YANG Chang-sen,YANG Chao-jun

(College of Mathematics and Information Science,Henan Normal University,Xinxiang 453007,China)

SOME OPERATOR INEQUALITIES OF MONOTONE FUNCTIONS CONTAINING FURUTA INEQUALITY

YANG Chang-sen,YANG Chao-jun

(College of Mathematics and Information Science,Henan Normal University,Xinxiang 453007,China)

In this paper,we study the relations between the operator inequalities and the operator monotone functions.By using the fundamental conclusions based on majorization,namely,product lemma and product theorem for operator monotone functions,we can give some operator inequalities.This result contains the Furuta inequality,which has a huge impact on positive operator theory.

operator monotone function;product lemma;product theorem；majorization

1 Introduction

LetJbe an interval such thatJ/(-∞,∞).P(J)denotes the set of all operator monotone functions onJ.We setP+(J)={f∈P(J)|f(t)≥0,t∈J}.Iff∈P+(a,b)and-∞＜a,thenfhas the natural extension to[a,b),which belongs toP+[a,b).We therefore identifyP+(a,b)withP+[a,b).

It is well-known that iff(t)∈P+(0,∞),thenare both inP+(0,∞),and that iff(t),φ(t),φ(t)are all inP+(0,∞),then so are

andf(t)αφ(t)1-αfor 0＜α＜1(see[1-5]).Throughout this work,we assume that a function is continuous and increasing means “strictly increasing”.Further more,for convenience,letB(H)denote theC?-algebra of all bounded linear operators acting on a Hilbert spaceH.A capital letterAmeans an element belongs toB(H),Φ means a positive linear map fromB(H)toB(H)and we assume Φ(I)=Ialways stand(see[7,8]).In this paper,we also assume thatJ=[a,b)orJ=(a,b)with-∞≤a＜b≤+∞.

De fi nition 1.1[9,10]Letdenote the following sets,respectively,

whereh-1stands for the inverse function ofh.

De fi nition 1.2Leth(t)andg(t)be functions defined onJ,andg(t)is increasing,thenhis said to be majorized byg,in symbolh≤gif the compositeh?g-1is operator monotone ong(J),which is equivalent to

Lemma 1.1(Product lemma)(see[9,10])Leth,gbe non-negative functions defined onJ.Suppose the producthgis increasing,(hg)(a+0)=0 and(hg)(b-0)=∞.Then

Moreover,for everyψ1,ψ2inP+[0,∞),

Theorem 1.1(Product theorem)(see[9,10])

Further,letgi(t)∈LP+(J)for 1≤i≤mandhj(t)∈P-1+(J)for 1≤j≤n.Then for everyψi,φj∈P+[0,∞),we have

2 Main Results

Before to prove our main results,we give the following lemmas.

Lemma 2.1(L-H inequality)(see[2,12])If 0≤α≤1,A≥B≥0,thenAα≥Bα.

Lemma 2.2(Furuta inequality)(see[6,9])LetA≥B≥0,then

wherer≥0,p≥1 with

Lemma 2.3(Hansen inequality)(see[13])LetXandAbe bounded linear operators onH,and such thatX≥ 0,‖A‖≤1.Iffis an operator monotone function on[0,∞),then

Theorem 2.1PutJ/=(-∞,∞),,fi∈P+(J),i=1,2,···,n,,andkn(t)=f1(t)f2(t)···fn(t).Ifh(t)is defined onJsuch that,then

(i)the functionφnon(0,∞)defined by

belongs toP+(0,∞);

(ii)ifA≤C≤B,then

Proof(i)Sincef1(t)f1(t)h(t),by product lemmah(t)f1(t)h(t),thereforeh(t)is nondecreasing.When,since,we haveη(t)g(t).Now puttingψ0(s)=s,ψ1(g)=η,ψ2(f1h)=f1,obviously,we haveψ0,ψ1,ψ2∈P+(0,∞).By takingsinψ0(s)asf2···fn,and from product theorem,we obtain

Therefore we haveφnbelongs toP+(0,∞)forφngiven in(i).

Wheng(t)=f1(t),by takingψ0(s)=s,ψ1(g(t)h(t))=η(t),we haveψ0,ψ1∈P+(0,∞),and thenφn∈P+(0,∞)by product theorem.

(ii)First we prove that

Sinceφn,kn,h,gare all nonnegative,nondecreasing functions andJis a right open interval,by consideringC+?,B+?,we may assume that,h(C),h(B),g(C),g(B)are positive semi-de fi nite and invertible.Through(i),

This implies the right part of(2.2)holds forn=1.Next we assume the right part of(2.2)holds forn-1.Sinceandand this means that there existssuch thatfn(t)= Ψn(kn-1(t)η(t)).Puts=kn-1(t)η(t),we can obtain.Since the following inequality holds

Denote the left side of the upper inequalities asH,the right one asK,we have

ByH=φn-1(kn-1(C)h(C)g(C))=kn-1(C)η(C),we obtain

By Lemma 2.3 again,we obtain

From the above inequalities and(2.4),we get

Therefore the right part of(2.2)holds forn,one can proof the left part of(2.2)similarly.

RemarkIn Theorem 2.1,letn=2,f1(t)=g(t)=1,f2(t)=tr(r≥0),h(t)=tp(p≥1),andη(t)=t,then we haveφ2(tp+r)=t1+r.So Furuta inequality can be obtained by(2.2)and L-H inequality.

Lemma 2.4(see[10,11])PutJ(-∞,∞),theng∈LP+(J)if and only if there exists a sequence{gn}of a fi nite product of functions inP+(J)which converges pointwise togonJ,further more,{gn}converges uniformly togon every bounded closed subinterval ofJ.

