WANG Yun,YAN Cheng

(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

Abstract:Using the method of majorization and the properties of quasi-norms,we give some quasi-norm inequalities related to Hayajneh and Kittaneh’s conjecture for operator in semifinite von Neumann algebras.Let E(M)be symmetric quasi-Banach space and let with xiyi=yixi,i=1,2,···, n,then.Some logarithmic submajorization inequalities for operator in semifinite von Neumann algebras are also considered.

Key words:logarithmic submajorisation inequalities;von Neumann algebras;noncommutative symmetric quasi-Banach space

0 Introduction

In order to solve Bourin’s Question,Hayajneh-Kittaneh[2]proposed the following stronger question

and

Obviously,replacingA1,A2,B1andB2byAp,Aq,BpandBq,respectively,we can get inequality(1)and(2).

A special case(i.e.,k=2)of inequalities(5)confirms a conjecture of Hayajneh and Kittaneh(3).

On the other hand,Liu-Wang-Sun[4]showed that ifAi,Bi∈such thatAiBi=BiAi,then

The inequality(4)can be derived from(6)by takingk=2.

In 2016,Han-Shao[5]proved the inequality(5)forp-norm on noncommutativeLpspaces associate with a semi-finite von Neumann algebra.The aim of this paper is to show some logarithmic submajorization and quasi-norm inequalities on noncommutative symmetric quasi-Banach spaces.By adopting the techniques in [6 −8],we obtain some logarithmic submajorization inequalities.As an application,we show that the inequalities(5)and(6)hold on noncommutative symmetric quasi-Banach spaces with order continuous quasi-norm.

1 Preliminaries

We denote by M a semi-finite von Neumann algebra on the Hilbert space H,with a fixed faithful and normal semi-finite trace τ,and M+its positive part.Let S+(M)={x∈M+:τ(s(x)<∞)}and S(M)be the liner span of S+(M).The elements of S(M)are said to be supported by projection of finite trace.We will often denote S+(M)and S(M)simply by S+and S,respectively.A closed densely defined operatorxon H is said to be affiliated with M ifux=xufor any unitaryuin the commutant M′of M.

An operatorxaffiliated with M is said to be measurable with respect to(M,τ)(or simply measurable)if for any δ>0 there existse∈P such that

We denote byL0(M,τ),or simplyL0(M),the family of all measurable operators.

Definition 1Letx∈L0(M,τ)andt>0.The generalized singular number ofxµt(x)is defined by

We denote simply byµ(x)the functiont→µt(x).Ifxis positive,then

See[7]for basic properties and detailed information onµt(x).

Forx∈L0(M)andt>0,we define Λt(x)by

To insure that Λt(x)is well-defined(i.e.∞−∞does not occur),we will consider,in the remainder of this paper,only the class of measurable operatorsxsatisfying:x∈M orµt(x)≤Ct−α,C,α>0.

Letx,ybe two τ−measurable operators.we have

For further details and proofs,we refer the reader to[7,9].

A quasi-Banach latticeEis called a symmetric quasi-Banach space iff∈E,g∈L0(0,∞)andµ(g)≤µ(f)implies thatg∈Eand‖g‖E≤‖f‖E(see[6,10]for more details).For 0

LetEbe a symmetric quasi-Banach space on (0,∞).We say thatEhas order continuous quasi-norm ‖·‖ if for every netfi,i∈IinEsuch thatfi↓0 implies‖fi‖↓0.We define

Then (E(M),‖·‖E(M)) is a noncommutative symmetric quasi-Banach space.IfE=Lp,then (E(M),‖·‖E(M)) is the usual noncommutativeLpspace(Lp(M),‖·‖p).For 0

As is shown in[6],ifEis a symmetric Banach space,thenE(r)(M)=E(M)(r),where

Further details may be found in[6,10].

Definition 2Letx,y∈M.We say thatxis logarithmic submajorization byyand writex≺logyif and only if

2 Main results

The following lemmas will be used to prove the main results in this section.The first lemma is from Theorem 5.11 of[11],the second lemma is from Remark 1 of[12].

