Extensions of interpolation between the arithmetic geometric mean inequality for matrices

R E S E A R C H

Open Access

Extensions of interpolation between the

arithmetic-geometric mean inequality for

matrices

Mojtaba Bakherad, Rahmatollah Lashkaripour

_{and Monire Hajmohamadi}

[email protected] Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

In this paper, we present some extensions of interpolation between the

arithmetic-geometric means inequality. Among other inequalities, it is shown that if A,B,Xaren×nmatrices, then

_AXB∗2

≤f1

(

)

(

)

(

)

(

)

wheref1,f2,g1,g2are non-negative continuous functions such thatf1(t)f2(t) =tand

g1(t)g2(t) =t(t≥0). We also obtain the inequality

_AB∗2

≤p

(

)

₍

₎

₍

₎

₍

₎

in whichm,n,s,tare real numbers such thatm+n=s+t= 1,||| · |||is an arbitrary unitarily invariant norm andp∈[0, 1].

MSC: Primary 47A64; secondary 15A60

Keywords: arithmetic-geometric mean; unitarily invariant norm; Hilbert-Schmidt norm; Cauchy-Schwarz inequality

1 Introduction and preliminaries

LetMnbe theC∗-algebra of alln×ncomplex matrices and·,·be the standard scalar

product inCn _{with the identity I. The Gelfand map f(t)→f(A) is an isometrical ∗}

-isomorphism between theC∗-algebraC(sp(A)) of continuous functions on the spectrum

sp(A) of a Hermitian matrixAand theC∗-algebra generated byAandI.

A norm ||| · |||onMn is said to be unitarily invariant norm if|||UAV|||=|||A|||, for all

unitary matricesUandV. ForA∈Mn, lets(A)≥s(A)≥ · · · ≥sn(A) denote the singular

values ofA,i.e.the eigenvalues of the positive semidefinite matrix|A|= (A∗A) arranged

in a decreasing order with their multiplicities counted. Note thatsj(A) =sj(A∗) =sj(|A|) (≤j≤n) andA=s(A). The Ky Fan norm of a matrixAis defined asA(k)=

k

j=sj(A)

(≤k≤n). The Fan dominance theorem asserts thatA(k)≤ B(k) fork= , , . . . ,n

if and only if|||A||| ≤ |||B|||for every unitarily invariant norm(see [], p.). The Hilbert-Schmidt norm is defined byA= (

n

j=sj(A))/, which is unitarily invariant.

The classical Cauchy-Schwarz inequality foraj≥,bj≥ (≤j≤n) states that

_n

j= ajbj



≤

_n

j= a_j

_n

j= b_j

with equality if and only if (a, . . . ,an) and (b, . . . ,bn) are proportional []. Bhatia and Davis gave a matrix Cauchy-Schwarz inequality as follows:

AXB∗≤A∗AXXB∗B, ()

whereA,B,X∈Mn. For further information as regards the Cauchy-Schwarz inequality,

see [–] and the references therein. Recently, Kittanehet al.[] extended inequality () to the form

AXB∗≤A∗A pXB∗B–pA∗A –pXB∗B p, ()

whereA,B,X∈Mnandp∈[, ]. Audenaert [] proved that, for allA,B∈Mnand all p∈[, ], we have

AB∗≤pA∗A+ ( –p)B∗B( –p)A∗A+pB∗B. ()

In [], the authors generalized inequality () for allA,B,X∈Mnand allp∈[, ] to the

form

AXB∗≤pA∗AX+ ( –p)XB∗B( –p)A∗AX+pXB∗B. ()

Inequality () interpolates between the arithmetic-geometric mean inequality. In [], the authors showed a refinement of inequality () for the Hilbert-Schmidt norm as follows:

AXB∗_≤pA∗AX+ ( –p)XB∗B_–rA∗AX–XB∗B_

×( –p)A∗AX+pXB∗B_–rA∗AX–XB∗B_ , ()

in whichA,B,X∈Mn, p∈[, ] andr=min{p,  –p}. The Young inequality for every

unitarily invariant norm states that|||ApB–p||| ≤ |||pA+ ( –p)B|||, whereA,Bare positive definite matrices andp∈[, ] (see [] and also [, ]). Kosaki [] extended the last inequality for the Hilbert-Schmidt norm as follows:

ApXB–p_≤pAX+ ( –p)XB_, ()

whereA,Bare positive definite matrices,Xis any matrix andp∈[, ]. In [], the authors considered as a refined matrix Young inequality for the Hilbert-Schmidt norm

ApXB–p_+rAX–XB_≤pAX+ ( –p)XB_, ()

Based on the refined Young inequality (), Zhao and Wu [] proved that

ApXB–p_+rA_XB_–AX

+ ( –p)

_AX–XB

≤pAX+ ( –p)XB  , ()

for  <p≤_ and

ApXB–p_+rA_XB_–XB

+p

_AX–XB

≤pAX+ ( –p)XB  ,

for_<p<  such thatr=min{p,  –p}andr=min{r,  – r}.

