Linear analysis of quadrature domains. III
Let \(T\) denote a hyponormal operator on a Hilbert space \(H\) with finite-rank self-commutator \(D\). Starting point is the observation that the spectrum of \(T\) is the closure of a quadrature domain if the span of all \(T^{*k}\text{Ran}(D)\), \(k= 0,1,\dots\), is finite-dimensional, additionally supposed that \(T\) is rationally cyclic and minimal. \(T\) can be identified with an operator matrix of two-diagonal shape with a sequence of operator pairs submitted to certain recurrence relations. When in this matrix representation \(H\) is decomposed as \(H_0\oplus H_1\oplus\cdots\) then subnormality of \(T\) is expressed by subnormality of any ``tail'' of the matrix defined by the compression of \(T\) to any subspace \(H_N\oplus H_{N+ 1}\oplus\cdots\)\ . If \(T\) is irreducible and has rank-one self-commutator \(\xi\otimes\xi\), if the principal function of \(T\) is the characteristic function of a bounded planar domain \(\Omega\) with piecewise continuous boundary, and if further \(C_0\) denotes the compression of \(T\) to the subspace generated by \(f(T)^*\xi\), where \(f\) ranges \(\text{Rat}(\sigma(T))\), then either \(\sigma(C_0)\cap \Omega\) is discrete in \(\Omega\) or \(\sigma(C_0)= \overline\Omega\). An essential mean for the proofs, a determining function, in its contractivity properties is interpreted in terms of partial differential equations. [For part I see Ark. Mat. 33, No. 2, 357-376 (1995; Zbl 0892.47025)].
scientific article; zbMATH DE number 1389238
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