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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

REFERENCES

1.Arveson, W., Unitary invariants for compact operators, Bull. Amer. Math. Soc. 76 (1970), 8–91.CrossRefGoogle Scholar
2.Arveson, W., Subalgebras of C*-algebras, Acta Math. 123 (1969), 141–224.CrossRefGoogle Scholar
3.Arveson, W., Subalgebras of C*-algebras II, Acta Math. 128 (1972), 271–308.CrossRefGoogle Scholar
4.Arveson, W., Notes on extensions of C*-algebras, Duke Math. J. 44 (1977), 329–355.CrossRefGoogle Scholar
5.Bonsall, F. F. and Duncan, J., Numerical Ranges II, London Math. Soc. Lecture Notes Ser. Vol. 10, (Cambridge Univ. Press, 1973).CrossRefGoogle Scholar
6.Brenner, J., The problem of unitary equivalence, Acta Math. 86 (1951), 297–308.CrossRefGoogle Scholar
7.Brown, A., The unitary equivalence of binormal operators, Amer. J. Math. 76 (1954), 414–434.CrossRefGoogle Scholar
8.Bunce, J. and Salinas, N., Completely positive maps on C*-algebras and the left matricial spectra of an operator, Duke Math. J. 43 (1976), 747–774.CrossRefGoogle Scholar
9.Chuan, W. F., Containment relations of operators and representations of C*-algebras, Dissertation, Univ. of California at Santa Barbara, 1979.Google Scholar
10.Deckard, D., Complete sets of unitary invariants for compact and trace-class operators, Acta Sci. Math. (Szeged) 28 (1967), 9–20.Google Scholar
11.Deckard, D. and Pearcy, C., On unitary equivalence of Hilbert-Schmidt operators, Proc. Amer. Math. Soc. 16 (1965), 671–675.CrossRefGoogle Scholar
12.Ernest, J., Charting the operator terrain, Mem. Amer. Math. Soc. 171 (1976).Google Scholar
13.Hadwin, D. W., An operator-valued spectrum, Indiana Univ. Math. J. 26 (1977), 329–340.CrossRefGoogle Scholar
14.Hadwin, D. W., Closures of unitary equivalence classes and completely positive maps, Notices Amer. Math. Soc. 23 (1976), A-587.Google Scholar
15.Halmos, P. R., Introduction to Hilbert space and the theory of spectral multiplicity, 2nd Ed., (Chelsea, New York, 1951).Google Scholar
16.Murnagham, F., On the unitary invariants of a square matrix, An. Acad. Brasil. Ci. 26 (1954), 1–7.Google Scholar
17.Pearcy, C., A complete set of unitary invariants for 3x3 complex matrices, Trans. Amer. Math. Soc. 104 (1962), 425–429.Google Scholar
18.Pearcy, C., A complete set of unitary invariants for operators generating finite W*-algebras of type I, Pacific. J. Math. 12 (1962), 1405–1416.CrossRefGoogle Scholar
19.Radjavi, H., On unitary equivalence of arbitrary matrices, Trans. Amer. Math. Soc. 104 (1962), 363–373.CrossRefGoogle Scholar
20.Segal, I. E., Decompositions of operator algebras II, Mem. Amer. Math. Soc. 9 (1951).Google Scholar
21.Specht, W., Zur Theorie der Matrizen, II, Jber. Dentsch. Math. Verein. 50 (1940), 19–23.Google Scholar
22.Tarn, P. K., On the unitary equivalence of certain classes of non-normal operators, I, Canad. J. Math. 23 (1971), 849–856.Google Scholar
23.Voiculescu, D., A non-commutative Weyl–von Neumann theorem, Rev. Roum. Math. Pures Appl. 21 (1976), 97–113.Google Scholar