Subspace gaps and Weyl's theorem for an elementary operator

Abstract

A range-kernal orthogonality property is established for the elementary operators ℰ(X)=∑i=1nAiXBi and ℰ*(X)=∑i=1nAi*XBi*, where A=(A1,A2,…,An) and B=(B1,B2,…,Bn) are n-tuples of mutually commuting scalar operators (in the sense of Dunford) in the algebra B(H) of operators on a Hilbert space H. It is proved that the operator ℰ satisfies Weyl's theorem in the case in which A and B are n-tuples of mutually commuting generalized scalar operators.