Some quantum f-divergence inequalities for convex functions of self-adjoint operators

Some Quantum f-divergence Inequalities for Convex Functions of Self-adjoint Operators

A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragomir

abstract: In this paper, some new inequalities for convex functions of self-adjoint operators are obtained. As applications, we present some inequalities for quantum f-divergence of trace class operators in Hilbert Spaces.

Key Words: Self-adjoint bounded linear operator, Trace inequality, Convex function, Quantumf-divergence.

Contents

1 Introduction 125

1.1 The Class ofχα_{-Divergences. . . 126}

1.2 Dichotomy Class . . . 126

1.3 Divergences of Arimoto-type. . . 127

2 Some inequalities for convex functions of self-adjoint operators 128

3 Some quantum f-divergence inequalities for convex functions 132

1. Introduction

Let (X,A_{) be a measurable space satisfying |}A_{| >2 and µ be a σ-finite}

mea-sure on (X,A_{). Let}P_{be the set of all probability measures on (X,}A_{) which are}

absolutely continuous with respect to µ. For P, Q _∈ P_{, let p=} dP

dµ and q = dQ dµ denote theRadon-Nikodym derivatives of P andQwith respect toµ.

Two probability measuresP, Q_∈ P _{are said to beorthogonal and we denote}

this byQ_⊥P if

P({q= 0}) =Q({p= 0}) = 1.

Letf : [0,∞) →(−∞,_∞] be a convex function that is continuous at 0, i.e., f(0) = limu↓0f(u).

In 1963, I. Csisz´ar [6] introduced the concept of f-divergence as follows.

Definition 1.1. Let P, Q_∈P_{. Then}

If(Q, P) =

Z

X p(x)f

q(x) p(x)

dµ(x), (1.1)

is called thef-divergence of the probability distributions QandP.

2010Mathematics Subject Classification:47A63, 47A99. Submitted March 26, 2016. Published March 10, 2017

125 Typeset by

BSP

Mstyle. c

Remark 1.2. Observe that, the integrand in the formula (1.1) is undefined when p(x) = 0.The way to overcome this problem is to postulate for f as above that

0f

q(x) 0

=q(x) lim u↓0 uf 1 u

, x_∈X. (1.2)

We now give some examples of f-divergences that are well-known and often used in the literature (see also [3]).

1.1. The Class of χα-Divergences

The f-divergences of this class, which is generated by the function χα, α ∈

[1,∞), defined by

χα(u) =|u₋1|α, u_∈[0,∞)

have the form

If(Q, P) =

Z X p q p−1

α dµ= Z X

p1−α_|q₋p_|αdµ. (1.3)

From this class only the parameter α = 1 provides a distance in the topologi-cal sense, namely the total variation distance V (Q, P) =R

X|q−p|dµ.The most prominent special case of this class is, however,Karl Pearson’s χ2-divergence

χ2(Q, P) =

Z

X q2

pdµ−1

that is obtained forα= 2.

1.2. Dichotomy Class

From this class, generated by the functionfα: [0,∞)→R

fα(u) =

          

u−1−lnu for α= 0;

1

α(1−α)[αu+ 1−α−u

α_{] for α∈R\ {0,1};}

1−u+ulnu for α= 1;

only the parameterα= 1 2

f1

2(u) = 2 (

√

u−1)2provides a distance, namely, the Hellinger distance

H(Q, P) =

Z

X

(√q−√p)2dµ

1₂

.

Another important divergence is the Kullback-Leibler divergence obtained for α= 1,

KL(Q, P) =

1.3. Divergences of Arimoto-type

This class is generated by the functions

Ψα(u) :=

      

     

α α−1

h

(1 +uα)α1 −2α1−1(1 +u)

i

for α∈(0,∞)\ {1};

(1 +u) ln 2 +ulnu₋(1 +u) ln (1 +u) for α= 1;

1

2|1−u| for α=∞.

