119892 (119909119896)) (22)

Proposition4 LetNM be semifinite vonNeumann algebrasand Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map from N into M (119896 = 1 2 119899) Suppose Φ =

sum119899

119895=1Φ119895is unital trace-preserving positive linear map fromM

into M If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (Φ119896(119909)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909)) ≼

119899

sum

119896=1

Φ119896(119892 (119909))

≼ 119892 (119909) forall119909 isin M+

(23)

Proof ByCorollary 29 in [15] we have thatsum119899119895=1

Φ119895is a trace-

preserving positive contraction Using Theorem 53 in [14]Lemma 31 in [16] (it is also holds for the semifinite case) andTheorem 2 we obtain the desired result

Corollary 5 Let 119886119896isin M (119896 = 1 2 119899) andsum119899

119896=1119886lowast

119896119886119896= 1

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin 1198710(M)+

(119896 = 1 2 119899)

119899

sum

119896=1

119886lowast

119896119891 (119909119896) 119886119896≼ 119891(

119899

sum

119896=1

119886lowast

119896119909119896119886119896) ≼

119899

sum

119896=1

119891 (119886lowast

119896119909119896119886119896) (24)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (119886lowast

119896119909119886119896) ≼ 119892(

119899

sum

119896=1

119886lowast

119896119909119886119896) ≼

119899

sum

119896=1

119886lowast

119896119892 (119909) 119886

119896

≼ 119892 (119909) forall119909 isin 1198710(M)+

(25)

LetM119899be von Neumann algebra of 119899 times 119899 complex matri-

ces and let 1198751 1198752 119875

119903be a family of mutually orthogonal

4 Journal of Function Spaces

projections in C119899 such that oplus119903119895=1

119875119895= 119868 where 119868 is unit matrix

inM119899 Then the operation of taking 119860 toC(119860) = sum

119903

119895=1119875119895119860119875119895

is called a pinching of 119860 The pinching C M119899

rarr M119899is a

trace-preserving positive map (see [17 18])

Proposition 6 LetC M119899rarr M

119899be a pinching

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

C (119891 (119860)) ≼ 119891 (C (119860)) forall119860 isin M+

119899 (26)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (C (119860)) ≼ C (119892 (119860)) ≼ 119892 (119860) forall119860 isin M+

119899 (27)

Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M) Define

diag (119909119896) =

((

(

1199091

0 sdot sdot sdot 0 sdot sdot sdot

0 1199092

sdot sdot sdot 0 sdot sdot sdot

d

0 0 sdot sdot sdot 119909119896

sdot sdot sdot

d

))

)

(28)

Proposition 7 Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M)

(i) If 119891 [0 +infin) rarr [0 +infin) where 119891(radic119905) is a concavefunction then

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

le (sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

forall0 lt 119901 le 1

(29)

(ii) If 119892 [0 +infin) rarr [0 +infin) where 119892(radic119905) is a convexfunction with 119892(0) = 0 then

(sum

119896ge1

1003817100381710038171003817119892 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

119892((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

forall1 le 119901 lt infin

(30)

Proof (i) Since

(

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

0 sdot sdot sdot

01003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

)

= C((

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

1199091119909lowast

2sdot sdot sdot

1199092119909lowast

1

1003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

))

= C((

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

))

(31)

and 119891(radic119905)119901 is concave by Lemma 1 we get

C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

≼ (

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

)

(32)

Hence

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

)))

le 120591((

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

))

= sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

(33)

Using the same arguments we can prove (ii)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Turdebek N Bekjan is partially supported by NSFC Grantno 11371304 Kordan N Ospanov is partially supportedby Project 3606GF4 of Science Committee of Ministry ofEducation and Science of the Republic of Kazakhstan

Journal of Function Spaces 5

References

[1] C Davis ldquoA Schwarz inequality for convex operator functionsrdquoProceedings of theAmericanMathematical Society vol 8 pp 42ndash44 1957

