ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 9 Issue 1(2017), Pages 65-73.
MORE OPERATOR INEQUALITIES FOR POSITIVE LINEAR MAPS
HAMID REZA MORADI, MOHSEN ERFANIAN OMIDVAR, SILVESTRU SEVER DRAGOMIR AND MOHAMMAD KAZEM ANWARY
Abstract. In this paper we present some new operator inequality for convex functions. We have obtained a number of Jensen’s type inequalities for convex and operator convex functions of self-adjoint operators for positive linear maps. Some results are exemplified for power and logarithmic functions.
1. Introduction and Preliminaries
As is customary, we reserveM, m for scalars. Other capital letters are used to denote general elements of theC∗-algebra B(H) of all bounded linear operators acting on a Hilbert space (H,h·,·i). An operator A∈ B(H) is called positive if hAx, xi ≥ 0 for all x∈ H, and we then writeA ≥ 0. For self-adjoint operators
A, B ∈ B(H) we say that A≤B ifB−A≥0. The Gelfand map establishes an isometrically∗-isomorphism Φ between the setC(sp(A)) of all continuous functions on the spectrum of A, denotedsp(A), and the C∗-algebra generated by Aand I
(see for instance [13, p. 15]). For anyf, g∈C(sp(A)) and anyα, β∈Cwe have • Φ (αf+βg) =αΦ (f) +βΦ (g);
• Φ (f g) = Φ (f) Φ (g); • kΦ (f)k=kfk:= sup
t∈sp(A) |f(t)|;
• Φ (f0) = 1H and Φ (f1) =A, wheref0(t) = 1 andf1(t) =t, fort∈sp(A). With this notation we definef(A) = Φ (f) for allf ∈C(sp(A)) and we call it the continuous functional calculus for a self-adjoint operatorA. It is well known that, ifAis a self-adjoint operator and f ∈C(sp(A)), thenf(t)≥0 for anyt∈sp(A) implies that f(A)≥0. It is extendible for two real valued functions onsp(A). A linear map φis positive if φ(A)≥0 wheneverA ≥0. It said to be normalized if
φ(I) = I. For more studies in this direction, we refer to [2, 6]. As is known to all, in [8, Theorem 1.2], the authors presented the operator version for the Jensen inequality. They proved the inequality
f(hAx, xi)≤ hf(A)x, xi,
2000Mathematics Subject Classification. Primary 47A63, Secondary 46L05, 47A60.
Key words and phrases. Positive map; operator convex; operator inequality. c
2017 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted September 13, 2016. Published January 15, 2017. Communicated by Takeaki Yamazzaki.
whereAis a self-adjoint operator withsp(A)⊆[m, M] andf(t) is a convex function on an interval [m, M]. Choi [3] and Davis [4] showed that if φ is a normalized positive linear map onB(H) and iff is an operator convex function on an interval [m, M], then the so-called Choi-Davis-Jensen inequality
f(φ(A))≤φ(f(A)), (1.1)
holds for every self adjoint operatorAonH whose spectrum is contained in [m, M]. As a special case of inequality (1.1), the authors in [8, Theorem 1.21] established the following generalization of Jensen’s inequality under the additional condition
pi,(i∈ {1, . . . , n}) are positive real numbers such that n P
i=1
pi= 1
f n X
i=1
piφi(Ai) !
≤ n X
i=1
piφi(f(Ai)).
The following variant of Jensen’s operator inequality for a convex function f ∈
C([m, M]), self-adjoint operatorsAi∈ B(H) with spectra in [m, M] and normal-ized positive linear mapsφi onB(H) was proved in [11]
f (m+M) 1H − n X
i=1
φi(Ai) !
≤(f(m) +f(M)) 1H − n X
i=1
φi(f(Ai)).
Moreover, in the same paper the following series of inequalities was proved
f(m+M) 1H − n X
i=1
φi(Ai)
≤
M1H − n P
i=1
φi(Ai)
M−m f(M) + n P
i=1
φi(Ai)−m1H
M −m f(M)
≤(f(m) +f(M)) 1H − n X
i=1
φi(f(Ai)).
There is considerable amount of literature devoting to the study of Jensen inequal-ity, we refer to [9, 10, 12] for a recent survey and references therein.
A function f : I → (0,+∞) is said to be log-convex if log(f) is convex, or, equivalently, if for allx, y∈Iandt∈[0,1] one has the inequality
f(tx+ (1−t)y)≤[f(x)]t[f(y)]1−t.
The functionf(x) =1x ,x∈(0,+∞) is log-convex on (0,+∞).
