Paul McGuire

Analytic Tridiagonal Reproducing Kernels

The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {( a n ,0 + a n ,1 z +…+ a n,J z J ) z n , n [ges ] 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, Mz is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so, and an example is given in which the commutant of M z on a tridiagonal space is strikingly different from that on any diagonal space.

On the spectral picture of an irreducible subnormal operator

In this paper we show that the spectral picture of an irreducible subnormal operator may be arbitrarily prescribed subject only to certain natural necessary conditions. This completes work begun by the second author.

A local functional calculus

This paper introduces a local functional calculus for a class of bounded operators on a Hilbert space. Properties of the local functional calculus are investigated and an application is given to subnormal operators. Also, the concept of a locally rationally multicyclic operator is introduced and the Berger-Shaw Theorem is discussed.

C*-algebras generated by subnormal operators

Abstract This paper gives necessary and sufficient conditions on a Hilbert space operator A in order that C ∗ (A) , the C ∗ -algebra generated by A , be generated by an irreducible, essentially normal, subnormal operator. Several specialized applications of this result are given and methods of constructing irreducible subnormal operators are developed.

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.