Donald Sarason

Functions of vanishing mean oscillation

A function of bounded mean oscillation is said to have vanish- ing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of functions of vanishing mean oscil- lation is characterized in several ways. This class is then applied to answer two questions in analysis, one involving stationary stochastic processes satisfying the strong mixing condition, the other involving the algebra H + C.

Products of Toeplitz operators

A sufficient condition is found for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator. The condition, which comprehends all previously known sufficient conditions, is shown to be necessary under additional hypotheses. The question whether the condition is necessary in general is left open.

Nevanlinna-Pick interpolation with boundary data

Versions of the Nevanlinna-Pick interpolation problem with boundary interpolation nodes and boundary interpolated values are investigated.

Nearly Invariant Subspaces of the Backward Shift

A theorem of D. Hitt describing certain subspaces of H2 that miss by one dimension being invariant under the backward shift operator is given a new approach and extended.

Local Dirichlet spaces as de Branges-Rovnyak spaces

The harmonically weighted Dirichlet spaces associated with unit point masses are shown to coincide with de Branges-Rovnyak spaces, with equality of norms. As a consequence it is shown that radial expansion operators act contractively in harmonically weighted Dirichlet spaces.

Harmonically weighted dirichlet spaces associated with finitely atomic measures

The space D(μ) associated with a positive measure μ on the unit circle is a Hilbert space made from the holomorphic functions in the unit disk whose derivatives are square integrable when weighted against the Poisson integral of μ. In this paper the structure of D(μ) is investigated for the case where μ is a finite sum of atoms. The wandering vectors of the shift operator on D(μ) are described.

Selecta : expository writing

A selection of the mathematical writings of Paul R. Halmos (1916 - 2006) is presented in two Volumes. Volume I consists of research publications plus two papers of a more expository nature on Hilbert Space. The remaining expository articles and all the popular writings appear in this second volume. It comprises 27 articles, written between 1949 and 1981, and also a transcript of an interview.

Representing measures for R(X) and their Green's functions

Abstract Let X be a compact subset of the complex plane with a nonempty interior, R ( X ) the uniform closure in C ( X ) of the rational functions with poles off X , and m a representing measure on ∂X for the functional on R ( X ) of evaluation at a point a in int X . Let N 2 be the space of functions f in L 2 ( m ) satisfying ∝ f dm = ∝ f K dm = 0 for all h in R ( X ), and let T be the operator on N 2 of multiplication by z followed by projection onto N 2 . The spectral properties of T are investigated and shown to depend in part on the behavior of the so-called Greens function of m . In case m is the harmonic measure on ∂X for a the latter function is the classical Greens function for int X with singularity at a . Special attention is paid to the case where X is the closure of a finitely connected Jordan domain whose boundary curves are analytic. In that context, new proofs are given of Beurlings invariant subspace theorem and of Forellis theorem on extreme points in the unit ball of the Hardy space H 1 ( m ).

Inverse problem for zeros of certain koebe-related functions

AbstractIf w1,…,wN is a finite sequence of nonzero points in the unit disk, then there are distinct points λ1,…, λN on the unit circle and positive numbers Μ1,…,ΜN such that