# Subnormal Operators, Self-Commutators and Pseudocontinuations

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We study pure subnormal operators S such that the self-commutator [S*,S] = S*S-SS* has zero as an eigenvalue; or equivalently, subnormal operators whose self-commutators do not have dense range. Examples of such operators include the unilateral shift and more generally any extension of the unilateral shift.

Other examples are operators with finite rank self-commutator. Subnormal (and hyponormal) operators with finite rank self-commutator have received a lot of attention, recently. Furthermore it is known that there are connections between these operators and quadrature domains. An important part of this paper is to find the appropriate generalization of quadrature domains for this larger class of operators.

There are several characterizations of quadrature domains, for example if the domain is simply connected, then it is a quadrature domain if and only if the Riemann map of the unit disk onto the domain is a rational function. In general, quadrature domains are characterized by the existence of a Schwarz function; that is, a meromorphic function that is continuous up to the boundary and agrees with z-bar on the boundary.

We shall say that a region G is a generalized quadrature domain if it has a generalized Schwarz function; that is, if there exists a Nevanlinna function R on G that has boundary values almost everywhere with respect to harmonic measure on the boundary of G, and these boundary values agree a.e. with z-bar. It is shown that a region G is a generalized quadrature domain precisely when the universal covering map from the unit disk onto G has a pseudocontinuation. Recall, that a Nevanlinna function f on the unit disk has a pseudocontinuation if there exists a Nevanlinna function F on the exterior of the unit disk such that the boundary values of f and F agree a.e. on the circle. It is known that the only functions analytic across the circle that have pseudocontinuations are the rational functions. However, the author shows how to construct several bounded univalent functions that are smooth on the closed unit disk, have a pseudocontinuation, and yet are not rational functions.

In this paper it is shown that there are relations between subnormal operators whose self-commutators do not have dense range and generalized quadrature domains. For example, if S is multiplication by z on the Hardy space H^2(G), then zero is an eigenvalue for the self commutator of S if and only if G is a generalized quadrature domain. On the other hand, multiplication by z on the Bergman space of a region G does not have zero as an eigenvalue for its self commutator for any region G.

We also consider cyclic subnormal operators and analytic Toepliz operators on the circle; as well as some general properties shared by all subnormal operators having zero as an eigenvalue for its self commutator.

## Sample Results

If T_{f} is an analytic Toeplitz operator with symbol f, then it is known that T_{f} has finite rank self commutator if and only if f is a rational function.

We show that the self commutator of T_{f} has zero as an eigenvalue precisely when f has a pseudocontinuation.

If S is an irreducible cyclic subnormal operator, then some partial results are given to support the (authors) conjecture that zero is an eigenvalue for the self commutator of S exactly when the set G of analytic bounded point evaluations form a generalized quadrature domain and the spectral measure is of the form Harmonic measure plus a sum of point masses in G where the points form an H^{infty}(G) zero set.

Several natural questions are raised and some examples are given using sampling sequences for Bergman spaces on the unit disk.