Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 131-144.

Constructing well-bounded operators not of type (B) on a class of inductive limits

Authors: Alan Stoneham
Author institution: School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, Australia

Summary: All well-bounded operators on reflexive Banach spaces are of type (B), and it is open and it is open whether there is a non-reflexive Banach space upon which every well-bounded operator is of type (B). It was suggested in Q. Cheng and I. Doust, \textit{Glasgow Math. J.} \textbf{43}(2001), 467--475, that the spaces constructed by G. Pisier, \textit{Acta Math.} \textbf{151}(1983), 181--208, could provide an example of such a space. In this paper, it will be shown that on a class of Banach spaces containing the spaces from G. Pisier, \textit{Acta Math.} \textbf{151}(1983), 181--208, there is always a well-bounded operator not of type (B).

DOI: http://dx.doi.org/10.7900/jot.2022aug18.2402
Keywords: Well-bounded operators, Banach space geometry, functional calculus

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