It has been shown in [19] that this class provides the distances [IΨα(Q, P)]

min₍α,_α)1 forα_∈(0,∞) and 1₂V (Q, P) forα=∞.

Forf continuous convex on [0,∞) we obtain the∗-conjugate function off by

f∗_{(u) =uf}

1 u

, u_∈(0,∞)

and

f∗(0) = lim u↓0f

∗_(u).

It is also known that iff is continuous convex on [0,∞) then so isf∗_.

The following two theorems contain the most basic properties off-divergences. For their proofs we refer the reader to Chapter 1 of [17] (see also [3]).

Theorem 1.3(Uniqueness and Symmetry Theorem). Letf, f1be continuous

con-vex on [0,∞).We have

If₁(Q, P) =If(Q, P),

for allP, Q_∈P _{if and only if there exists a constantc∈}R _{such that}

f1(u) =f(u) +c(u−1),

for any u_∈[0,∞).

Theorem 1.4 (Range of Values Theorem). Let f : [0,∞)→ R _{be a continuous}

convex function on [0,∞).

For anyP, Q∈P_{, we have the double inequality}

f(1)≤If(Q, P)≤f(0) +f∗(0). (1.4)

(i) IfP =Q,then the equality holds in the first part of (1.4).

If f is strictly convex at 1, then the equality holds in the first part of (1.4) if and only ifP =Q;

(ii) IfQ_⊥P, then the equality holds in the second part of (1.4).

Iff(0) +f∗_{(0)<∞,then equality holds in the second part of (1.4) if and only}

The following result is a refinement of the second inequality in Theorem 1.4

(see [3, Theorem 3]).

Theorem 1.5. Let f be a continuous convex function on[0,∞)with f(1) = 0 (f is normalised) andf(0) +f∗(0)<∞. Then

0≤If(Q, P)≤ 1

2[f(0) +f

∗_{(0)]V (Q, P) (1.5)}

for any Q, P _∈P_.

For other inequalities for f-divergence see [2], [7]- [12] and [13] Motivated by the above results, in this paper we obtain some new inequalities for quantum f -divergence of trace class operators in Hilbert spaces. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantumf-divergence in terms of variational andχ2_{-distance are provided. Applications for some classes}

of divergence measures such as Umegaki and Tsallis relative entropies are also given.

In what follows we recall some facts we need concerning the trace of operators and quantumf-divergence for trace class operators in infinite dimensional complex Hilbert spaces.

2. Some inequalities for convex functions of self-adjoint operators

Suppose thatI is an interval of real numbers with interior ˚I and f :I→R_is

a convex function on I. Then f is continuous on ˚I and has finite left and right derivatives at each point of ˚I. Moreover, if x, y ∈ ˚I and x < y, then f₋′ (x) ≤

f+′ (x) ≤ f−′ (y) ≤ f+′ (y), which shows that both f−′ and f+′ are nondecreasing

function on ˚I. It is also known that a convex function must be differentiable except for at most countably many points.

Let A be a self-adjoint operator on the complex Hilbert space (H,h·,·i) with Sp(A)⊆[m, M] for some real numbersm < Mand let{Eλ}λbe its spectral family. Then for any continuous function f : [m, M]→ R_{, it is known that we have the}

following spectral representation in terms of the Riemann-Stieltjes integral (see for instance [14, p. 257]):

hf(A)x, yi=

Z M

m−0

f(λ)d(hEλx, yi),

and

kf(A)xk2=

Z M

m−0|

f(λ)|2d_kEλx_k2,

for anyx, y ∈H.

is convex on I, then

hf(A)x, xi − f(M)−f(m)

M ₋m hAx, xi

≥

"

2 M −m

Z M

m

f(t)dt−M f(M)−mf(m)

M−m

#

kxk2.

Proof. Without loosing the generality, we can assume thatf is differentiable on ˚I. By the convexity off onI, we have

f(t)−f(s)≥f′(s)(t−s), (2.1)

for allt, s∈[m, M]⊂˚I.