This paper include but are not restricted to the positive linear map version of the Dragomir-Ionescu inequality, Slater’s type inequalities for operators and its inverses, Jensen’s inequality for differentiable functions, Jensen’s type inequalities for log-convex functions and for differentiable log-convex functions.
2. Inequalities for Differentiable and Convex Functions
Theorem 2.1. Let A be a self adjoint operator on the Hilbert space H with
sp(A)⊆Iandφbe a normalized positive linear map onB(H)and letf :I→Rbe a convex and differentiable function on intervalIwhose derivative f0 is continuous onI. Then
0≤ hφ(f(A))x, xi −f(hφ(A)x, xi)
≤ hφ(f0(A)A)x, xi − hφ(A)x, xi hφ(f0(A))x, xi (2.1)
for eachx∈H withkxk= 1.
Proof. First we remark that, the first inequality in (2.1) has been proven for convex functions with no differentiability assumption in [5]. As in [5] and for the sake of completeness, we give here a short proof of the second inequality from (2.1). Since
f is convex and differentiable, by the gradient inequality we have that
f(t)−f(s)≤f0(t) (t−s),
for anyt, s∈[m, M]. Sincesp(A)⊆[m, M] by substitutes=hφ(A)x, xi ∈[m, M], we have
f(t)−f(hφ(A)x, xi)≤f0(t) (t− hφ(A)x, xi).
Now, by the functional calculus fort∈[m, M] we deduce
f(A)−f(hφ(A)x, xi) 1H ≤f0(A) (A− hφ(A)x, xi1H) and sinceφis a normalized positive linear map we have
hφ(f(A))x, xi −f(hφ(A)x, xi)
≤ hφ(f0(A)A)x, xi − hφ(A)x, xi hφ(f0(A))x, xi
for eachx∈H withkxk= 1.
Remark. There are several examples of normalized positive linear maps. But for our application we consider among them φ : Mn(C) → Mn(C), which φ(A) =
1
ntr(A)
1H for all Hermitian matrices A∈ Mn(C). From inequality (2.1), we have
0≤ 1
ntr(f(A))−f 1
ntr(A)
≤ 1
ntr(f
0(A)A)− 1
n2tr(A)tr(f 0(A)).
If we take f(t) =t2, then 1
ntr
2(A)≤tr A2 .
In the following we give more interesting situations.
Corollary 2.2. Let Ai be self-adjoint operators with sp(Ai) ⊆ I, i ∈ {1, . . . , n} and φi are normalized positive linear map on B(H) (i∈ {1, . . . , n}) and assume that f is as in the Theorem 2.1. Then
0≤
* n X
i=1
φi(f(Ai))x, x +
−f * n
X
i=1
φi(Ai)x, x +
≤
* n X
i=1
φi(f0(Ai)Ai)x, x +
−
* n X
i=1
φi(Ai)x, x +
×
* n X
i=1
φi(f0(Ai))x, x +
,
for eachx∈H withkxk= 1.
Proof. If we put
e A:=
A1 . . . 0 ..
. . .. ... 0 · · · An
,
then we havespAe
⊆[m, M]⊆Io, moreover, define
φ:B(H)⊕. . .⊕ B(H)→ B(H) as
φ(A1⊕. . .⊕An) = n X
i=1
φi(Ai)
in this case, we have
φfAe
= n X
i=1
φi(f(Ai)), φ
e A=
n X
i=1
φi(Ai).
Applying Theorem 2.1 forAewe deduce the desired result (2.2).
Corollary 2.3. Notation as in above. Ifpiare positive real numbers with n P
i=1
pi= 1,
then
0≤
* n X
i=1
pif(φi(Ai))x, x +
−f * n
X
i=1
piφi(Ai)x, x +!
≤
* n X
i=1
pif0(φi(Ai)Ai)x, x +
−
* n X
i=1
pi(φi(Ai))x, x +
×
* n X
i=1
pif0(φi(Ai))x, x +
,
for eachx∈H withkxk= 1.
Remark. The following particular cases are of interest (see also[5]): (a) Ifp≥1 andAis a positive operator onH, then we have the inequality
0≤ hφ(Ap)x, xi − hφ(A)x, xip
≤phhφ(Ap)x, xi − hφ(A)x, xiDφAp−1x, xEi (2.3)
for eachx∈H with kxk= 1.
(b) If A is a positive definite operator on H, then the inequality (2.3) holds for
p <0.
(c) If0< p <1 andA is a positive definite operator onH, then hφ(Ap)x, xi − hφ(A)x, xip
≥p
hφ(Ap)x, xi − hφ(A)x, xi
φ Ap−1 x, x
≥0
for eachx∈H with kxk= 1.