[2] M D Choi ldquoA Schwarz inequality for positive linear maps onClowast-algebrasrdquo Illinois Journal of Mathematics vol 18 pp 565ndash574 1974

[3] M Khosravi J S Aujla S S Dragomir and M S MoslehianldquoRefinements of Choi-Davis-Jensenrsquos inequalityrdquo Bulletin ofMathematical Analysis and Applications vol 3 no 2 pp 127ndash133 2011

[4] J Antezana P Massey and D Stojanoff ldquoJensenrsquos inequality forspectral order and submajorizationrdquo Journal of MathematicalAnalysis and Applications vol 331 no 1 pp 297ndash307 2007

[5] T Ando Topics on Operator Inequalities Hokkaido UniversitySapporo Japan 1978

[6] L G Brown and H Kosaki ldquoJensenrsquos inequality in semi-finitevon Neumann algebrasrdquo Journal of OperatorTheory vol 23 no1 pp 3ndash19 1990

[7] FHansen andGK Pedersen ldquoJensenrsquos operator inequalityrdquoTheBulletin of the London Mathematical Society vol 35 no 4 pp553ndash564 2003

[8] F Hansen and G K Pedersen ldquoJensenrsquos inequality for operatorsand Lownerrsquos theoremrdquoMathematische Annalen vol 258 no 3pp 229ndash241 1982

[9] G Pisier andQ Xu ldquoNoncommutative L119901-spacesrdquo inHandbook

of the Geometry of Banach Spaces vol 2 pp 1459ndash1517 2003[10] Q Xu Noncommutative L

119901-Spaces and Martingale Inequalities

2007[11] T Fack and H Kosaki ldquoGeneralized 119904-numbers of 120591-measure

operatorsrdquo Pacific Journal of Mathematics vol 123 no 2 pp269ndash300 1986

[12] P G Dodds T K Dodds and B de Pagter ldquoNoncommutativeBanach function spacesrdquoMathematische Zeitschrift vol 201 no4 pp 583ndash597 1989

[13] P G Dodds T K Dodds and B de Pager ldquoNoncommutativeKothe dualityrdquo Transactions of the American MathematicalSociety vol 339 pp 717ndash750 1993

[14] P G Dodds and F A Sukochev ldquoSubmajorisation inequalitiesfor convex and concave functions of sums of measurable opera-torsrdquo Positivity vol 13 no 1 pp 107ndash124 2009

[15] V Paulsen Completely Bounded Maps and Operator Algebrasvol 78 of Cambridge Studies in Advanced Mathematics Cam-bridge University Press 2002

[16] T N Bekjan and M Raikhan ldquoAn Hadamard-type inequalityrdquoLinear Algebra and its Applications vol 443 pp 228ndash234 2014

[17] R Bhatia M D Choi and C Davis ldquoComparing a matrix to itsoff-diagonal partrdquoOperatorTheory Advances and Applicationsvol 40 pp 151ndash163 1989

[18] J-C Bourin andMUchiyama ldquoAmatrix subadditivity inequal-ity for 119891(119860 + 119861) and 119891(119860) + 119891(119861)rdquo Linear Algebra and itsApplications vol 423 no 2-3 pp 512ndash518 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

Using (vi) of Lemma 25 in [11] we obtain that

int

119905

0

120583119904(119891 (Φ (119909))) 119889119904 = int

119905

0

119891 (120583119904(Φ (119909))) 119889119904

= lim119898rarrinfin

int

119905

0

119891 (120583119904(Φ (119909 and 1198981))) 119889119904

= lim119898rarrinfin

int

119905

0

120583119904(119891 (Φ (119909 and 1198981))) 119889119904

ge lim119898rarrinfin

int

119905

0

120583119904(Φ (119891 (119909 and 1198981))) 119889119904

= int

119905

0

120583119904(Φ (119891 (119909))) 119889119904 forall119905 gt 0

(16)

that is (10) holds(ii) The proof is similar to the proof of (i)

