Some Vector Inequalities for Continuous Functions of Self-Adjoint Operators in Hilbert Spaces

Ss Dragomir

doi:10.1155/2011/564836

Abstract

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator , where and for various classes of continuous functions are given. Applications for the power function and the logarithmic function are also provided.

Highlights

Let U be a self-adjoint operator on the complex Hilbert space H, ·, · with the spectrum Sp U included in the interval m, M for some real numbers m < M and let {Eλ}λ be its spectral family
The function gx,y λ : Eλx, y is of bounded variation on the interval m, M and gx,y m − 0 0, gx,y M x, y, 1.3 for any x, y ∈ H
If f : m, M → C is a continuous function of bounded variation on m, M, one has the inequality f s x, y − f A x, y s

Summary

Introduction

Let U be a self-adjoint operator on the complex Hilbert space H, ·, · with the spectrum Sp U included in the interval m, M for some real numbers m < M and let {Eλ}λ be its spectral family. ≤ x y f, m for any x, y ∈ H and for any s ∈ m, M Another result that compares the function of a self-adjoint operator with the integral mean is embodied in the following theorem 2. We investigate the quantity f M 1H − f A f A − f m 1H x, y , 1.7 where x, y are vectors in the Hilbert space H and A is a self-adjoint operator with Sp A ⊆ m, M , and provide different bounds for some classes of continuous functions f : m, M → C. Applications for some particular cases including the power and logarithmic functions are provided as well

Some Vector Inequalities

Applications