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Published online by Cambridge University Press: 25 June 2025
Subnormal operators arise naturally in complex function theory, differential geometry, potential theory, and approximation theory, and their study has rich applications in many areas of applied sciences as well as in pure mathematics. We discuss here some research problems concerning the structure of such operators: subnormal operators with finite-rank self-commutator, connections with quadrature domains, invariant subspace structure, and some approximation problems related to the theory. We also present some possible approaches for the solution of these problems.
The operator S is pure if S has no normal summand and is irreducible if S is not unitarily equivalent to a direct sum of two nonzero operators. The theory of subnormal operators provides rich applications in many areas, since many natural operators that arise in complex function theory, differential geometry, potential theory, and approximation theory are subnormal operators. Many deep results have been obtained since Halmos introduced the concept of a subnormal operator. In particular, Thomson's solution of the long-standing problem on the existence of bounded point evaluations reveals a structure theory of cyclic subnormal operators. Thomson's work answers many questions that had been open for a long time and promises to enable researchers to answer many more; see [Thomson 1991] or [Conway 1991]. The latter is a general reference for the theory of subnormal operators.
Here we will present some research problems on subnormal operators and discuss some possibilities for their solution.
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