R E S E A R C H
Open Access
Operator
P
-class functions
Mojtaba Bakherad
1, Hassane Abbas
2, Bassam Mourad
2*and Mohammad Sal Moslehian
3*Correspondence: [email protected] 2Department of Mathematics, Faculty of Sciences, Lebanese University, Hadath, Beirut, Lebanon Full list of author information is available at the end of the article
Abstract
We introduce and investigate the notion of an operatorP-class function. We show that every nonnegative operator convex function is of operatorP-class, but the converse is not true in general. We present some Jensen type operator inequalities involvingP-class functions and some Hermite-Hadamard inequalities for operator
P-class functions.
MSC: 47A63; 47A60; 26D15
Keywords: P-class function; Jensen operator inequality; positive linear map; Hermite-Hadamard inequality
1 Introduction and preliminaries
LetB(H) denote theC∗-algebra of all bounded linear operators on a complex Hilbert spaceHwith its identity denoted byI. WhendimH=n, we identifyB(H) with the ma-trix algebraMnof alln×nmatrices with entries in the complex fieldC. We denote by
σ(J) the set of all self-adjoint operators onHwhose spectra are contained in an intervalJ. An operatorA∈B(H) is called positive (positive semidefinite for a matrix) ifAx,x ≥ for allx∈Hand in such a case we writeA≥. For self-adjoint operatorsA,B∈B(H), we writeB≥AifB–A≥. The Gelfand mapf →f(A) is an isometrical∗-isomorphism be-tween theC∗-algebraC(σ(A)) of a complex-valued continuous functions on the spectrum
σ(A) of a self-adjoint operatorAand theC∗-algebra generated byIandA. Iff,g∈C(σ(A)), thenf(t)≥g(t) (t∈σ(A)) implies thatf(A)≥g(A). A real-valued continuous functionf
on an intervalJis called operator increasing (operator decreasing, resp.) ifA≤Bimplies
f(A)≤f(B) (f(B)≤f(A), resp.) for allA,B∈σ(J). We recall that a real-valued continuous functionfdefined on an intervalJis operator convex iff(λA+(–λ)B)≤λf(A)+(–λ)f(B) for allA,B∈σ(J) and allλ∈[, ].
A functionf :J−→Ris said to be ofP-class onJor is aP-class function onJif
fλx+ ( –λ)y≤f(x) +f(y), ()
wherex,y∈Jandλ∈[, ]; see []. Many properties ofP-class functions can be found in [–]. Note that the set of allP-class functions contains all convex functions and also all nonnegative monotone functions. Every non-zeroP-class function is nonnegative valued. In fact, chooseλ= and fixx∈J. It follows from () that
f(x)≤f(x) +f(y),
wherey∈J. Thus ≤f(y) for ally∈J.
For aP-class functionf on an interval [a,b],
f
a+b
≤
fta+ ( –t)bdt≤f(a) +f(b),
which is known as the Hermite-Hadamard inequality for theP-class continuous functions; see [].
In this paper, we introduce and investigate the notion of an operatorP-class function and give several examples. We show that iff is aP-class function on (,∞) such that limt→∞f(t) = , then it is operator decreasing. We also prove that iff is an operatorP -class function on an intervalJ, then
fC∗AC≤C∗f(A)C,
whereA∈σ(J) andC∈B(H) is an isometry. In addition, we present a Hermite-Hadamard inequality for operatorP-class functions.
2 OperatorP-class functions
In this section, we investigate operatorP-class functions and study some relations between the operatorP-class functions and the operator monotone functions.
We start our work with the following definition.
Definition Letf be a real-valued continuous function defined on an intervalJ. We say thatf is of operatorP-class onJif
fλA+ ( –λ)B≤f(A) +f(B) for allA,B∈σ(J) and allλ∈[, ].
Clearly every nonnegative operator convex function is of operatorP-class.
Example Letf(t) =t–r(≤r≤) be defined on (,∞). It follows from the operator concavity oftr(≤r≤) [] and the arithmetic-harmonic mean inequality that
λA+ ( –λ)B–r≤λAr+ ( –λ)Br– by the concavity oftr
≤λA–r+ ( –λ)B–r (by the arithmetic-harmonic mean inequality)
≤A–r+B–r,
whereA,B∈σ((,∞)) andλ∈[, ]. Thusf is an operatorP-class function on (,∞). In addition, every operatorP-classf on an intervalJis of operatorQ-class in the sense that
fλA+ ( –λ)B≤f(A) λ +
f(B) –λ
Example The functionf(t) = –tdefined on [–√,√] is of operatorQ-class; see [,
Example .]. We putλ=,A=
andB=–
–
. Thenf(λA+ ( –λ)B) =
f(A) +f(B) =
. Hencef is not of operatorP-class.
Example Letα> andf be a continuous function on the interval [α, α] into itself. It follows from
fλA+ ( –λ)B≤α≤f(A) +f(B) A,B∈σ[α, α],λ∈[, ] thatf is of operatorP-class on [α, α].