If {Et}t∈Ris the spectral family of A then the mapping [m, M]∋t7→ hEtx, xiis

monotonic real mapping for anyx∈H.

If we integrate (2.1) overt∈[m, M] with the integralerhEtx, xithen we get

Z M

m−0

f(t)dhEtx, x_{i −} f(s)

Z M

m−0

d_hEtx, x_i

≥f′(s)

" Z M

m−0

td_hEtx, x_{i −}s

Z M

m−0

d_hEtx, x_i

#

for anys_∈[m, M] andx_∈H.

Using the spectral representation theorem for self-adjoint operators we get

hf(A)x, xi −f(s)kx_k2_≥f′(s)

hAx, x_{i −}s_kx_k2

(2.2)

for anys∈[m, M] andx∈H.

Now if we integrate (2.2) overson [m, M] and divide by M−m, we get

hf(A)x, xi − kx_k2 1 M ₋m

Z M

m

f(s)ds

≥ hAx, xi 1

M ₋m

Z M

m

sf′(s)ds

!

=:K

(2.3)

However, we know that

1 M −m

Z M

m

f′(s)ds= f(M)−f(m) M −m

and

1 M −m

Z M

m

sf′(s)ds= 1 M−m

"

M f(M)−mf(m)−

Z M

m

f(s)ds

#

Therefore

K = hAx, x_if(M)−f(m)

M−m −

kxk2h_{M f(M)−mf(m)−}RM

m f(s)ds

i

M−m

= hAx, xif(M)−f(m)

M−m −

kx_k2hM f(M)−mf(m)−RM

m f(s)ds

i

M−m

+ kxk

2

M ₋m

Z M

m

f(s)ds

By (2.3) we then get

hf(A)x, xi − f(M)−f(m)

M −m hAx, xi

≥ 2kxk

2

M₋m

Z M

m

f(s)ds−kxk

2_{[M f(M)−mf(m)]}

M ₋m

=kxk2

"

2 M₋m

Z M

m

f(s)ds−M f(M)−mf(m)

M ₋m

#

.

✷

Remark 2.2. Since f(t) =t−1 is convex function for t >0, by above theorem we have

hmM A−1x, x_i+hAx, x_{i ≥}2mM ln( M m) M−m for kxk= 1.

Theorem 2.3. Let A and B be two self-adjoint operators on the Hilbert space (H,h·,_·i)with Sp(A),Sp(B)⊆[m, M]⊂˚I andf :I _→R _{a continuously}

differen-tiable convex function on I. Then

hf(A)x, xikyk2 − hf(B)y, yikxk2

≥DhAx, xif′(B)− kxk2f′(B)By, yE

for any x, y_∈H.

Proof. Sincef :I_→R_{is convex onI we have}

f(t)−f(s)≥f′(s)(t−s), for anyt, s_∈˚I. (2.4)

Let{Et}t∈Rbe the spectral family ofAand{Ft}t∈Rthe spectral family ofB. The

Integrating (2.4) overton [m, M] with integralerhEtx, xiwe get

Z M

m−0

f(t)dhEtx, x_{i −} f(s)

Z M

m−0

d_hEtx, x_i

≥f′(s)

Z M

m−0

td_hEtx, x_{i −}s

Z M

m−0

d_hEtx, x_i

!

which is equivalent to

hf(A)x, xi − kxk2f(s)≥f′(s)hAx, xi −sf′(s)kxk2,

for anys_∈[m, M] andx_∈H.

Integrating above inequality overs on [m, M] with integralerhFsx, xiwe then get

hf(A)x, xi

Z M

m−0

d_hFsy, y_{i − k}x_k2

Z M

m−0

f(s)dhFsy, y_i

≥ hAx, x_i

Z M

m−0

f′(s)dhFsy, y_{i − k}x_k2

Z M

m−0

sf′(s)dhFsy, y_i

which is equivalent to

hf(A)x, xikyk2 − hf(B)y, yikxk2

≥ hAx, x_ihf′(B)y, yi − kx_k2_hf′(B)By, yi.

