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

Authors

Mohammad Sal Moslehian, Ferdowsi University of Mashhad
Fuzhen Zhang, Nova Southeastern UniversityFollow

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 (A_t)_t∈T be a continuous field of operators in A such that the function t↦A_t is norm continuous on T and the function t↦∥A_t∥ is integrable. Then the following equality including Bouchner integrals holds equation

∫_T∣A^t−∫_TA_sdP∣∣2dP=∫_T|A_t|²dP−∣∫_TA_tdP∣².

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

NSUWorks Citation

Moslehian, Mohammad Sal and Zhang, Fuzhen, "An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities" (2008). Mathematics Faculty Articles. 27.
https://nsuworks.nova.edu/math_facarticles/27

DOI

10.1016/j.laa.2008.06.010