Theorem 2 Let N M be semifinite von Neumann algebrasΦ1 Φ

119899positive linear continuous maps from 119871

0(N) into

1198710(M) such that the restriction of Φ

119896onN is a positive linear

map from N into M (119896 = 1 2 119899) and Φ = sum119899

119895=1Φ119895

unital(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

for 119909119896isin 1198710(N)+

(119896 = 1 2 119899)119899

sum

119896=1

Φ119896(119891 (119909119896)) ⪯ 119891(

119899

sum

119896=1

Φ119896(119909119896)) (17)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin 1198710(N)+

(119896 = 1 2 119899)

119892(

119899

sum

119896=1

Φ119896(119909119896)) ⪯

119899

sum

119896=1

Φ119896(119892 (119909119896)) (18)

Proof LetN119899be the von Neumann algebra

N119899=

(

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

) 119909119896isinN 119896

= 1 2 119899

(19)

on Hilbert spaceH oplus sdot sdot sdot oplusH Define Φ N119899rarr M by

Φ((

1199091

0 sdot sdot sdot 0

0 1199092

sdot sdot sdot 0

d

0 0 sdot sdot sdot 119909119899

)) =

119899

sum

119896=1

Φ119896(119909119896) (20)

then Φ is a unital positive linear map from N119899into M By

Lemma 1 we obtain desired result

Using Theorem 53 in [14] and Theorem 2 we obtain thefollowing

Proposition 3 Let N119872 be semifinite von Neumann alge-bras Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map fromN intoM (119896 = 1 2 119899) andΦ = sum119899

119895=1Φ119895

unital

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

Φ119896(119891 (119909119896)) ≼ 119891(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

119891 (Φ119896(119909119896)) (21)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then for 119909

119896isin N+ (119896 = 1 2 119899)

119899

sum

119896=1

119892 (Φ119896(119909119896)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909119896)) ≼

119899

sum

119896=1

Φ119896(119892 (119909119896)) (22)

Proposition4 LetNM be semifinite vonNeumann algebrasand Φ

1 Φ

119899positive linear continuous maps from 119871

0(N)

into 1198710(M) such that the restriction of Φ

119896on N is a positive

linear map from N into M (119896 = 1 2 119899) Suppose Φ =

sum119899

119895=1Φ119895is unital trace-preserving positive linear map fromM

into M If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (Φ119896(119909)) ≼ 119892(

119899

sum

119896=1

Φ119896(119909)) ≼

119899

sum

119896=1

Φ119896(119892 (119909))

≼ 119892 (119909) forall119909 isin M+

(23)

Proof ByCorollary 29 in [15] we have thatsum119899119895=1

Φ119895is a trace-

preserving positive contraction Using Theorem 53 in [14]Lemma 31 in [16] (it is also holds for the semifinite case) andTheorem 2 we obtain the desired result

Corollary 5 Let 119886119896isin M (119896 = 1 2 119899) andsum119899

119896=1119886lowast

119896119886119896= 1

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function thenfor 119909119896isin 1198710(M)+

(119896 = 1 2 119899)

119899

sum

119896=1

119886lowast

119896119891 (119909119896) 119886119896≼ 119891(

119899

sum

119896=1

119886lowast

119896119909119896119886119896) ≼

119899

sum

119896=1

119891 (119886lowast

119896119909119896119886119896) (24)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119899

sum

119896=1

119892 (119886lowast

119896119909119886119896) ≼ 119892(

119899

sum

119896=1

119886lowast

119896119909119886119896) ≼

119899

sum

119896=1

119886lowast

119896119892 (119909) 119886

119896

≼ 119892 (119909) forall119909 isin 1198710(M)+

(25)