We know that

D

hAx, x_if′(B)− kx_k2f′(B)By, yE= hAx, x_iI_{− k}x_k2B

f′(B)hy, y_i

and

hf(A)x, xiky_k2_{− h}f(B)y, yikx_k2=h(hf(A)x, xiI_{− h}x, x_if(B))y, y_i

On the other hand, we have

hAx, x_iI_{− k}x_k2B

f′(B)≤ hf(A)x, xiI_{− k}x_k2f(B)

and

hf(A)x, xiI− kxk2f(B)≥f′(B)(hAx, xiI− kxk2B)

for allx_∈H. We obtain the desired result. ✷

Remark 2.4. Let B =Pn

j=1αjXj in above theorem and αj ≥0,j ∈ {1, . . . , n}

such that Pn

hf(A)x, xiI _{− k}x_k2f   n X j=1 αjXj  

≥f′

  n X j=1 αjXj   

hAx, xiI− kxk2

n X j=1 αjXj  .

Multiply above inequality by αk and summing onk∈ {1, . . . , n}, we get

* n

X

i=1

αkf(Xk)x, x

+

I_{− k}x_k2f

  n X j=1 αjXj  

≥f′

  n X j=1 αjXj     * _n

X k=1 αkXk ! x, x +

I− kxk2

n X j=1 αjXj  . Equivalently, * n X i=1

αkf(Xk)x, x

+

I₋f

  n X j=1 αjXj  

≥f′

  n X j=1 αjXj     * _n

X k=1 αkXk ! x, x + I₋ n X j=1 αjXj  

for kx_k= 1which is a Jensen type inequality.

3. Some quantum f-divergence inequalities for convex functions

Let (H,h·,·i) be a complex Hilbert space and{ei}i∈I anorthonormal basis of H.We say thatA_∈B_{(H) is aHilbert-Schmidt operator if}

X

i∈I

kAeik2<∞. (3.1)

It is well know that, if {ei}i∈I and {fj}j∈J are orthonormal bases for H and A_∈B_{(H) then}

X

i∈I

kAei_k2=X

j∈I

kAfj_k2=X

j∈I

kA∗fj_k2 (3.2)

showing that the definition (3.1) is independent of the orthonormal basis andAis a Hilbert-Schmidt operator iffA∗ _{is a Hilbert-Schmidt operator.}

LetB_{2(H) the set of Hilbert-Schmidt operators in}B_{(H). ForA∈}B_{2(H) we}

define

kA_k2:= X

i∈I

kAei_k2

!1/2

for {ei}i∈I an orthonormal basis of H. This definition does not depend on the choice of the orthonormal basis.

Using the triangle inequality inl2(I),one checks thatB_{2(H) is avector space}

and that k·k2is a norm on B2(H),which is usually called in the literature as the

Hilbert-Schmidt norm.

Denotethe modulus of an operatorA∈B_{(H) by|A|:= (A}∗_A)1/2_.

Becausek|A_|x_k =kAx_kfor all x_∈H, Ais Hilbert-Schmidt iff|A_| is Hilbert-Schmidt and kA_k2 =k|A_|k2.From (3.2) we have that if A_∈B_{2(H), then A}∗ _∈ B_{2(H) andkAk}

2=kA∗k2.

We define thetrace of a trace class operatorA_∈B_{1(H) to be}

Tr (A) :=X i∈I

hAei, eii, (3.4)

where {ei_}i∈I an orthonormal basis of H.Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (3.4) converges absolutely and it is independent from the choice of basis.

Utilising the trace notation we obviously have that

hA, B_i2= Tr (B∗A) = Tr (AB∗) and kA_k2₂= Tr (A∗A) = Tr|A_|2

for anyA, B_∈B_2(H).

