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Operator P-class functions
Journal of Inequalities and Applications volume 2014, Article number: 451 (2014)
Abstract
We introduce and investigate the notion of an operator P-class function. We show that every nonnegative operator convex function is of operator P-class, but the converse is not true in general. We present some Jensen type operator inequalities involving P-class functions and some Hermite-Hadamard inequalities for operator P-class functions.
MSC:47A63, 47A60, 26D15.
1 Introduction and preliminaries
Let denote the -algebra of all bounded linear operators on a complex Hilbert space ℋ with its identity denoted by I. When , we identify with the matrix algebra of all matrices with entries in the complex field ℂ. We denote by the set of all self-adjoint operators on ℋ whose spectra are contained in an interval J. An operator is called positive (positive semidefinite for a matrix) if for all and in such a case we write . For self-adjoint operators , we write if . The Gelfand map is an isometrical ∗-isomorphism between the -algebra of a complex-valued continuous functions on the spectrum of a self-adjoint operator A and the -algebra generated by I and A. If , then () implies that . A real-valued continuous function f on an interval J is called operator increasing (operator decreasing, resp.) if implies (, resp.) for all . We recall that a real-valued continuous function f defined on an interval J is operator convex if for all and all .
A function is said to be of P-class on J or is a P-class function on J if
where and ; see [1]. Many properties of P-class functions can be found in [1–4]. Note that the set of all P-class functions contains all convex functions and also all nonnegative monotone functions. Every non-zero P-class function is nonnegative valued. In fact, choose and fix . It follows from (1) that
where . Thus for all .
For a P-class function f on an interval ,
which is known as the Hermite-Hadamard inequality for the P-class continuous functions; see [3].
In this paper, we introduce and investigate the notion of an operator P-class function and give several examples. We show that if f is a P-class function on such that , then it is operator decreasing. We also prove that if f is an operator P-class function on an interval J, then
where and is an isometry. In addition, we present a Hermite-Hadamard inequality for operator P-class functions.
2 Operator P-class functions
In this section, we investigate operator P-class functions and study some relations between the operator P-class functions and the operator monotone functions.
We start our work with the following definition.
Definition 1 Let f be a real-valued continuous function defined on an interval J. We say that f is of operator P-class on J if
for all and all .
Clearly every nonnegative operator convex function is of operator P-class.
Example 1 Let () be defined on . It follows from the operator concavity of () [5] and the arithmetic-harmonic mean inequality that
where and . Thus f is an operator P-class function on .
In addition, every operator P-class f on an interval J is of operator Q-class in the sense that
for all and ; see [6]. In the next example, we show the converse is not true, in general.
Example 2 The function defined on is of operator Q-class; see [[7], Example 2.1]. We put , and . Then . Hence f is not of operator P-class.
Example 3 Let and f be a continuous function on the interval into itself. It follows from
that f is of operator P-class on .
Example 4 Let g be a nonnegative continuous function on an interval and . We put . Then
where and . Hence f is an operator P-class function.
Next, we explore some relations between operator P-class functions and operator monotone functions. In fact, we have the following.
Theorem 1 If f is an operator P-class function on the interval such that , then f is operator decreasing.
Proof Let . Fix . We put . Let . It follows from that there exists such that for all . We may assume that the spectrum of the strictly positive operator C is contained in for some . It follows from that there exists such that for all . Hence for all . Now, by the functional calculus for the positive operator , we have for all . Thus for all and . Since and f is P-class we have
for all . Hence
where and . As and then we obtain for all . As , we conclude that . □
3 Jensen operator inequality for operator P-class functions
In this section, we present a Jensen operator inequality for operator P-class functions. We start with the following result in which we utilized the well-known technique of [8].
Theorem 2 Let f be an operator P-class function on an interval J, , and be an isometry. Then
Proof Let for some and let and , where . Now we can easily conclude from the two facts and that U and V are unitary operators in . Further,
and
Using the operator P-class property of f we have
Therefore
□
Applying Theorem 2 we have some inequalities including the subadditivity.
Corollary 1 Let f be operator P-class on an interval J, (), and (), where . Then
Proof Let
It follows from and (2) that
□
Corollary 2 Let f be operator P-class on such that , , and be a contraction. Then
Proof For every contraction , we put . It follows from and (2) that
□
Corollary 3 Let f be operator P-class on such that and such that . Then
Proof Let such that . We put . Then , so C is a contraction. It follows from (2) that
Therefore
□
In the following theorem, we obtain the Choi-Davis-Jensen type inequality for operator P-class functions.