Theorem 2.2PutJ(-∞,∞),f(t)＞0 fort∈Jandη(t),h(t),k(t),g(t)are nonnegative functions onJsuch that,then

(i)the functionφon(0,∞)defined by

belongs toP+(0,∞);

(ii)IfA≤C≤B,then forφ∈P(0,∞)such thatφ≤φon(0,∞),

Proof(i)First consider,thenk=lfand

Letψ0(s)=s,ψ1(f(t)h(t))=f(t),ψ2(g(t))=η(t),thenψ0,ψ1,ψ2∈P+(0,∞).By takings=l(t)and applying product theorem,we get

which equals tok(t)η(t)≤k(t)h(t)g(t).So we haveφ∈P+(0,∞)forφsuch that

Ifg=f,takingψ0(s)=s,ψ1(h(t)g(t))=η(t),obviously,we haveψ0,ψ1∈P+(0,∞),and thenψ0(k)ψ1(hg).Hence we also haveφ∈P+(0,∞)from product theorem.

(ii)From Lemma 2.4,we obtain there exists a sequence{ln},whereln(t)is a fi nite product of functions inP+(J),such thatln(t)converges ponitwise tol(t).Putkn(t)=f(t)ln(t)then we easily getkn(t)converges tok(t)=f(t)l(t).De fi neφn(kn(t)h(t)g(t))=kn(t)η(t)(t∈J),φn∈P+(0,∞).By Theorem 2.1,we have

Lemma 2.5(Choi inequality)(see[6,7])Let Φ be a positive unital linear map,then

(C1)whenA＞0 and-1≤p≤0,then Φ(A)p≤Φ(Ap);

(C2)whenA≥ 0 and 0≤p≤1,then Φ(A)p≥ Φ(Ap);

(C3)whenA≥ 0 and 1≤p≤2,then Φ(A)p≤Φ(Ap).

Corollary 2.1PutJ/=(-∞,∞),f(t)＞0 fort∈Jandη(t),h(t),k(t),g(t)are nonnegative functions onJsuch that,the functionφon(0,∞)defined as(2.5),Φ is a positive unital

linear map.If

then forφ∈P(0,∞)such thatφ≤φ,

ProofBy Choi inequality and L-H inequality,we obtain

Corollary 2.2Put

such that.Then(2.5)and(2.6)in Theorem 2.2 hold.

ProofPutc=min{ 1,p},thenf(t)=t1-c∈P+(0,∞).Thus we get

which means the conditions of Theorem 2.2 is satis fi ed.Therefore(2.5)and(2.6)in Theorem 2.2 hold.

Corollary 2.3Put,p,r≥0 andp+r≥1,s≥1,we obtain

ProofPutg(t)=ts(s≥1),η(t)=tin Corollary 2.2.Then we only need to show logs≤φ(s),s∈(0,∞).The de fi nition ofφis given in(2.5).The upper majorization relationship is equivalent to

It is obviously that logk(t),logh(t),logtsare operator monotone on(0,∞)and,then

Therefore(2.8)holds.

[1]Bhatia R.Matrix analysis[M].New York：Springer,1996.

[3]Horn R A,Johnson C R.Matrix analysis[M].Cambridge：Cambridge Univ.Press,1985.

[4]Rosenblum M,Rovenyak J.Hardy classes and operator theory[M].Oxford：Oxford Univ.Press,1985.

[5]Pedersen G K.Some operator monotone functions[J].Proc.Amer.Math.Soc.,1972,36：309-310.

[6]Choi M D.Some assorted inequalities for positive linear map onC?-algebras[J].J.Oper.The.,1980,4：271-285.

[7]Choi M D.A Schwarz inequality for positive linear maps onC?-algebras[J].Illinois.J.Math.,1974,18：565-574.

[8]Ando T.Concavity of certain maps on postive de fi nite matrices and applications to hadamard products[J].Linear Alg.Appl.,1976,26：203-241.

[9]Uchiyama M.A new majorization between functions,polynomials,and operator inequalities[J].J.Funct.Anal.,2006,231：221-244.

[10]Uchiyama M.A new majorization between functions,polynomials,and operator inequalities II[J].J.Math.Soc.Japan,2008,60：291-310.

[11]Uchiyama M.Operator inequalities：from a general theorem to concrete inequalities[J].Linear Alg.Appl.,2015,465：161-175.

[12]Yang C.Inequalities relating to means of positive operators[J].J.Math.,1996,16(4)：467-474.

[13]Hensen F.An operator inequality[J].Math.Ann.,1980,246：249-250.

一些蘊含F(xiàn)uruta不等式的算子單調(diào)函數(shù)的算子不等式

楊長森,楊朝軍

(河南師范大學數(shù)學與信息科學學院,河南新鄉(xiāng) 453007)

本文研究了算子不等式與算子單調(diào)函數(shù)之間的聯(lián)系.利用關于算子單調(diào)函數(shù)的乘積引理,乘積定理等基本控制原理,給出許多算子不等式,這些不等式可包含正算子理論中應有十分廣泛的Furuta不等式.

算子單調(diào)函數(shù);積引理;積定理;控制

O177.1

on：47A62;47A63

A Article ID： 0255-7797(2017)04-0698-07

date：2015-09-21Accepted date：2015-12-11

Supported by National Natural Science Foundation of China(11271112;11201127)and Technology and the Innovation Team in Henan Province(14IRTSTHN023).

Biography：Yang Changsen(1965-),male,born at Xinxiang,Henan,professor,major in functional analysis.