LetM119899be von Neumann algebra of 119899 times 119899 complex matri-

ces and let 1198751 1198752 119875

119903be a family of mutually orthogonal

4 Journal of Function Spaces

projections in C119899 such that oplus119903119895=1

119875119895= 119868 where 119868 is unit matrix

inM119899 Then the operation of taking 119860 toC(119860) = sum

119903

119895=1119875119895119860119875119895

is called a pinching of 119860 The pinching C M119899

rarr M119899is a

trace-preserving positive map (see [17 18])

Proposition 6 LetC M119899rarr M

119899be a pinching

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

C (119891 (119860)) ≼ 119891 (C (119860)) forall119860 isin M+

119899 (26)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (C (119860)) ≼ C (119892 (119860)) ≼ 119892 (119860) forall119860 isin M+

119899 (27)

Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M) Define

diag (119909119896) =

((

(

1199091

0 sdot sdot sdot 0 sdot sdot sdot

0 1199092

sdot sdot sdot 0 sdot sdot sdot

d

0 0 sdot sdot sdot 119909119896

sdot sdot sdot

d

))

)

(28)

Proposition 7 Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M)

(i) If 119891 [0 +infin) rarr [0 +infin) where 119891(radic119905) is a concavefunction then

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

le (sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

forall0 lt 119901 le 1

(29)

(ii) If 119892 [0 +infin) rarr [0 +infin) where 119892(radic119905) is a convexfunction with 119892(0) = 0 then

(sum

119896ge1

1003817100381710038171003817119892 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

119892((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

forall1 le 119901 lt infin

(30)

Proof (i) Since

(

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

0 sdot sdot sdot

01003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

)

= C((

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

1199091119909lowast

2sdot sdot sdot

1199092119909lowast

1

1003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

))

= C((

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

))

(31)

and 119891(radic119905)119901 is concave by Lemma 1 we get

C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

≼ (

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

)

(32)

Hence

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

)))

le 120591((

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

))

= sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

(33)

Using the same arguments we can prove (ii)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Turdebek N Bekjan is partially supported by NSFC Grantno 11371304 Kordan N Ospanov is partially supportedby Project 3606GF4 of Science Committee of Ministry ofEducation and Science of the Republic of Kazakhstan

Journal of Function Spaces 5

References

[1] C Davis ldquoA Schwarz inequality for convex operator functionsrdquoProceedings of theAmericanMathematical Society vol 8 pp 42ndash44 1957

[2] M D Choi ldquoA Schwarz inequality for positive linear maps onClowast-algebrasrdquo Illinois Journal of Mathematics vol 18 pp 565ndash574 1974

[3] M Khosravi J S Aujla S S Dragomir and M S MoslehianldquoRefinements of Choi-Davis-Jensenrsquos inequalityrdquo Bulletin ofMathematical Analysis and Applications vol 3 no 2 pp 127ndash133 2011

[4] J Antezana P Massey and D Stojanoff ldquoJensenrsquos inequality forspectral order and submajorizationrdquo Journal of MathematicalAnalysis and Applications vol 331 no 1 pp 297ndash307 2007

[5] T Ando Topics on Operator Inequalities Hokkaido UniversitySapporo Japan 1978

[6] L G Brown and H Kosaki ldquoJensenrsquos inequality in semi-finitevon Neumann algebrasrdquo Journal of OperatorTheory vol 23 no1 pp 3ndash19 1990

[7] FHansen andGK Pedersen ldquoJensenrsquos operator inequalityrdquoTheBulletin of the London Mathematical Society vol 35 no 4 pp553ndash564 2003

[8] F Hansen and G K Pedersen ldquoJensenrsquos inequality for operatorsand Lownerrsquos theoremrdquoMathematische Annalen vol 258 no 3pp 229ndash241 1982

[9] G Pisier andQ Xu ldquoNoncommutative L119901-spacesrdquo inHandbook

of the Geometry of Banach Spaces vol 2 pp 1459ndash1517 2003[10] Q Xu Noncommutative L