Theorem 3 Let Φ be a unital positive linear map on , and f be operator P-class on an interval J. Then
Proof Let . We put Ψ the restriction of Φ to the -algebra generated by I and A. Then Ψ is a unital completely positive map on . The celebrated Stinespring dilation theorem [[9], Theorem 1] states that there exist an isometry and a unital ∗-homomorphism such that . Hence
□
We will show that the constant 2 is the best possible such one in the following example.
Example 5 Let for . Then and
where . Hence f is of operator P-class on . Now, consider that the unital positive map is defined by . Then for the Hermitian matrix we have , , , and . Therefore . This shows that the coefficient 2 in (2) and (3) is the best.
Example 6 Consider (the nonnegative increasing function and so) P-class function where . Let the unital positive map be defined by with and let . Then and . Hence . It follows from (3) that f is not of operator P-class.
We present a Hermite-Hadamard inequality for operator P-class functions in the next theorem.
Theorem 4 Let Φ be a unital positive linear map on and f be operator P-class on J. Then
where and .
Proof Let and . Then
Integrating both sides of (4) over we obtain
□
4 Some inequalities for P-class functions involving continuous operator fields
Let be a -algebra of operators acting on a Hilbert space and let T be a locally compact Hausdorff space. A field of operators in is called a continuous field of operators if the mapping is norm continuous on T. If is a Radon measure on T and the function is integrable, one can form the Bochner integral , which is the unique element in such that
for every linear functional φ in the norm dual of .
Let denote the set of bounded continuous functions on T with values in . It is easy to see that the set is a -algebra under the pointwise operations and the norm ; cf. [10].
Assume that there is a field of positive linear mappings from to another -algebra ℬ. We say that such a field is continuous if the mapping is continuous for every . If the -algebras are unital and the field is integrable with integral I, we say that is unital; see [10].
Theorem 5 Let be an operator P-class function defined on an interval J, and let and ℬ be unital -algebras. If is a unital field of positive linear mappings defined on a locally compact Hausdorff space T with a bounded Radon measure μ, then
holds for every bounded continuous field of self-adjoint elements in with spectra contained in J.
Proof We consider the unital positive linear map defined by . Let . It follows from and (3) that
□
In the discrete case, in Theorem 5, we get the following result.
Corollary 4 Let be an operator P-class function defined on an interval J, let () and () be unital positive linear maps on . Then
References
Dragomir SS, Pečarić J, Persson LE: Some inequalities of Hadamard type. Soochow J. Math. 1995,21(3):335–341.
Dragomir SS, Pearce CEM: Quasi-convex functions and Hadamard’s inequality. Bull. Aust. Math. Soc. 1998,57(3):377–385. 10.1017/S0004972700031786
Pearce CEM: P -Functions, quasi-convex functions, and Hadamard-type inequalities. J. Math. Anal. Appl. 1999, 240: 92–104. 10.1006/jmaa.1999.6593
Sarikaya MZ, Set E, Ozdemir ME: On some new inequalities of Hadamard type involving h -convex functions. Acta Math. Univ. Comen. 2010, 2: 265–272.
Pečarić JE, Furuta T, Mićić Hot J, Seo Y: Mond Pečarić Method in Operator Inequalities. ELEMENT, Zagreb; 2005.
Fujii JI, Kian M, Moslehian MS: Operator Q -class functions. Sci. Math. Jpn. 2010,73(1):75–80.
Moslehian MS, Kian M: Jensen type inequalities for Q -class functions. Bull. Aust. Math. Soc. 2012,85(1):128–142. 10.1017/S0004972711002863
Hansen F, Pedersen GK: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 1982,258(3):229–241. 10.1007/BF01450679
Stinespring WF:Positive functions on -algebras. Proc. Am. Math. Soc. 1955, 6: 211–216.
Hansen F, Perić I, Pečarić J: Jensen’s operator inequality and its converses. Math. Scand. 2007,100(1):61–73.
Acknowledgements
The second and the third authors are supported by the Lebanese University grants program for the Discrete Mathematics and Algebra group.
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Bakherad, M., Abbas, H., Mourad, B. et al. Operator P-class functions. J Inequal Appl 2014, 451 (2014). https://doi.org/10.1186/1029-242X-2014-451
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DOI: https://doi.org/10.1186/1029-242X-2014-451