The Unitary Equivalence of Binormal Operators

Abstract

the object of investigation has the advantage of immediacy, the more convenient definition for the purposes of this paper, and the one given below, is in terms of the *-ring it generates. The exact relation between these two aspects of a binormal operator is the first problem to which we address our attention, and the desired formulation is stated in Theorems 1 and 2. The central result of the paper may be paraphrased as follows: We attach to a binormal operator four commuting normal operators (fixed polynomial functions of the operator and its adjoint) and show that except for comparatively uninteresting normal direct summands which each posseses, two binormal operators are unitarily equivalent if and only if these four functions of each are simultaneously. unitarily equivalent in pairs (Theorem 5). In the course of proving this it is necessary to obtain considerable information about the structure of a binormal operator, and in particular a sort of standard form is given for any such operator (? 9). Also we show that Dixmier's solution of the two projection problem [2] is subsumed under the theory of binormal operators in a natural and satisfactory manner.