Vol. 18, No. 3, 1966

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Operators with finite ascent and descent

Selwyn Ross Caradus

Vol. 18 (1966), No. 3, 437–449

Let X be a Banach space and T a closed linear operator with range and domain in X. Let α(T) and δ(T) denote, respectively, the lengths of the chains of null spaces N(TK) and ranges R(TK) of the iterates of T. The Riesz region RT of an operator T is defined as the set of λ such that α(T λ) and δ(T λ) are finite. The Fredholm region FT is defined as the set of λ such that n(T λ) and d(T λ) are finite, n(T) denoting the dimension of N(T) and d(T) the codimension of R(T). It is shown that FT JT is an open set on the components of which α(T λ) and δ(T λ) are equal, when T is densely defined, with common value constant except at isolated points. Moreover, under certain other conditions, RT is shown to be open. Finally, some information about the nature of these conditions is obtained.

Mathematical Subject Classification
Primary: 47.10
Received: 5 August 1965
Published: 1 September 1966
Selwyn Ross Caradus