Mathematics Faculty Articles

Title

An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities

Document Type

Article

Publication Date

10-1-2008

Publication Title

Linear Algebra and its Applications

Keywords

Bounded linear operator, Characterization of inner product space, Hilbert space, Q-Norm, Norm inequality, Schatten p-norm, Continuous filed of operators, Bouchner integral

ISSN

0024-3795

Volume

429

Issue/No.

8-9

First Page

2159

Last Page

2167

Abstract

Let A be a C∗ -algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)t∈T be a continuous field of operators in A such that the function t↦At is norm continuous on T and the function t↦∥At∥ is integrable. Then the following equality including Bouchner integrals holds equation

T∣At−∫TAsdP∣∣2dP=∫T|At|2dP−∣∫TAtdP2.

This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten p-norm inequalities.

Comments

Under an Elsevier user license

DOI

10.1016/j.laa.2008.06.010