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.
