Bounded point evaluation for a finitely multicyclic commuting tuple of operators

Abstract

We generalize the notion of bounded point evaluation introduced by Williams for a cyclic operator to a finitely multicyclic commuting d-tuple T=(T1,…,Td) of bounded linear operators on a complex separable Hilbert space. We show that the set bpe(T) of all bounded point evaluations for T is a unitary invariant and we characterize it in terms of the dimension of the joint cokernel of T. Using this, we show that if bpe(T) has non-empty interior, then T can be realized as the d-tuple Mz=(Mz1,…,Mzd) of multiplication operators on a reproducing kernel Hilbert space H of functions on bpe(T). We further characterize the largest open subset of bpe(T) on which all the elements of H are analytic, which we refer to as the set of all analytic bounded point evaluations. As an application, we describe the set of all analytic bounded point evaluations for toral and spherical isometries, and also, derive an analytic model of a commuting d-tuple of composition operators.