Corollary 2.4. LetAbe a strictly positive operator on the Hilbert spaceH. Then 0≤lnhφ(A)x, xi − hφ(lnA)x, xi ≤ hφ(A)x, xi
φ A−1 x, x
−1
for eachx∈H withkxk= 1.
Now, from a different view point we may state:
Theorem 2.5. All assumptions as in Theorem 2.1. Then hφ(f(A)−f(B))x, xi
≤ hφ(f0(A)A)x, xi − hφ(f0(A))x, xi hφ(B)x, xi. (2.4)
for eachx∈H withkxk= 1.
Proof. Sincef is convex and differentiable we have
f(t)−f(s)≤f0(t) (t−s)
for any t, s ∈I. Fix s ∈I and apply the functional calculus for the operator A, then
f(A)−f(s) 1H ≤f0(A) (A−s).
Sinceφis normalized positive linear map we get
φ(f(A))−f(s) 1H ≤φ(f0(A)A)−sφ(f0(A)) (2.5)
therefore
hφ(f(A))x, xi −f(s)≤ hφ(f0(A)A)x, xi −shφ(f0(A))x, xi
for eachx∈H withkxk= 1. Apply again functional calculus for the operatorB
to obtain
hφ(f(A))x, xi1H −f(B)
≤ hφ(f0(A)A)x, xi1H − hφ(f0(A))x, xiB. (2.6)
Sinceφis normalized positive linear map we have hφ(f(A))x, xi1H −φ(f(B))
≤ hφ(f0(A)A)x, xi1H − hφ(f0(A))x, xiφ(B) (2.7)
therefore
hφ(f(A))x, xi − hφ(f(B))y, yi
≤ hφ(f0(A)A)x, xi − hφ(f0(A))x, xi hφ(B)y, yi. (2.8)
for each x, y∈H withkxk =kyk = 1. This is an inequality of interest in itself. Finally, on makingy=xin (2.8) we deduce the desired result (2.4).
By settingφ(A) = n1tr(A)1H in (2.4), we find the following result.
Remark. Let A, B∈Mn(C)be two Hermitian matrices, then 1
ntr(f(A)−f(B))≤
1
ntr(f
0(A)A)− 1
n2tr(f
0(A))tr(B).
Remark. If we chooseA=B in (2.4), we get
hφ(f0(A)A)x, xi ≥ hφ(f0(A))x, xi hφ(A)x, xi (2.9)
The following result is known in the literature as the Holder-McCarthy inequality.
Remark. If we takef(t) =t2 in (2.9), we deduce
φ A2
x, x
≥ hφ(A)x, xi2
for eachx∈H withkxk= 1.
The case of norm operator may be of interest and is embodied in the following remark.
Remark. Let A be a positive operator on B(H) and f0 ∈C(sp(A)). By taking supremum overx∈H withkxk= 1, andy∈H withkyk= 1in (2.8)respectively, we obtain
kφ(f0(A)A)k ≥ kφ(f0(A))k kφ(A)k.
The following result concerning Slater type inequality holds:
Theorem 2.6. Let A be a self-adjoint operator on the Hilbert space H with
sp(A) ⊆ [m, M] ⊆ I and φ be a normalized positive linear map on B(H) and letf :I→Rbe a convex and differentiable function on intervalI whose derivative
f0 is continuous and strictly positive on I. Then
0≤f
hφ(Af0(A))x, xi hφ(f0(A))x, xi
− hφ(f(A))x, xi
≤f0
hφ(Af0(A))x, xi hφ(f0(A))x, xi
hφ(Af0(A))x, xi
hφ(f0(A))x, xi − hφ(A)x, xi
(2.10)
for eachx∈H withkxk= 1.
Proof. Sincef is convex and differentiable onI, then we have that
f0(s) (t−s)≤f(t)−f(s)≤f0(t) (t−s)
for anyt, s∈[m, M]. If we fixt∈[m, M] and apply the functional calculus for the operatorA, then we obtain
f0(A) (t−A)≤f(t) 1H −f(A)≤f0(t) (t−A) (2.11)
for anyt∈[m, M]. Sinceφis normalized positive linear map we have
tφ(f0(A))−φ(f0(A)A)≤f(t) 1H −φ(f(A))≤f0(t)t1H −f0(t)φ(A) (2.12) for anyt∈[m, M]. The inequality (2.12) is equivalent with
thφ(f0(A))x, xi − hφ(f0(A)A)x, xi ≤f(t)− hφ(f(A))x, xi
≤f0(t)t−f0(t)hφ(A)x, xi (2.13)
for anyt∈[m, M]. Now, sinceAis self-adjoint withm1H ≤A≤M1H andf0(A) is strictly positive, thenmf0(A)≤Af0(A)≤M f0(A). Also, φis a unital positive linear map, thereforemφ(f0(A))≤φ(Af0(A))≤M φ(f0(A)), i.e.,
mhφ(f0(A))x, xi ≤ hφ(Af0(A))x, xi ≤Mhφ(f0(A))x, xi,
hencehφ(Af0(A))x, xi hφ(f0(A))x, xi−1∈[m, M]. If we put
t:=hφ(Af
0(A))x, xi hφ(f0(A))x, xi
The following results follows from the inequality (2.10) for the convex function
f(t) =tp, p∈(−∞,0)∪[1,∞) by performing the required calculation. The details are omitted.
