Some generalizations of operator inequalities for positive linear maps

R E S E A R C H

Open Access

Some generalizations of operator

inequalities for positive linear maps

Jianming Xue

_{and Xingkai Hu}

*_{Correspondence:} [email protected] 1_{Oxbridge College, Kunming}

University of Science and Technology, Kunming, Yunnan 650106, P.R. China

Full list of author information is available at the end of the article

Abstract

In this paper, we generalize some operator inequalities for positive linear maps due to Lin (Stud. Math. 215:187-194, 2013) and Zhang (Banach J. Math. Anal. 9:166-172, 2015).

MSC: 47A63; 47A30

Keywords: operator inequalities; Kantorovich inequalities; positive linear maps

1 Introduction

Throughout this paper, letM,M,m,mbe scalars,Ibe the identity operator, andB(H) be the set of all bounded linear operators on a Hilbert space (H,·,·). The operator norm is denoted by · . We writeA≥ if the operatorAis positive. IfA–B≥, then we say thatA≥B. ForA,B> , we use the following notation:

A∇μB= ( –μ)A+μB,AμB=A



(A–BA–)μA, where≤_μ≤.

Whenμ=_ we writeA∇BandABfor brevity forA∇

BandA B, respectively; see

Kubo and Ando [].

A linear mapis positive if(A)≥ wheneverA≥. It is said to be unital if(I) =I. We say thatis -positive if whenever the × operator matrix_BA B∗_C

is positive, then so is_(B(A)∗_)(C)(B).

Let  <m≤A,B≤Mandbe positive unital linear map. Lin [], Theorem ., proved the following reversed operator AM-GM inequalities:



A+B



≤K(h)(AB), (.)



A+B



≤K(h)(A) (B), (.)

whereK(h) =(h+)_h withh=M_m is the Kantorovich constant.

Can the inequalities (.) and (.) be improved? Lin [], Conjecture ., conjectured

that the constantK(h) can be replaced by the Specht ratioS(h) = h



h– elogh



h–

in (.) and (.),

which remains as an open question.

Zhang [], Theorem ., generalized (.) and (.) whenp≥:

p

A+B



≤(K(M+m))p

Mp_mp

p_{(AB), (.)}

p

A+B



≤(K(M+m))p

Mp_mp

(A) (B)p. (.)

We will present some operator inequalities which are generalizations of (.), (.), (.), and (.) in the next section.

Bhatia and Davis [] proved that if  <m≤A≤MandXandYare two partial isometries onHwhose final spaces are orthogonal to each other. Then for every -positive unital linear map,

X∗AYY∗AY–Y∗AX≤

M–m M+m



X∗AX. (.)

Lin [], Conjecture ., conjectured that the following inequality could be true:

X∗AYY∗AY–Y∗AXX∗AX–≤

M–m M+m



.

Recently, Fu and He [], Theorem , proved

X∗AYY∗AY–Y∗AXX∗AX–≤ 



M–m M+m



M+ 

m



. (.)

We will get a stronger result than (.).

2 Main results

We begin this section with the following lemmas.

Lemma [] Let A,B> .Then the following norm inequality holds:

AB ≤ 

A+B

_{. (.)}

Lemma [] Let A> .Then for every positive unital linear map,

A–≥–(A). (.)

Lemma [] Let A,B> .Then,for≤r<∞,

Ar+Br≤(A+B)r. (.)

Lemma ([], Theorem ) Suppose that two operators A,B and positive real numbers m,m,M,Msatisfy either of the following conditions:

()  <m≤A≤m<M≤B≤M, ()  <m≤B≤m<M≤A≤M.

Then

A∇μB≥Kr

hAμB

Theorem  Let <m≤A≤m<M≤B≤M.Then A+B

 +MmK



_h(AB)–≤_{M+m, (.)}

where K(h) =(h_h+) with h=M

m.

Proof It is easy to see that 

(M–A)(m–A)A

–_≤,

then

MmA

–

 +

A

 ≤

M+m

 .

Similarly,

MmB

–

 +

B

 ≤

M+m

 .

Summing up the above two inequalities, we get

A+B

 +Mm

A–_+B–

 ≤M+m.

