A classification theorem for essentially binormal operators

Abstract

An essentially binormal operator on Hilbert space is an operator which is unitarily equivalent to a 2 × 2 matrix of essentially commuting, essentially normal operators. A natural invariant of essentially binormal operators up to unitary equivalence in the Calkin Algebra is the reducing essential 2 × 2 matricial spectrum. A nonempty compact subset X of the set of 2 × 2 matrices is called hypoconvex, if it is the reducing essential 2 × 2 matricial spectrum of an operator on Hilbert space. The set EN 2( X) is then defined to be the set of all equivalence classes (up to unitary equivalence in the Calkin algebra) of essentially binormal operators whose reducing essential 2 × 2 matricial spectrum coincides with X. The aim of this paper is to prove a result that enables one to compute EN 2( X) in terms of the topological structure of the space X ̃ of unitary orbits of X. Indeed, it is shown that for every hypoconvex subset X of the set of 2 × 2 matrices, there exists a natural homomorphism from Ext( X ̃ ) onto EN 2( X). Also, a six term cyclic exact sequence is obtained, which produces a characterization of the kernel of the above-mentioned homomorphism.