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12 - An introduction to Read-type operators

Published online by Cambridge University Press:  10 December 2009

Frédéric Bayart
Affiliation:
Université de Clermont-Ferrand II (Université Blaise Pascal), France
Étienne Matheron
Affiliation:
Université d'Artois, France
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Summary

Introduction

In this final chapter, our aim is to give a short and gentle introduction to the kind of operator constructed by C. J. Read in the 1980s.

In his 1987 paper [202], Read solved in the negative the invariant subset problem for the space l1(N), and in fact for any separable Banach space containing a complemented copy of l1. In other words, he was able to produce on such a space an operator for which every non-zero vector is hypercyclic.

The construction carried out in [202] is something of a tour de force. Moreover, its understanding requires some familiarity with earlier constructions by the same author relating to the invariant subspace problem, which are already quite involved (see e.g. [201]). This convinced us that we should not be overly ambitious regarding the material presented in this chapter. Thus, we have chosen to concentrate on the simplest example of a “Read-type” operator, i.e. an operator T acting on l1(N) for which every non-zero vector x is cyclic. This is a counter-example to the invariant subspace problem for the space l1.

As already said, we have tried to give a helpful presentation, meaning that we have made some effort to explain the underlying ideas as we understand them, inserting heuristic comments whenever this seemed necessary. Some parts of the discussion are deliberately informal, but the construction is nevertheless complete and selfcontained. We hope that this chapter will be useful for people interested in that kind of question.

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Publisher: Cambridge University Press
Print publication year: 2009

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