Example Letg be a nonnegative continuous function on an interval [a,b] and α= supx,y∈[a,b],t∈[x,y]|g(t) –g(x) –g(y)|. We putf(t) =g(t) +α. Then
fλA+ ( –λ)B=gλA+ ( –λ)B+α
≤g(A) +α+g(B) +α=f(A) +f(B),
whereA,B∈σ([a,b]) andλ∈[, ]. Hencef is an operatorP-class function.
Next, we explore some relations between operatorP-class functions and operator mono-tone functions. In fact, we have the following.
Theorem If f is an operator P-class function on the interval (,∞) such that
limt→∞f(t) = ,then f is operator decreasing.
Proof Let <A ≤B. Fix ε> . We put C =B–A+ε. Let θ > . It follows from limt→∞f(t) = that there existsM> such thatf(t)≤θfor allt≥M. We may assume that the spectrum of the strictly positive operatorCis contained in [α,β] for some <α<β. It follows fromlimλ→– λ
–λ=∞that there existsδ> such that λ
–λ≥ M
α for allλ∈( –δ, ). Henceσ( λ
–λC)⊆[M,∞) for allλ∈( –δ, ). Now, by the functional calculus for the pos-itive operator –λλC, we havef(–λλC)≤θ for allλ∈( –δ, ). Thusf(–λλC)x,x ≤θx
for allλ∈( –δ, ) andx∈H. Sinceλ(B+ε) =λA+ ( –λ)(–λλ)Candf isP-class we have
fλ(B+ε)≤f(A) +f
λ
–λ
C
for allλ∈( –δ, ). Hence
fλ(B+ε)x,x≤f(A)x,x+ f
λ
–λC
x,x
≤f(A)x,x+θx,
where λ∈( –δ, ) andx∈H. As λ→– and thenθ →+ we obtainf(B+ε)x,x ≤
f(A)x,xfor allx∈H. Asε→+, we conclude thatf(B)≤f(A).
3 Jensen operator inequality for operatorP-class functions
Theorem Let f be an operator P-class function on an interval J,A∈σ(J),and C∈B(H)
be an isometry.Then
fC∗AC≤C∗f(A)C. ()
Proof Let X=AB∈B(H⊕H) for some B∈ σ(J) and let U =C –DC∗ and V = C–D
C∗
, whereD=√H–CC∗. Now we can easily conclude from the two facts C∗D=
√
H–CC∗C= and DC=C√H–C∗C = that U andV are unitary operators in B(H⊕H). Further,
U∗XU=
C∗AC C∗AD DAC DAD+CBC∗
and
V∗XV=
C∗AC –C∗AD
–DAC DAD+CBC∗
.
Using the operatorP-class property off we have
f(C∗AC) f(DAD+CBC∗)
=f
C∗AC DAD+CBC∗
=f
U∗XU+V∗XV
≤fU∗XU+fV∗XV
=
C∗f(A)C
Df(A)D+Cf(B)C∗
.
Therefore
fC∗AC≤C∗f(A)C. Applying Theorem we have some inequalities including the subadditivity.
Corollary Let f be operator P-class on an interval J,Aj∈σ(J) (≤j≤n),and Cj∈B(H) (≤j≤n),wherejn=C∗jCj= .Then
f n
j= C∗jAjCj
≤ n
j=
Cj∗f(Aj)Cj.
Proof Let
˜
A=A˜ =
⎛ ⎜ ⎜ ⎜ ⎝ A A · · · An ⎞ ⎟ ⎟ ⎟
⎠∈B(H⊕ · · · ⊕H), C˜= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ C C .. . Cn ⎞ ⎟ ⎟ ⎟ ⎟
It follows fromC˜∗C˜ = and () that
f n
j= C∗jAjCj
=fC˜∗A˜C˜≤C˜∗f(A˜)C˜ = n
j=
Cj∗f(Aj)Cj.
Corollary Let f be operator P-class on[,∞)such that f() = ,A∈σ([,∞)),and C∈B(H)be a contraction.Then
fC∗AC≤C∗f(A)C.
Proof For every contractionC∈B(H), we putD=√H–C∗C. It follows fromC∗C+
D∗D= Hand () that
fC∗AC=fC∗AC+D∗D≤fC∗AC+ fD∗D= C∗f(A)C. Corollary Let f be operator P-class on[,∞)such that f() = and A,B∈σ((,∞))
such that A≤B.Then
A–f(A)≤B–f(B).
Proof Let A,B∈ σ((,∞)) such that < A≤B. We put C=B–/A/. Then CC∗ =
B–/AB–/≤
H, soCis a contraction. It follows from () that
f(A) =fC∗BC≤C∗f(B)C= A/B–/f(B)B–/A/.
Therefore
A–f(A)≤B–f(B). In the following theorem, we obtain the Choi-Davis-Jensen type inequality for operator
P-class functions.