In this paper, we obtain some operator and unitarily invariant norms inequalities. Among other results, we obtain a refinement of inequality () and we also extend inequal-ities (), () and () to the functionf(t) =tp_(p∈R).

2 Main results

In this section, using some ideas of [, ], we extend the Audenaert results for the operator norm.

Theorem  Let A,B,X∈Mnand f,f,g,gbe non-negative continuous functions such

that f(t)f(t) =t and g(t)g(t) =t(t≥).Then

AXB∗≤f

A∗A Xg

B∗B f

A∗A Xg

B∗B . ()

Proof It follows from

AXB∗ =BX∗A∗AXB∗

=sBX∗A∗AXB∗

=λmax

BX∗A∗AXB∗ sinceBX∗A∗AXB∗is positive semidefinite

=λmax

X∗A∗AXB∗B

=λmax

X∗fA∗A fA∗A XgB∗B gB∗B

=λmax

gB∗B X∗fA∗A fA∗A XgB∗B

≤gB∗B X∗fA∗A fA∗A XgB∗B

≤g

B∗B X∗f

A∗A f

A∗A Xg

B∗B

=fA∗A XgB∗B fA∗A XgB∗B

that we get the desired result.

Corollary  If A,B,X∈Mnand m,n,s,t are real numbers such that m+n=s+t= , then

AXB∗≤A∗A mXB∗B sA∗A nXB∗Bt. ()

Corollary  Let A,B∈Mnand let f,f,g,gbe non-negative continuous functions such that f(t)f(t) =t and g(t)g(t) =t(t≥).Then

AB∗≤pfA∗A



p_{+ ( –p)gB}∗_B –p_{( –p)fA}∗_A –p_+pgB∗_B p_,

where p∈[, ].

Proof Applying Theorem  forX=I, we have

AB∗≤f

A∗A g

B∗B f

A∗A g

B∗B

=fA∗A p p_gB∗_B –p –p_fA∗_A –p –p_gB∗_B p p

(by Theorem )

≤pfA∗A



p_{+ ( –p)gB}∗_B –p_{( –p)fA}∗_A –p_+pgB∗_B p

(by the Young inequality).

Corollary  Let A,B∈Mnand let f,g be non-negative continuous functions such that f(t)g(t) =t_{(t≥).Then}

AB∗≤pfA∗A + ( –p)gB∗B _{( –p)fA}∗_{A +pgB}∗_B 

×pgA∗A + ( –p)fB∗B _{( –p)gA}∗_{A +pfB}∗_B _,

where p∈[, ].

Proof Applying Theorem  and the Young inequality we get

AB∗≤fA∗A



_gB∗_B _gA∗_A _fB∗_B 

(by Theorem  forf and√g)

≤fA∗A pgB∗B–pfA∗A –pgB∗Bp

×gA∗A pfB∗B–pgA∗A –pfB∗B p

(by inequality ())

≤pfA∗A + ( –p)gB∗B ( –p)fA∗A +pgB∗B

×pgA∗A + ( –p)fB∗B ( –p)gA∗A +pfB∗B

(by the Young inequality).

3 Some interpolations for unitarily invariant norms

In this section, applying some ideas of [], we generalize some interpolations for an arbi-trary unitarily invariant norm.

LetQk,ndenote the set of all strictly increasingk-tuples chosen from , , . . . ,n,i.e. I∈ Qk,nifI= (i,i, . . . ,ik), where ≤i<i<· · ·<ik≤n. The following lemma gives some

Lemma  Let A,B∈Mn.Then

(a) (kA)(kB) =k(AB)fork= , . . . ,n. (b) (kA)∗=kA∗fork= , . . . ,n.

(c) (kA)–₌k

A–_{fork= , . . . ,n.}

(d) Ifs,s, . . . ,snare the singular values ofA,then the singular values of

k_Aare

si,si, . . . ,sik,where(i,i, . . . ,ik)∈Qk,n.

Now, we show inequality () for an arbitrary unitarily invariant norm.

