research article choi-davis-jensen inequalities in...
Post on 05-Oct-2020
0 Views
Preview:
TRANSCRIPT
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
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
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
top related