119901-Spaces and Martingale Inequalities

2007[11] T Fack and H Kosaki ldquoGeneralized 119904-numbers of 120591-measure

operatorsrdquo Pacific Journal of Mathematics vol 123 no 2 pp269ndash300 1986

[12] P G Dodds T K Dodds and B de Pagter ldquoNoncommutativeBanach function spacesrdquoMathematische Zeitschrift vol 201 no4 pp 583ndash597 1989

[13] P G Dodds T K Dodds and B de Pager ldquoNoncommutativeKothe dualityrdquo Transactions of the American MathematicalSociety vol 339 pp 717ndash750 1993

[14] P G Dodds and F A Sukochev ldquoSubmajorisation inequalitiesfor convex and concave functions of sums of measurable opera-torsrdquo Positivity vol 13 no 1 pp 107ndash124 2009

[15] V Paulsen Completely Bounded Maps and Operator Algebrasvol 78 of Cambridge Studies in Advanced Mathematics Cam-bridge University Press 2002

[16] T N Bekjan and M Raikhan ldquoAn Hadamard-type inequalityrdquoLinear Algebra and its Applications vol 443 pp 228ndash234 2014

[17] R Bhatia M D Choi and C Davis ldquoComparing a matrix to itsoff-diagonal partrdquoOperatorTheory Advances and Applicationsvol 40 pp 151ndash163 1989

[18] J-C Bourin andMUchiyama ldquoAmatrix subadditivity inequal-ity for 119891(119860 + 119861) and 119891(119860) + 119891(119861)rdquo Linear Algebra and itsApplications vol 423 no 2-3 pp 512ndash518 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

projections in C119899 such that oplus119903119895=1

119875119895= 119868 where 119868 is unit matrix

inM119899 Then the operation of taking 119860 toC(119860) = sum

119903

119895=1119875119895119860119875119895

is called a pinching of 119860 The pinching C M119899

rarr M119899is a

trace-preserving positive map (see [17 18])

Proposition 6 LetC M119899rarr M

119899be a pinching

(i) If 119891 [0 +infin) rarr [0 +infin) is a concave function then

C (119891 (119860)) ≼ 119891 (C (119860)) forall119860 isin M+

119899 (26)

(ii) If 119892 [0 +infin) rarr [0 +infin) is a convex function with119892(0) = 0 then

119892 (C (119860)) ≼ C (119892 (119860)) ≼ 119892 (119860) forall119860 isin M+

119899 (27)

Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M) Define

diag (119909119896) =

((

(

1199091

0 sdot sdot sdot 0 sdot sdot sdot

0 1199092

sdot sdot sdot 0 sdot sdot sdot

d

0 0 sdot sdot sdot 119909119896

sdot sdot sdot

d

))

)

(28)

Proposition 7 Let 119909 = (119909119896)119896ge1

be a sequence in 1198710(M)

(i) If 119891 [0 +infin) rarr [0 +infin) where 119891(radic119905) is a concavefunction then

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

le (sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

forall0 lt 119901 le 1

(29)

(ii) If 119892 [0 +infin) rarr [0 +infin) where 119892(radic119905) is a convexfunction with 119892(0) = 0 then

(sum

119896ge1

1003817100381710038171003817119892 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

)

1119901

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

119892((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817119901

forall1 le 119901 lt infin

(30)

Proof (i) Since

(

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

0 sdot sdot sdot

01003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

)

= C((

1003816100381610038161003816119909lowast

1

1003816100381610038161003816

2

1199091119909lowast

2sdot sdot sdot

1199092119909lowast

1

1003816100381610038161003816119909lowast

2

1003816100381610038161003816

2

sdot sdot sdot

d

))

= C((

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

))

(31)

and 119891(radic119905)119901 is concave by Lemma 1 we get

C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

≼ (

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

)

(32)

Hence

10038171003817100381710038171003817100381710038171003817100381710038171003817

119891((sum

119896ge1

10038161003816100381610038161199091198961003816100381610038161003816

2

)