Theorem  Let A,B,X∈Mnand||| · |||be an arbitrary unitarily invariant norm.Then

AXB∗≤A∗A mXB∗BsA∗A nXB∗Bt, ()

where m,n,s,t are real numbers such that m+n=s+t= .In particular,if A,B are positive definite,then

A_XB≤_Ap_XB–p_A–p_XBp_{, ()}

where p∈[, ].

Proof If we replaceA,BandXbykA,kBandkX, theirkth antisymmetric tensor powers in inequality () and apply Lemma , then we have

k AXB∗



≤

k

A∗A mXB∗B s k

A∗A nXB∗Bt,

which is equivalent to

s_

k

AXB∗

≤s

k

A∗A mXB∗B s

s

k

A∗A nXB∗B t

.

Applying Lemma (d), we have

k

j=

sjAXB∗ ≤ k j= s   j

A∗AmXB∗B s k j= s   j

A∗A nXB∗B t

≤ k j= s   j

A∗AmXB∗B s s

 

j

A∗A nXB∗B t , ()

wherek= , . . . ,n. Inequality () implies that

k

j=

sjAXB∗ ≤ k j= s   j

A∗A mXB∗B s s

 

j

A∗A nXB∗Bt

≤ _k j= sj

A∗AmXB∗B s

 k

j= sj

A∗A nXB∗B t

 

wherek= , . . . ,n. Hence

AXB∗_(k)≤A∗A mXB∗B s_(k)A∗AnXB∗B t_(k).

Now, using the Fan dominance theorem [], p., we get the desired result.

Now, using inequality (), Theorem  and the same argument in the proof of Corol-laries  and , we get the following results; these inequalities are generalizations of the Audenaert inequality ().

Corollary  Let A,B∈Mn,m,n,s,t be real numbers such that m+n=s+t= and let

||| · |||be an arbitrary unitarily invariant norm.Then

AB∗≤pA∗A mp _{+ ( –p)B}∗_B –sp_{( –p)A}∗_A –np_+pB∗_B tp_,

where p∈[, ].

Corollary  Let A,B∈Mn,m,n,s,t be real numbers such that m+n=s+t= and let

||| · |||be an arbitrary unitarily invariant norm.Then

AB∗≤pA∗Am+ ( –p)B∗B s_{( –p)A}∗_A m_+pB∗_B s

×pA∗A n+ ( –p)B∗Bt



_{( –p)A}∗_A n_+pB∗_B t _()

in which p∈[, ].

Remark  If we put n=m=s=t=  in inequality (), then we obtain the Audenaert inequality (). Also, if we use inequality (), Corollaries  and , then similar to Corollaries  and  we get the following inequalities:

AXB∗_≤pA∗A m

p_{X+ ( –p)XB}∗_B –sp 

×( –p)A∗A –np_X+pXB∗_B pt

, ()

whereA,B∈Mn,m,n,s,tare real numbers such thatm+n=s+t= ,p∈[, ] and

_AXB∗ ≤p

A∗A mX+ ( –p)XB∗B s

 

( –p)

A∗AmX+pXB∗B s

 



×pA∗AnX+ ( –p)XB∗B t

 

( –p)

A∗AnX+pXB∗Bt

 



forA,B∈Mn, real numbersm,n, s,t, in whichm+n=s+t=  andp∈[, ]. These

inequalities are generalizations of () for the Hilbert-Schmidt norms.

Theorem  Let A,B,X∈Mn.Then

AXB∗_≤pA∗A m

p_{X+ ( –p)XB}∗_B –sp –r

_A∗_A mp_X–XB∗_B –sp 

×( –p)A∗A –np_X+pXB∗_B pt –r

_A∗_A –np_X–XB∗_B pt  ,

in which m,n,s,t are real numbers such that m+n=s+t= ,p∈[, ]and r=min{p,  –p}.

Proof Applying inequality (), we deduce that

AXB∗_≤A∗A mXB∗Bs_A∗A nXB∗Bt_

=A∗A mp p_XB∗_B –sp –p A

∗_A –np –p_XB∗_B pt p 

≤pA∗A mp_{X+ ( –p)XB}∗_B –sp –r

_A∗_A mp_X–XB∗_B –sp 

×( –p)A∗A –np_X+pXB∗_B tp –r

_A∗_A –np_X–XB∗_B tp  ,

wherep∈[, ] andr=min{p,  –p}, and the proof is complete.

Theorem  includes a special case as follows.