IfA_≥0 andP _∈B_{1(H) withP ≥0,then}

0≤Tr (P A)≤ kAkTr (P). (3.5)

Indeed, sinceA_≥0,thenhAx, x_{i ≥}0 for anyx_∈H.If{ei_}i∈I an orthonormal basis ofH, then

0≤DAP1/2ei, P1/2eiE≤ kAk P

1/2

ei

2

=kAk hP ei, eii

for anyi∈I.Summing overi∈Iwe get

0≤X

i∈I

D

AP1/2ei, P1/2eiE≤ kAkX

i∈I

hP ei, eii=kAkTr (P)

and since

X

i∈I

D

AP1/2ei, P1/2eiE=X i∈I

D

P1/2AP1/2ei, eiE= TrP1/2AP1/2= Tr (P A)

we obtain the desired result (3.5).

This obviously imply the fact that, ifAand B are self-adjoint operators with A_≤B andP_∈B_{1(H) withP ≥0, then}

Now, ifAis a self-adjoint operator, then we know that

|hAx, x_{i| ≤ h|}A_|x, x_i for anyx_∈H.

This inequality follows by Jensen’s inequality for the convex function f(t) = |t_| defined on a closed interval containing the spectrum ofA.

If{ei_}i∈I is an orthonormal basis ofH, then

|Tr (P A)| =

X

i∈I

D

AP1/2ei, P1/2eiE

≤X

i∈I

D

AP1/2ei, P1/2eiE

(3.7)

≤ X

i∈I

D

|A|P1/2ei, P1/2eiE= Tr (P|A|),

for anyA a self-adjoint operator andP _∈B+_{1 (H) :={}P _∈B_{1(H) with}P _≥0}. For the theory of trace functionals and their applications the reader is referred to [22].

For some classical trace inequalities see [4], [5], [18] and [23], which are con-tinuations of the work of Bellman [1].

On complex Hilbert space (B_2(H),h·,·i

2), where the Hilbert-Schmidt inner

product is defined by

hU, Vi2:= Tr (V∗U), U, V ∈B2(H),

for A, B _∈ B+_{(H) consider the operators L}

A : B2(H) → B2(H) and RB :

B_2(H)→B_{2(H) defined by}

L_{AT :=AT and}R_{BT :=T B.}

We observe that they are well defined and since

hLAT, T_i2=hAT, T_i2= Tr (T∗_{AT) = Tr|T}∗_|2

A_≥0

and

hR_{BT, Ti}

2=hT B, Ti2= Tr (T∗T B) = Tr

|T|2B≥0

for any T ∈ B_{2(H), they are also positive in the operator order of} B₍B_2(H)),

the Banach algebra of all bounded operators onB_{2(H) with the normk·k} 2 where

kTk2= Tr

|T|2, T ∈B_2(H).

Since Tr|X∗_|2

= Tr|X_|2for anyX _∈B_{2(H),then also}

Tr (T∗AT) = TrT∗A1/2A1/2T= TrA1/2T∗A1/2T

= Tr

A

1/2_T

2

= Tr

A1/2T∗

2

= Tr

T

∗_A1/2

forA≥0 andT ∈B_2(H).

We observe that L_{A and} R_{B are commutative, therefore the product} L_AR_B

is a self-adjoint positive operator inB₍B_{2(H)) for any positive operatorsA, B ∈} B_(H).

For A, B _∈ B+_{(H) with B invertible, we define the Araki transform A}

A,B :

B_2(H)→B_{2(H) by}A_{A,B :=}L_AR_B−1.We observe that for T ∈B2(H) we have AA,BT =AT B−1 _and

hA_{A,BT, Ti} 2=

AT B−1, T

2= Tr T

∗_{AT B}−1

.

Observe also, by the properties of trace, that

Tr T∗AT B−1= TrB−1/2T∗A1/2A1/2T B−1/2

= TrA1/2T B−1/2∗A1/2T B−1/2= Tr

A

1/2_{T B}−1/2

2

giving that

hA_{A,BT, Ti} 2= Tr

A

1/2_{T B}−1/2

2

≥0 (3.8)

for anyT _∈B_2(H).