12

)

10038171003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

1199091

0 sdot sdot sdot

1199092

0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

))

= 120591(C(119891119901

(

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

119909lowast

1119909lowast

2sdot sdot sdot

0 0 sdot sdot sdot

d

)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

)))

le 120591((

119891(1003816100381610038161003816119909lowast

1

1003816100381610038161003816)119901

0 sdot sdot sdot

0 119891 (1003816100381610038161003816119909lowast

2

1003816100381610038161003816)119901

sdot sdot sdot

d

))

= sum

119896ge1

1003817100381710038171003817119891 (1003816100381610038161003816119909119896

1003816100381610038161003816)1003817100381710038171003817

119901

(33)

Using the same arguments we can prove (ii)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Turdebek N Bekjan is partially supported by NSFC Grantno 11371304 Kordan N Ospanov is partially supportedby Project 3606GF4 of Science Committee of Ministry ofEducation and Science of the Republic of Kazakhstan

Journal of Function Spaces 5

References

[1] C Davis ldquoA Schwarz inequality for convex operator functionsrdquoProceedings of theAmericanMathematical Society vol 8 pp 42ndash44 1957

[2] M D Choi ldquoA Schwarz inequality for positive linear maps onClowast-algebrasrdquo Illinois Journal of Mathematics vol 18 pp 565ndash574 1974

[3] M Khosravi J S Aujla S S Dragomir and M S MoslehianldquoRefinements of Choi-Davis-Jensenrsquos inequalityrdquo Bulletin ofMathematical Analysis and Applications vol 3 no 2 pp 127ndash133 2011

[4] J Antezana P Massey and D Stojanoff ldquoJensenrsquos inequality forspectral order and submajorizationrdquo Journal of MathematicalAnalysis and Applications vol 331 no 1 pp 297ndash307 2007

[5] T Ando Topics on Operator Inequalities Hokkaido UniversitySapporo Japan 1978

[6] L G Brown and H Kosaki ldquoJensenrsquos inequality in semi-finitevon Neumann algebrasrdquo Journal of OperatorTheory vol 23 no1 pp 3ndash19 1990

[7] FHansen andGK Pedersen ldquoJensenrsquos operator inequalityrdquoTheBulletin of the London Mathematical Society vol 35 no 4 pp553ndash564 2003

[8] F Hansen and G K Pedersen ldquoJensenrsquos inequality for operatorsand Lownerrsquos theoremrdquoMathematische Annalen vol 258 no 3pp 229ndash241 1982

[9] G Pisier andQ Xu ldquoNoncommutative L119901-spacesrdquo inHandbook

of the Geometry of Banach Spaces vol 2 pp 1459ndash1517 2003[10] Q Xu Noncommutative L

119901-Spaces and Martingale Inequalities

2007[11] T Fack and H Kosaki ldquoGeneralized 119904-numbers of 120591-measure

operatorsrdquo Pacific Journal of Mathematics vol 123 no 2 pp269ndash300 1986

[12] P G Dodds T K Dodds and B de Pagter ldquoNoncommutativeBanach function spacesrdquoMathematische Zeitschrift vol 201 no4 pp 583ndash597 1989

[13] P G Dodds T K Dodds and B de Pager ldquoNoncommutativeKothe dualityrdquo Transactions of the American MathematicalSociety vol 339 pp 717ndash750 1993

[14] P G Dodds and F A Sukochev ldquoSubmajorisation inequalitiesfor convex and concave functions of sums of measurable opera-torsrdquo Positivity vol 13 no 1 pp 107ndash124 2009

[15] V Paulsen Completely Bounded Maps and Operator Algebrasvol 78 of Cambridge Studies in Advanced Mathematics Cam-bridge University Press 2002

[16] T N Bekjan and M Raikhan ldquoAn Hadamard-type inequalityrdquoLinear Algebra and its Applications vol 443 pp 228ndash234 2014

