Operator P-class functions
© Bakherad et al.; licensee Springer. 2014
Received: 27 May 2014
Accepted: 23 October 2014
Published: 6 November 2014
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 .
where . Thus for all .
which is known as the Hermite-Hadamard inequality for the P-class continuous functions; see .
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.
for all and all .
Clearly every nonnegative operator convex function is of operator P-class.
where and . Thus f is an operator P-class function on .
for all and ; see . 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 [, Example 2.1]. We put , and . Then . Hence f is not of operator P-class.
that f is of operator P-class on .
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.
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 .
Applying Theorem 2 we have some inequalities including the subadditivity.
In the following theorem, we obtain the Choi-Davis-Jensen type inequality for operator P-class functions.
We will show that the constant 2 is the best possible such one in the following example.
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.
where and .
4 Some inequalities for P-class functions involving continuous operator fields
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. .
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 .
holds for every bounded continuous field of self-adjoint elements in with spectra contained in J.
In the discrete case, in Theorem 5, we get the following result.
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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