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The unitary equivalence of compact operators

Published online by Cambridge University Press:  18 May 2009

Wai-Fong Chuan
Wai-Fong Chuan
Affiliation:
Department of Mathematics, Chung-Yuan Christian University, Chung Li, Taiwan 320, Republic ofChina
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The problem of determining whether or not two operators are unitarily equivalent has been around for many years and considerable work has been done in attempting to solve this problem (see for example [1], [3], [5], [6], [7], [10], [11], [12], [15], [16], [17], [18], [19], [20], [21] and [22]). In many cases, a complete set of unitary invariants is furnished for a certain class of operators. Here we just mention two of such results which are related to what we are going to discuss. The first one was due to Arveson, who showed that two irreducible compact operators are unitarily equivalent if and only if they have the same nth algebraic matricial ranges, for each n≧1 ([1] and Theorem 2.4.3 of [3]). The second one was due to Parrott, who showed that two compact operators with zero reducing null spaces are unitarily equivalent if and only if they have the same nth spatial matricial ranges, for each n≧1 ([5, p. 146]). In this paper, we investigate the closures of the spatial matricial ranges of compact operators and obtain a complete set of unitary invariants for compact operators, from which Parrott's result follows easily.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 2 , July 1985 , pp. 145 - 149
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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