Remark. Let φbe a positive linear map onB(H)and letAbe a strictly positive operator onH.
(a) Ifp≥1, we have that
0≤ hφ(Ap)x, xip−1−
φ Ap−1x, xp
≤phφ(Ap)x, xip−2
hφ(Ap)x, xi − hφ(A)x, xi
φ Ap−1 x, x
.
for anyx∈H with kxk= 1. (b) Ifp <0, we have that
0≤ hφ(Ap)x, xip−1−
φ Ap−1 x, xp
≤(−p)hφ(Ap)x, xip−2
hφ(A)x, xi −
φ Ap−1 x, x
hφ(Ap)x, xi .
for anyx∈H with kxk= 1.
Remark. Notice that we can obtain other results by taking various functions. The details are omitted.
3. Some Inequalities Involving States
In this section, we apply the continuous functional calculus to convex and differ-entiable functions and present some reverses Jensen’s inequalities involving states onC∗-algebras. Our main result of this section reads as follows.
Theorem 3.1. Let τ be a state on B(H)and f :I →Rbe a convex and differ-entiable function on intervalI whose derivative f0 is continuous on I. Then
0≤τ(f(A))−f(τ(A))≤τ(f0(A)A)−τ(A)τ(f0(A)).
Proof. Sincef is convex and differentiable, we have that
f(t)−f(s)≤f0(t) (t−s) if we chose in this inequalitys=τ(A) we obtain
f(t)−f(τ(A))≤f0(t) (t−τ(A)).
Applying functional calculus for the operatorA
f(A)−f(τ(A))≤f0(A)A−f0(A)τ(A).
For the stateτ we have
τ(f(A))−f(τ(A))≤τ(f0(A)A)−τ(A)τ(f0(A)).
Apply Theorem 3.1 to the stateτ defined byτ(A) =hAx, xifor fixed unit vector
x∈H. We have the following result:
Corollary 3.2. Let A be a self adjoint operator on the Hilbert space H with
sp(A)⊆I and let f :I→Rbe a convex and differentiable function on interval I whose derivativef0 is continuous onI. Then
This inequality was obtained by Dragomir (see [1, Theorem 5]).
By Theorem 2.5 and making use of a similar argument to the one in the proof of Theorem 3.1 we can state the following result as well. However, the details are nor provided here.
Theorem 3.3. Let τ be a state on B(H)and f :I →Rbe a convex and differ-entiable function on intervalI whose derivative f0 is continuous on I. Then
τ(f(A))−τ(f(B))≤τ(f0(A)A)−τ(f0(A))τ(B). (3.1)
Remark. If we chooseA=B in (3.1), we obtain
τ(f0(A))τ(A)≤τ(f0(A)A).
Remark. Let τ be a state on B(H)andf(t) =t2 in (3.1), then
τ(A)2≤τ A2
(A≥0).
In a similar fashion, for self-adjoint operatorsA∈ B(H)
τ eαA2
≤τ e2αA
(α≥0).
By applying Theorem 2.6 and using the same strategy as in the proof of Theorem 3.1 we get the next result.
Theorem 3.4. Let τ be a state onB(H)andf :I→Rbe a convex and differen-tiable function on intervalI whose derivativef0 is continuous and strictly positive onI. Then
0≤f
τ(Af0(A))
τ(f0(A))
−τ(f(A))
≤f0
τ(Af0(A))
τ(f0(A))
τ(Af0(A))
τ(f0(A)) −τ(A)
.
Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.
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Department of Mathematics
Mashhad Branch, Islamic Azad University, Mashhad, Iran E-mail address:[email protected]
Department of Mathematics
Mashhad Branch, Islamic Azad University, Mashhad, Iran E-mail address:[email protected]
Department of Mathematics
College of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne City, MC 8001, Australia
E-mail address:[email protected]
Department of Pure Mathematics
Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi Univer-sity of Mashhad, P.O. Box 1159, Mashhad 91775, Iran
