While real orthogonal polynomials and orthogonal polynomials on the unit circle are very well understood, relatively little is known about systems of polynomials (in one complex variable) orthogonal with respect to a measure supported on an arbitrary closed, but not necessarily compact, subset of the complex plane. Every such orthonormalizing measure
can be obtained from the spectral measure of a normal extension of a Hessenberg operator, namely multiplication by the independent variable in C
[Z], in some abstract Hilbert space. Therefore, the theory of orthogonal polynomials is closely related to spectral theory of subnormal Hessenberg operators.
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