The oscillation of an operator

Abstract

Foia~ and Singer introduced the oscillation of a bounded linear operator mapping C(S) into a Banach space. Using this concept we define a generalization of the Fredholm operators T with YiK(T) Xbe a bounded linear operator, the oscillation of Tat a point s in S, coT(s) = wo(T, s), is the supremum over all positive a satisfying the following: for every neighborhood U of s there is a function f of norm one in C(S), which vanishes outside of U, with 11 Tf 11 ? a. The operator T is almost diffuse if the set of diffusion points of T, D(T) =co 1(), is dense and T is countably almost diffuse if the set of concentration points, y(T) og '(0, so), is countable. It is easy to show that the set wUT '([a, oo)) is a closed set, so w,T(s) is a nonnegative upper semicontinuous function of s which is bounded by 11T 1. Consequently D(T) =co l(O) is a G,6 and thus, if D(T) is dense, i.e. if T is almost diffuse, its complement y(T) is of first category [6, Theorem 1, p. 437]. In general any nonnegative upper semicontinuous function may occur as the oscillation of a linear operator. For example let h be a nonnegative upper semicontinuous function on [0, 1]. Define A mapping Cr0, 1] to B[O, 1], the bounded functions on [0, 1] with Received by the editors March 5, 1971. AMS 1970 subject classifications. Primary 47A55, 47B30, 47B99, 40H05.