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In the present paper we investigate generalized Fredholm operators (see [15], [16] and [17]) on a complex Hilbert space H. The main results of this paper read as follows: ## T € $ g (H) if and only if T = T\ © T2 , where T\ is a Fredholm operator such that at{T\ -A) and /?(Ti -A) are constant for |A| small and T2 is a finite-dimensional nilpotent operator. ## If T € $>g(H) is not finite-dimensional, then dist(0, 00 where te(T) denotes the essential minimum modulus of T. ## Preliminaries and notations The present paper is a continuation of our previous papers [15], [16] and [17]. Notations and definitions not explicitly given are taken from [15], [16] and [17]. In this section, X always denotes a complex, infinite-dimensional Banach space. C{X) denotes the Banach algebra of all bounded linear operators on X. We will use the following notations: It is well-known (see [5]) that there are 6 > 0 and integers ci,c 2 > 0 such that T-Xle \*(X) for |A| < 6, MSC classification: 47A11.

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Publisher

Walter de Gruyter GmbH

Published

1999

Language

EN

Field

Mathematics (Physical Sciences)