[17] R Bhatia M D Choi and C Davis ldquoComparing a matrix to itsoff-diagonal partrdquoOperatorTheory Advances and Applicationsvol 40 pp 151ndash163 1989

[18] J-C Bourin andMUchiyama ldquoAmatrix subadditivity inequal-ity for 119891(119860 + 119861) and 119891(119860) + 119891(119861)rdquo Linear Algebra and itsApplications vol 423 no 2-3 pp 512ndash518 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

References

[1] C Davis ldquoA Schwarz inequality for convex operator functionsrdquoProceedings of theAmericanMathematical Society vol 8 pp 42ndash44 1957

[2] M D Choi ldquoA Schwarz inequality for positive linear maps onClowast-algebrasrdquo Illinois Journal of Mathematics vol 18 pp 565ndash574 1974

[3] M Khosravi J S Aujla S S Dragomir and M S MoslehianldquoRefinements of Choi-Davis-Jensenrsquos inequalityrdquo Bulletin ofMathematical Analysis and Applications vol 3 no 2 pp 127ndash133 2011

[4] J Antezana P Massey and D Stojanoff ldquoJensenrsquos inequality forspectral order and submajorizationrdquo Journal of MathematicalAnalysis and Applications vol 331 no 1 pp 297ndash307 2007

[5] T Ando Topics on Operator Inequalities Hokkaido UniversitySapporo Japan 1978

[6] L G Brown and H Kosaki ldquoJensenrsquos inequality in semi-finitevon Neumann algebrasrdquo Journal of OperatorTheory vol 23 no1 pp 3ndash19 1990

[7] FHansen andGK Pedersen ldquoJensenrsquos operator inequalityrdquoTheBulletin of the London Mathematical Society vol 35 no 4 pp553ndash564 2003

[8] F Hansen and G K Pedersen ldquoJensenrsquos inequality for operatorsand Lownerrsquos theoremrdquoMathematische Annalen vol 258 no 3pp 229ndash241 1982

[9] G Pisier andQ Xu ldquoNoncommutative L119901-spacesrdquo inHandbook

of the Geometry of Banach Spaces vol 2 pp 1459ndash1517 2003[10] Q Xu Noncommutative L

119901-Spaces and Martingale Inequalities

2007[11] T Fack and H Kosaki ldquoGeneralized 119904-numbers of 120591-measure

operatorsrdquo Pacific Journal of Mathematics vol 123 no 2 pp269ndash300 1986

[12] P G Dodds T K Dodds and B de Pagter ldquoNoncommutativeBanach function spacesrdquoMathematische Zeitschrift vol 201 no4 pp 583ndash597 1989

[13] P G Dodds T K Dodds and B de Pager ldquoNoncommutativeKothe dualityrdquo Transactions of the American MathematicalSociety vol 339 pp 717ndash750 1993

[14] P G Dodds and F A Sukochev ldquoSubmajorisation inequalitiesfor convex and concave functions of sums of measurable opera-torsrdquo Positivity vol 13 no 1 pp 107ndash124 2009

[15] V Paulsen Completely Bounded Maps and Operator Algebrasvol 78 of Cambridge Studies in Advanced Mathematics Cam-bridge University Press 2002

[16] T N Bekjan and M Raikhan ldquoAn Hadamard-type inequalityrdquoLinear Algebra and its Applications vol 443 pp 228ndash234 2014

[17] R Bhatia M D Choi and C Davis ldquoComparing a matrix to itsoff-diagonal partrdquoOperatorTheory Advances and Applicationsvol 40 pp 151ndash163 1989

[18] J-C Bourin andMUchiyama ldquoAmatrix subadditivity inequal-ity for 119891(119860 + 119861) and 119891(119860) + 119891(119861)rdquo Linear Algebra and itsApplications vol 423 no 2-3 pp 512ndash518 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of