The following corollary can be obtained from Lemma 3 and we give its proof separately for easy reference.

ProofThrough Lemma 1,Lemma 3.4 of[13]and Theorem 2 of[12],we obtain

ProofAccording to Theorem 3 of[14](see also[15]),we obtainxy∈E(r)(M).On the other hand,we know from the fact thatE(M)is an α-convex symmetric quasi-Banach space thatE(r)isrα-convex.Then

whereVis a family of positive measures on(0,∞)(Page 500 of[16]).It is easy to see that

We can now formulate our main result.

ProofIt follows from Theorem 3 of[14](see also[15])that

Let 0 ≤xi∈E(M)(p)∩M,0 ≤yi∈E(M)(q)∩M,i=1,2,···,k.Then

which implies thatAis PPT.Consequently,

is also PPT.From Corollary 2,we get

According to Lemma 2.6 of[7],we have

Then

On the other hand,we can see from Page 271 of[7]that τ(e(n,∞)(x))→0 asn→∞.Therefore,we obtain

Lettingg(t)=,Theorem 1.1 of[17],Theorem 2.1 of[18]and Lemma 2.5(iv)of[7]now yields

This implies that

Similarly,

We can conclude from Theorem 3 of[14]that

By an argument similar to the one presented above,we obtain

asn→∞.It follows that

asn→∞.Hence

asn→∞.Combining(14)and(15)with(16),we have

This completes the proof.

Lemma 4LetEbe a symmetric Banach space and let 0 ≤xi,i=1,2,···,k∈E(M).Then

ProofAccording to Theorem 5.3 of[19],letf(t)=tr,r≥1,we get

Combining(17)with(13)gives the lemma.

ProofFirst,it follows from Theorem 3 of[14]that

Note that if we setx1=xt,y1=x1−t,x2=yt,y2=y1−t,in Theorem 1,and setn=2,we get the following inequality.

Corollary 4Letx,y∈E(M)+.Then

3 Logarithmic submajorization inequalities

In this section,we will give some logarithmic submajorization inequalities related to log convex function.In the following we need the definition of log convex function.Letfbe a non-negative real function[0,∞).The functionfis called log convex if

where 0<α<1 ands,t∈[0,∞).

Proposition 2Letfbe a continuous increasing function on[0,+∞]such thatf(0)=0 and lett→f(et)be convex.Ifx,y∈andp≥q>0,then

Moreover,ifx,y∈M,

That is to say,

Takingf(t)=ta,a>0 in the above inequality,we have

The following result may be proved in much the same way as matrix.

Lemma 5Letx,y∈M be self-adjoint.Then

Theorem 2Letx,y∈M be self-adjoint.Then

By the Lemma 2.5(i)(v) of [7],we know µt+s(T)≤µs(T−S)+µt(S) and µt+s(S)≤µs(S−T)+µt(T).Therefore,µt+s(T)a≤µs(T−S)a+µt(S)aandµt+s(S)a≤µs(S−T)a+µt(T)afor 0

It follows from(20)and Lemma 5 that

hence that

Lettinga↓0.Then the assertion follows(see Page 288 of[7]).

Corollary 5Ifx,y∈M+are invertible,then

ProofFrom Theorem 2,we deduce

Therefore,this corollary is proved by Lemma 2.5(v)of[7].

Lemma 6Let f be a convex continuous non-negative increasing function on(0,∞)andx,y∈M+.Then

ProofIfx≥0,we can getf(〈xξ,ξ〉)≤〈f(x)ξ,ξ〉,where ξ ∈H.

Sincex,y∈M+,by(7),we obtain

Therefore,

Theorem 3Let logfbe a convex continuous non-negative increasing function on(0,∞).Ifx,y∈M+are invertible,then

ProofIt follows from Lemma 2,Corollary 5 and Lemma 6 that