Sheldon Axler

Commuting Toeplitz operators with harmonic symbols

This paper shows that on the Bergman space, two Toeplitz operators with harmonic symbols commute only in the obvious cases. The main tool is a characterization of harmonic functions by a conformally invariant mean value property.

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.

Finite-codimensional invariant subspaces of Bergman spaces

ABSTRACT. For a large class of bounded domains in C, we describe thosefinite codimensional subspaces of the Bergman space that are invariant undermultiplication by z. Using different techniques for certain domains in CN, wedescribe those finite codimensional subspaces of the Bergman space that areinvariant under multiplication by all the coordinate functions. Fix a positive integer N, and let V denote Lebesgue volume measure on CN (sothat if N = 1, then V is just area measure). Let Q c C^ be a domain, which, asusual, means that 0 is a nonempty open connected subset of C^. For / an analyticfunction from 0 to C and 1 < p < oo, the norm ||/||n,P is defined by Wfh,r=(jn\f\pdV The Bergman space L^(Q) is defined to be the set of analytic functions from fi toC such that ||/||n,P < oo. Our goal in this paper is to describe the closed finite codimensional subspaces ofLpa (0) that are invariant under multiplication by the coordinate functions zi,...,

Harmonic Functions from a Complex Analysis Viewpoint

Etude des fonctions harmoniques en liaison avec les fonctions analytiques. Theoreme de conjugaison logarithmique

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in R n can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

Harmonic Polynomials and Dirichlet-Type Problems

We take a new approach to harmonic polynomials via differ- entiation. Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x| 2−n and observing the patterns that emerge. This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollary 1.8) and a new proof of the identity in Corollary 1.10. We then discuss a fast algorithm for computing the Poisson integral of any polynomial. (Note: The algorithm involves differentiation, but no integration.) We show how this algorithm can be used for many other Dirichlet-type problems with polynomial data. Finally, we show how Lemma 1.4 leads to the identity in (3.2), yielding a new and simple proof that the Kelvin transform preserves harmonic functions. 1. Derivatives of |x| 2−n Unless otherwise stated, we work in R n ,n >2; the function |x| 2−n is then har- monic and nonconstant on R n \{ 0}. (When n = 2 we need to replace |x| 2−n with log |x|; the minor modifications needed in this case are discussed in Section 4.) Letting Dj denote the partial derivative with respect to the j th coordinate vari- able, we list here some standard differentiation formulas that will be useful later: Dj|x| t = txj|x| t−2 ∆|x| t = t(t + n − 2)|x| t−2

Sequences in the maximal ideal space of

This paper studies the behavior of sequences in the maximal ideal space of the algebra of bounded analytic functions on an arbitrary domain. The main result states that for any such sequence, either the sequence has an interpolating subsequence or infinitely many elements of the sequence lie in the same Gleason part.

Bôcher's Theorem

The usual proofs of Bochers Theorem rely either on the theory of superharmonic functions ([4], Theorem 5.4) or series expansions using spherical harmonics ([5], Chapter X, Theorem XII). (The referee has called our attention to the proof given by G. E. Raynor [7]. Raynor points out that the original proof of Maxime Bocher [2] implicitly uses some non-obvious properties of the level surfaces of a harmonic function.) In this, paper we offer a different and simpler approach to this theorem. The only results about harmonic functions needed are the minimum principle, Harnacks Inequality, and the solvability of the Dirichlet problem in Bn. We will investigate a harmonic function by studying its dilates. For u a function defined on Bn \ {0} and r E (0, 1), the dilate ur is the function defined on (l/r)Bn \ {0} by

The Neumann problem on ellipsoids

The Neumann problem on an ellipsoid in