Theorem Let be a unital positive linear map onB(H),A∈σ(J)and f be operator P-class on an interval J.Then
f (A)≤ f(A). ()
Proof LetA∈σ(J). We putthe restriction of to theC∗-algebraC∗(A,I) generated byI
andA. Thenis a unital completely positive map onC∗(A,I). The celebrated Stinespring dilation theorem [, Theorem ] states that there exist an isometryV :H−→Hand a unital∗-homomorphismπ:C∗(A,I)−→B(H) such that(A) =V∗π(A)V. Hence
f (A)=f(A)=fV∗π(A)V≤V∗fπ(A)V (by ())
= V∗πf(A)V= f(A)= f(A).
Example Letf(t) = –tfort∈[–, ]. Then ≤f(t)≤ and
fλA+ ( –λ)B= –λA+ ( –λ)B≤≤ –A+ –B=f(A) +f(B),
whereA,B∈σ([–, ]). Hencef is of operatorP-class on [–, ]. Now, consider that the unital positive map :M →M is defined by (A) = tr(A)I. Then for the Hermitian
matrixA=– we have (A) = ,f( (A)) = ,f(A) =I, and (f(A)) =I. Therefore
f( (A)) = (f(A)). This shows that the coefficient in () and () is the best.
Example Consider (the nonnegative increasing function and so)P-class functionf(t) =
√
twheret∈(,∞). Let the unital positive map:M(C)→Cbe defined by(A) = a withA= (aij)≤i,j≤ and letA=
. Then(f(A)) = andf((A)) =√. Hence
f((A))(f(A)). It follows from () thatf is not of operatorP-class.
We present a Hermite-Hadamard inequality for operatorP-class functions in the next theorem.
Theorem Let be a unital positive linear map onB(H)and f be operator P-class on J.
Then
f
(A) + (B)
≤
fλ (A) + ( –λ) (B)dλ≤ f(A)+ f(B),
where A,B∈σ(J)andλ∈[, ].
Proof LetA,B∈σ(J) andλ∈[, ]. Then
f
(A) + (B)
=f
λ (A) + ( –λ) (B) + ( –λ) (A) +λ (B)
≤fλ (A) + ( –λ) (B)+f( –λ) (A) +λ (B)
≤f (A)+f (B). ()
Integrating both sides of () over [, ] we obtain
f
(A) + (B)
≤
fλ (A) + ( –λ) (B)dλ
+
f( –λ) (A) +λ (B)dλ
=
fλ (A) + ( –λ) (B)dλ
≤f (A)+f (B)
≤ f(A)+ f(B) (by ()).
4 Some inequalities forP-class functions involving continuous operator fields
if the mappingt→Atis norm continuous onT. Ifμ(t) is a Radon measure onTand the functiont→ Atis integrable, one can form the Bochner integral
TAtdμ(t), which is the unique element inAsuch that
ϕ
T
Atdμ(t)
=
T
ϕ(At)dμ(t)
for every linear functionalϕin the norm dualA∗ofA.
LetC(T,A) denote the set of bounded continuous functions onTwith values inA. It is easy to see that the setC(T,A) is aC∗-algebra under the pointwise operations and the norm(At)t∈T=supt∈TAt;cf.[].
Assume that there is a field ( t)t∈Tof positive linear mappings t:A−→BfromAto anotherC∗-algebraB. We say that such a field is continuous if the mappingt→ t(A) is continuous for every A∈A. If the C∗-algebras are unital and the fieldt→ t(I) is integrable with integralI, we say that ( t)t∈Tis unital; see [].
Theorem Let f :J−→Rbe an operator P-class function defined on an interval J,and let AandBbe unital C∗-algebras.If( t)t∈Tis a unital field of positive linear mappings t:
A−→Bdefined on a locally compact Hausdorff space T with a bounded Radon measureμ,
then
f
T
t(At)dμ(t)
≤
T t
f(At)
dμ(t)
holds for every bounded continuous field(At)t∈Tof self-adjoint elements inAwith spectra
contained in J.
Proof We consider the unital positive linear map : C(T,A) −→ B defined by
((At)t∈T) =
T t(At)dμ(t). LetA˜ = (At)t∈T∈C(T,A). It follows fromσ(A˜)⊆Jand () that
f(At)t∈T
=f(A˜)≤f(A˜)= f(At)t∈T
= f(At)
t∈T
.
In the discrete case,T={, . . . ,n}in Theorem , we get the following result.
Corollary Let f :J−→Rbe an operator P-class function defined on an interval J,let Aj∈σ(J) (≤j≤n)and j(≤j≤n)be unital positive linear maps onB(H).Then
f n
j=
j(Aj)
≤ n
j=
j
f(Aj).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to the manuscript and read and approved the final manuscript.
Author details
Acknowledgements
The second and the third authors are supported by the Lebanese University grants program for the Discrete Mathematics and Algebra group.
Received: 27 May 2014 Accepted: 23 October 2014 Published:06 Nov 2014
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