Corollary ([], Theorem .) Let A,B,X∈Mn.Then

AXB∗_≤pA∗AX+ ( –p)XB∗B_–rA∗AX–XB∗B_

×( –p)A∗A X+pXB∗B_–rA∗AX–XB∗B_ ,

where p∈[, ]and r=min{p,  –p}.

Proof Forp∈[, ], if we putm=t=pandn=s=  –pin Theorem , then we get the

desired result.

The next result is a refinement of inequality ().

Theorem  Let A,B,X∈Mn(C)and let p∈(, ).Then

(i) For <p≤_,

AXB∗_≤pA∗AX+ ( –p)XB∗B_–rA∗A



_XB∗_B _–A∗_AX



– ( –p)A∗AX–XB∗B_ 

×( –p)A∗AX+pXB∗B_–rA∗A _XB∗_B _–A∗_AX



–pA∗AX–XB∗B_ _{. ()}

(ii) For  <p< ,

AXB∗_≤pA∗AX+ ( –p)XB∗B_–rA∗A



_XB∗_B _–XB∗_B 



– ( –p)A∗AX–XB∗B_

×( –p)A∗AX+pXB∗B_–rA∗A



_XB∗_B _–XB∗_B 



–pA∗AX–XB∗B_



_{, ()}

wherer=min{p,  –p}andr=min{r,  – r}.

Proof The proof of inequality () is similar to that of inequality (). Thus, we only need to prove the inequality ().

If  <p≤_, replacingAandBbyA∗AandB∗Bin inequality (), respectively, we have

A∗A pXB∗B –p_≤pA∗AX+ ( –p)XB∗B_–rA∗A



_XB∗_B  _–A∗_AX



– ( –p)A∗AX–XB∗B_ _{. ()}

Interchanging the roles ofpand  –pin the inequality (), we get

A∗A –pXB∗B p_≤( –p)A∗AX+pXB∗B_–rA∗A



_XB∗_B  _–A∗_AX



–pA∗AX–XB∗B_



_{. ()}

Applying inequalities (), () and (), we get the desired result.

Corollary  Let A,B∈Mn(C)and p∈(, ).Then

(i) For <p≤_,

AB∗_≤pA∗A+ ( –p)B∗B_–rA∗A



_B∗_B _–A∗_A



– ( –p)A∗A–B∗B_

 

×( –p)A∗A+pB∗B_–rA∗A



_B∗_B _–A∗_A



–pA∗A–B∗B_

 _.

(ii) For_<p< ,

AB∗_≤pA∗A+ ( –p)B∗B_–rA∗A



_B∗_B _–B∗_B 



– ( –p)_A∗_A–B∗_B 

 

×( –p)A∗A+pB∗B_–rA∗A



_B∗_B _–B∗_B 



–pA∗A–B∗B_

 _,

wherer=min{p,  –p}andr=min{r,  – r}.

Through the following, we would like to obtain upper bound for |||AXB∗|||, for every unitary invariant norm.

Lemma  Let A,B,X∈Mnsuch that A,B are positive semidefinite.Then,for≤p≤, we have

ApXB–p+r|||AX|||–|||XB|||≤p|||AX|||+ ( –p)|||XB|||, ()

where r=min{p,  –p}.

Proposition  Let A,B,X∈Mn.Then

AXB∗≤pA∗AX+ ( –p)XB∗B –r_A∗AX–XB∗B  

×( –p)A∗AX+pXB∗B –r_A∗AX–XB∗B 

 _,

where p∈[, ]and r=min{p,  –p}.

Proof In inequality (), if we replaceAbyA∗AandBbyB∗B, then we have

A∗A pXB∗B–p≤pA∗AX+ ( –p)XB∗B 

–r_A∗AX–XB∗B 



_{. ()}

Interchangingpwith  –pin inequality (), we get

A∗A –pXB∗B p≤( –p)A∗AX+pXB∗B 

–r_A∗AX–XB∗B 



_{. ()}

Now applying inequalities (), () and () we get the desired inequality.

4 Conclusions

Our application of the methods based on the Audenaert results is presented in this paper to the operator norm and so are some interpolations for an arbitrary unitarily invariant norm. Moreover, we refine some previous inequalities as regards the Cauchy-Schwarz in-equality for the operator and Hilbert-Schmidt norms.

Acknowledgements

The first author would like to thank the Tusi Mathematical Research Group (TMRG).

Funding

The authors declare that there is no source of funding for this research.

Competing interests

The authors declare that they have no competing interests.

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The authors contributed equally to the manuscript and read and approved the final manuscript.

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