Unitary equivalence of sets of self-adjoint operators

Abstract

In this paper we show that the same problem contains the problem of the classification of triples {Pl, P2, P3} of orthogonal projections, where Pl ± P2- We show, moreover, that classification up to unitary equivalence of pairs of bounded self-adjoint operators would permit giving a classification up to unitary equivalence of countable sets {Ak}k~__l of bounded operators in H whose norms are jointly bounded. This confirms once again the complexity of the classification of pairs of self-adjoint operators ,$ and hence also of the corresponding triples of orthogonal projections. 1. Given any set of self-adjoint operators {Ak}k~=l whose norms are jointly bounded, we construct a pair of bounded self-adjoint operators ~lA~l, ~l.4k} in the space $~ -- ~e// as follows: 1