Parareflexive operators on Banach spaces (Q1812917)

The objective of this paper is to extend to a class of Banach spaces the results of \textit{C. Apostol}, \textit{R. G. Douglas} and \textit{C. Foiaş} on nilpotent operators and parareflexive operators on Hilbert spaces [Trans. Am. Math. Soc. 224 (1976), No. 2, 407-415 (1977; Zbl 0342.47008)]. The quasi-similarity of nilpotent operators to Jordan models and the consequent relations for adjoints are obtained. A paraclosed subspace is, essentially, the range of a bounded operator. For parareflexivity, the results include: (a) If \(S\) and \(T\) are two operators on a Banach space \(X\), then \(T= u(S)\) for an analytic entire function \(u\) (subject to another condition), if and only if \(T\) leaves invariant every \(S\)-invariant paraclosed subspace of \(X\). (b) If \(S\) is a quasi-affine transform of \(T\) or if \(T\) is a quasi-affine transform of \(S\), then if \(S\) is parareflexive then \(T\) is parareflexive. (c) A necessary and sufficient condition for a nilpotent operator to be parareflexive is obtained. It is admitted that, in some instances, the proofs are ``sketchy''.