Hypoconvexity and essentially 𝑛-normal operators

Salinas, Norberto

doi:10.1090/S0002-9947-1979-0546921-4

Hypoconvexity and essentially $n$-normal operators
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Trans. Amer. Math. Soc. 256 (1979), 325-351 Request permission

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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Über beschrankte quadratischen Formen deren Differenz vollstetig ist

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