We observe that, by the definition of operator order and by (3.8) we have r1B₂(H)≤AA,B≤R1B₂(H) for someR≥r≥0 if and only if

rTr|T_|2_≤Tr

A

1/2_{T B}−1/2

2

≤RTr|T_|2 (3.9)

for anyT ∈B_2(H).

We also notice that a sufficient condition for (3.9) to hold is that the following inequality in the operator order ofB_{(H) is satisfied}

r|T|2≤ A

1/2

T B−1/2 2

≤R|T|2 (3.10)

for anyT ∈B_2(H).

LetU be a self-adjoint linear operator on a complex Hilbert space (K;h·,·i). The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set C(Sp (U)) of all continuous functions defined on the spectrum of U, denoted Sp (U), and the C∗-algebraC∗(U) generated byU and the identity operator 1K onK as follows:

For anyf, g_∈C(Sp (U)) and anyα, β_∈C_{we have}

(i) Φ (αf+βg) =αΦ (f) +βΦ (g) ; (ii) Φ (f g) = Φ (f) Φ (g) and Φ ¯f

= Φ (f)∗; (iii) kΦ (f)k=kf_k:= supt∈Sp(U)|f(t)|;

(iv) Φ (f0) = 1Kand Φ (f1) =U,wheref0(t) = 1 andf1(t) =t,fort∈Sp (U).

With this notation we define

and we call it thecontinuous functional calculus for a self-adjoint operatorU. If U is a self-adjoint operator and f is a real valued continuous function on Sp (U), then f(t)≥ 0 for any t ∈ Sp (U) implies that f(U) ≥0, i.e. f(U) is a positive operator on K. Moreover, if both f and g are real valued functions on Sp (U) then the following important property holds:

f(t)≥g(t) for any t_∈Sp (U) implies that f(U)≥g(U) (P)

in the operator order ofB(K).

Letf : [0,∞)→R_{be a continuous function. Utilising the continuous functional}

calculus for the Araki self-adjoint operator A_{Q,P ∈}B₍B_{2(H)) we can define the}

quantum f-divergence for

Q, P _∈D1(H) :={P ∈B1(H), P ≥0 with Tr (P) = 1}

andP invertible, by

Df(Q, P) :=Df(A_Q,P)P1/2_{, P}1/2E 2= Tr

P1/2f(A_Q,P)P1/2_{. (3.11)}

If we consider the continuous convex function f : [0,∞)→R_{, with}f(0) := 0 andf(t) =tlntfort >0 then forQ, P _∈D1(H) andQ, P invertible we have

Df(Q, P) = Tr [Q(lnQ₋lnP)] =:U(Q, P),

which is the Umegaki relative entropy.

If we take the continuous convex function f : [0,∞) → R_{, f(t) = |t−1| for}

t≥0 then forQ, P ∈D1(H) withP invertible we have

Df(Q, P) = Tr (|Q₋P_|) =:V(Q, P),

whereV (Q, P) is the variational distance.

If we takef : [0,∞)→R_{,f(t) =t}2_{−1 fort≥0 then forQ, P ∈D1(H) with}

P invertible we have

Df(Q, P) = Tr Q2P−1−1 =:χ2(Q, P),

which is called theχ2_-distance

Letq_∈(0,1) and define the convex functionfq : [0,∞)→R_byfq(t) = 1−₁₋t_qq. Then

Dfq(Q, P) =1−Tr Q q_P1−q

1−q , which isTsallis relative entropy.

If we consider the convex functionf : [0,∞)→R_{byf(t) =}1 2

√

t−12,then

Df(Q, P) = 1−TrQ1/2P1/2=:h2(Q, P),

If we take f : (0,∞) → R_{, f(t) = −lnt then for Q, P ∈ D1(H) and Q, P}

invertible we have

Df(Q, P) = Tr [P(lnP−lnQ)] =U(P, Q).

