Abstract
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.
Highlights
Introduction and preliminaries LetB(H) denote the C∗-algebra of all bounded linear operators on a complex Hilbert space H with its identity denoted by I
We recall that a real-valued continuous function f defined on an interval J is operator convex if f (λA+( –λ)B) ≤ λf (A)+( –λ)f (B) for all A, B ∈ σ (J) and all λ ∈ [, ]
3 Jensen operator inequality for operator P-class functions we present a Jensen operator inequality for operator P-class functions
Summary
Example Let α > and f be a continuous function on the interval [α, α] into itself. It follows from f λA + ( – λ)B ≤ α ≤ f (A) + f (B) A, B ∈ σ [α, α] , λ ∈ [ , ]. Example Let g 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)|. Theorem If f is an operator P-class function on the interval ( , ∞) such that limt→∞ f (t) = , f is operator decreasing. We may assume that the spectrum of the strictly positive operator C is contained in [α, β] for some < α < β
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