Index Theory for Perturbations of Direct Sums of Normal Operators and Weighted Shifts

Abstract

In this paper we consider the construction of a norm-continuous index theory for unitary equivalence up to commutative normality of operators on a separable Jlilhert space, and establish some essentially best possible results in this direction for operators which can be written as direct sums of weighted shifts and normal operators. Indeed, for such operators we establish both a normcontinuous analogue and a norm-continuous extension of the elegant results of L. Brown, R. Douglas and P. Fillmore [4].