The reader can obtain other particular quantumf-divergence measures by utilizing the normalized convex functions from Introduction, namely the convex functions defining the dichotomy class, Matsushita’s divergences, Puri-Vincze divergences or divergences of Arimoto-type. We omit the details.

In the important case of finite dimensional spaceH and the generalized inverse P−1, numerous properties of the quantumf-divergence, mostly in the case when f isoperator convex,have been obtained in the recent papers [15], [16], [20], [21] and the references therein.

In the following theorem we apply the same proof of Theorem2.3.

Theorem 3.1. LetA_{Q,P be Araki self-adjoint operator on} B₍B_2(H))and

Sp(A_{Q,P)⊆[m, M]⊂I}˚_{whereI is an interval. If the functionf :I→}R_{is convex}

on I, then

hf(A_{Q,P)T, Ti2 −} f(M)−f(m)

M₋m hAQ,PT, Ti2

≥

"

2 M −m

Z M

m

f(t)dt−M f(M)−mf(m)

M −m

#

kT_k2₂.

for T _∈B2(H).

Remark 3.2. Let T =P12 in above theorem we get

hf(A_Q,P)P12, P21i₂ − f(M)−f(m)

M ₋m hAQ,PP 1 2, P12i₂

≥

"

2 M −m

Z M

m

f(t)dt−M f(M)−mf(m)

M −m

#

.

for kP12k₂= 1.

On the other hand, by (3.11) we have

Df(Q, P)≥f(M_M)−f(m)

−m +

"

2 M ₋m

Z M

m

f(t)dt−M f(M_M)−mf(m)

−m

#

(3.12)

since Q∈D1(H).

If we take the continuous convex function f : [0,∞) → R_{, f(t) = |t−1| for}

t≥0 then forP, Q∈D1(H) withP invertible we have the following form≥1

Df(Q, P) = Tr(|Q₋P_|)

= V(Q, P)≥2(M+m₋1).

Theorem 3.3. LetA_{Q,P and}B_{V,U be two Araki self-adjoint operators on}B₍B_2(H))

withSp(A_Q,P),Sp(B_{V,U)⊆[m, M]⊂}˚_{Iandf :I→}R_{a continuously differentiable}

convex function on I. Then

hf(A_{Q,P)T, Ti2k}S_k2 _{− h}f(B_{V,U)S, Si2k}T_k2

≥ DhA_{Q,PT, Ti2f}′₍B_{V,U)− kTk}2_f′₍B_V,U)B_{V,US, S}E

2

for any x, y_∈H.

Remark 3.4. Let T =P12 andS=U12 in above theorem we get

hf(A_Q,P)P12, P12i₂ − hf(B_V,U)U12, U12i₂

≥ DhA_Q,PP12, P12i₂f

′

(B_V,U)−f′(B_V,U)B_V,UU12, U12

E

2

for kP12k₂=kU12k₂= 1.

From above inequality we have

Df(Q, P)−Df(V, U)≥Trf′(B_V,U)U−f′₍B_{V,U)V (3.13)}

sinceQ∈D1(H).

Putf(t) =t2_{−1 for t≥0 in above inequality (3.13) we get}

χ2(Q, P)−χ2(V, U)≥2 Tr (B_V,UU−B_V,UV)

whereP, Q, U, V _∈D1(H) andP, U are invertible. Equivalently, we have

χ2(Q, P)−χ2(V, U)≥2 Tr V −V2U−1

.

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A. Taghavi, V. Darvish, H. M. Nazari,

Department of Mathematic,s Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-146, Babolsar, Iran.

E-mail address:[email protected] [email protected], [email protected]

and

S. S. Dragomir,

Mathematics, School of Engineering and Science, Victoria University

P. O. Box 14428, Melbourne City, MC 8001, Australia.