By (AB)–_=A–_B–_{and Lemma , we have}

A+B

 +MmK



_h(AB)–₌ A+B

 +MmK



_hA–_B– ≤A+B

 +Mm

A–_+B–



≤M+m.

This completes the proof.

Theorem  Let <m≤A≤m<M≤B≤M.Then for every positive unital linear map

,



A+B



≤K(h)

K(h)

_{(AB) (.)}

and



A+B



≤K(h)

K(h)

(A) (B), (.)

where K(h) =(h+)_h,K(h) = (h_h+),h=M_m,and h=M

Proof The inequality (.) is equivalent to

A+B



–(AB)≤ K(h)

K(h)

. (.)

Compute

A+B



MmK_h–_(AB)

≤ 



A+B



+MmK_h–_(AB) 

(by (.))

≤ 



A+B



+MmK_h(AB)–



(by (.))

= 

A+B

 +MmK



_h(AB)– 

≤ 

(M+m)



(by (.))

=

(M+m)

_.

That is,

A+B



–(AB)≤ (M+m)



MmK_(h)

= K(h)

K_(h)

.

Thus, (.) holds. The proof of (.) is similar, we omit the details.

This completes the proof.

Remark  Because of K_K(h(h)₎ <K(h), inequalities (.) and (.) are refinements of (.) and (.), respectively.

Theorem  Let <m≤A≤m<M≤B≤M and≤p<∞.Then for every positive unital linear map,

p

A+B



≤ 



K_(h)(M_+m₎

K(h)M_m

p

p(AB) (.)

and

p

A+B



≤ 



K(h)(M+m)

K(h)M_m

p

(A) (B)p, (.)

where K(h) =(h+)_h,K(h) = (h_h+),h=M_m,and h=M_m.

Proof By the operator reverse monotonicity of inequality (.), we have

–(AB)≤L–

A+B



, (.)

whereL= K(h)

Compute p

A+B



Mpmp–p(AB)

≤ 

 Lpp

A+B



+

M_m

L

p



–p(AB)



(by (.))

≤ 

 L

A+B



+M

_m

L

–_(AB) p

(by (.))

≤ 

 L

A+B



+LMm–

A+B



p (by (.))

≤ 



LM+mp (by [], (.)). That is,

p

A+B



–p(AB)≤  

L(M_+m₎

Mm

p

= 

K_(h)(M_+m₎

K(h)M_m

p



.

Thus, (.) holds. By inequality (.), the proof of (.) is similar, we omit the details.

This completes the proof.

Remark  SinceK(h) > , inequalities (.) and (.) are sharper than (.) and (.), respectively.

Theorem  Let <m≤A≤M and let X,Y be two isometries onHwhose final spaces are orthogonal to each other.Then for every-positive unital linear map,

X∗AYY∗AY–Y∗AXX∗AX–≤(M–m)



Mm . (.)

Proof Since X is isometric and  < m ≤ A ≤ M, m ≤ (X∗AX) ≤ M and _M ≤

(X∗AX)–_≤  m.

Compute

M–m M+m



MmX∗AYY∗AY–Y∗AXX∗AX–

≤ 



X∗AYY∗AY–Y∗AX+

M–m M+m



MmX∗AX–



(by (.))

≤ 



M_M–₊m_mX∗AX+

M–m M+m



MmX∗AX–



(by (.))

≤ 



M–m M+m



(M+m).

Hence,

X∗AYY∗AY–Y∗AXX∗AX–≤(M–m)



Mm .

Remark  Since  <m≤M, 



M–m M+m



M+ 

m

 ≥

M–m M+m



M m ≥

M–m M+m



(M+m)

Mm =

(M–m)

Mm .

Thus, (.) is tighter than (.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{Oxbridge College, Kunming University of Science and Technology, Kunming, Yunnan 650106, P.R. China.}2_{Faculty of}

Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P.R. China.

Acknowledgements

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript. This research was supported by Scientific Research Fund of Yunnan Provincial Education Department (No. 2014C206Y).

Received: 9 October 2015 Accepted: 15 January 2016 References

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