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

Mohammad Sal Moslehian; Fuzhen Zhang

doi:10.1016/j.laa.2008.06.010

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

Mohammad Sal Moslehian
, Fuzhen Zhang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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.

Original language	American English
Pages (from-to)	2159-2167
Number of pages	9
Journal	Linear Algebra and its Applications
Volume	429
Issue number	8-9
DOIs	https://doi.org/10.1016/j.laa.2008.06.010
State	Published - Oct 16 2008

ASJC Scopus Subject Areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Keywords

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

Disciplines

Mathematics

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10.1016/j.laa.2008.06.010

http://www.sciencedirect.com/science/article/pii/S0024379508003157

Cite this

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title = "An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities",

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−∣∫TAtdP∣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.",

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

author = "Moslehian, \{Mohammad Sal\} and Fuzhen Zhang",

year = "2008",

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day = "16",

doi = "10.1016/j.laa.2008.06.010",

language = "American English",

volume = "429",

pages = "2159--2167",

journal = "Linear Algebra and its Applications",

issn = "0024-3795",

publisher = "Elsevier Inc.",

number = "8-9",

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TY - JOUR

T1 - An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities

AU - Moslehian, Mohammad Sal

AU - Zhang, Fuzhen

PY - 2008/10/16

Y1 - 2008/10/16

N2 - 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−∣∫TAtdP∣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.

AB - 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−∣∫TAtdP∣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.

KW - Bouchner integral

KW - Bounded linear operator

KW - Characterization of inner product space

KW - Continuous filed of operators

KW - Hilbert space

KW - Norm inequality

KW - Q-Norm

KW - Schatten p-norm

UR - https://nsuworks.nova.edu/math_facarticles/27

UR - http://www.sciencedirect.com/science/article/pii/S0024379508003157

UR - https://www.scopus.com/pages/publications/49249127747

UR - https://www.scopus.com/pages/publications/49249127747#tab=citedBy

U2 - 10.1016/j.laa.2008.06.010

DO - 10.1016/j.laa.2008.06.010

M3 - Article

SN - 0024-3795

VL - 429

SP - 2159

EP - 2167

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

IS - 8-9

ER -

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

Abstract

ASJC Scopus Subject Areas

Keywords

Disciplines

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