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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hypoconvexity and essentially $ n$-normal operators


Author: Norberto Salinas
Journal: Trans. Amer. Math. Soc. 256 (1979), 325-351
MSC: Primary 47B15; Secondary 46M20
DOI: https://doi.org/10.1090/S0002-9947-1979-0546921-4
MathSciNet review: 546921
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Abstract: In this paper a classifying structure for the class of essentially n-normal operators on a separable Hilbert space is introduced, and various invariance properties of this classifying structure are studied. The notion of a hypoconvex subset of the algebra $ {\mathcal{M}_n}$ of all complex $ n\, \times \,n$ matrices is defined, and it is shown that the set of all equivalence classes of essentially n-normal operators (under a natural equivalence relation), whose reducing essential $ n\, \times \,n$ matricial spectrum is a given hypoconvex set, forms an abelian group. It is also shown that this correspondence between hypoconvex subsets of $ {\mathcal{M}_n}$ and abelian groups is a homotopy invariant, covariant functor. This result is then used to prove that Toeplitz operators (on strongly pseudoconvex domains) which have homotopic continuous matricial symbols, are unitarily equivalent up to compact perturbation.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0546921-4
Keywords: Unitary equivalence up to compact perturbation, the reducing essential $ n\, \times \,n$ matricial spectrum, extensions of $ {C^\ast}$-algebras, the classification problem for essentially n-normal operators, Toeplitz operators on strongly pseudoconvex domains
Article copyright: © Copyright 1979 American Mathematical Society