## 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 (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}−∫_{T}A_{s}dP∣∣2d*P*=∫_{T}|A_{t}|^{2}d*P*−∣∫_{T}A_{t}d*P*∣^{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.

#### 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.

http://nsuworks.nova.edu/math_facarticles/27

#### DOI

10.1016/j.laa.2008.06.010

## Comments

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