Nathan S. Feldman

Hypercyclicity and supercyclicity for invertible bilateral weighted shifts

We give a characterization of the invertible bilateral weighted shifts that are hypercyclic or supercyclic. Although there is a general characterization due to H. Salas, in the invertible case the conditions simplify greatly.

Perturbations of hypercyclic vectors

We show that a linear operator can have an orbit that comes within a bounded distance of every point, yet is not dense. We also prove that such an operator must be hypercyclic. This gives a more general form of the hypercyclicity criterion. We also show that a sufficiently small perturbation of a hypercyclic vector is still hypercyclic.

The Hardy space of a slit domain

Preface Notation List of Symbols Preamble 1 Introduction 2 Preliminaries 3 Nearly invariant subspaces 4 Nearly invariant and the backward shift 5 Nearly invariant and de Branges spaces 6 Invariant subspaces of the slit disk 7 Cyclic invariant subspaces 8 The essential spectrum 9 Other applications 10 Domains with several slits 11 Final thoughts 12 Appendix

ESSENTIALLY SUBNORMAL OPERATORS

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form “Normal plus Compact”. The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam’s Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

Computing the Fredholm index of Toeplitz operators with continuous symbols

We show how to compute the Fredholm index of a Toeplitz operator with a continuous symbol constructed from any subnormal operator with compact self-commutator. We also show that the essential spectral pictures of such Toeplitz operators can be prescribed arbitrarily.

The State of Subnormal Operators

In this paper we present the highlights of the theory of subnormal operators that was initiated by Paul Halmos in 1950. This culminates in Thomson’s Theorem on bounded point evaluations where several applications are presented. Throughout the paper are some open problems.

SUBNORMAL OPERATORS, SELF-COMMUTATORS, AND PSEUDOCONTINUATIONS

AbstractWe study pure subnormal operators whose self-commutators have zero as an eigenvalue. We show that various questions in this are closely related to questions involving approximation by functions satisfying

Nearly invariant and de Branges spaces

\bar \partial ^2 f = 0