An approximate, multivariable version of Specht's theorem

In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 55; no. 2; pp. 159 - 173
Main Authors Marcoux, L. W., Mastnak, M., Radjavi, H.
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.03.2007
Subjects
2000 Mathematics Subject Classifications: Primary 15A21
Approximation
Secondary15A04
Semigroups
Specht's theorem
Trace
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ISSN0308-1087
1563-5139
DOI10.1080/03081080600693285

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Summary:In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A*) and (B, B*) to coincide, but only to be close.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081080600693285