Finite Central Truncations of Linear Operators

Gustafsson, B; Putinar, M

doi:10.1007/978-3-319-65810-0_5

Finite Central Truncations of Linear Operators

Lookup NU author(s): Professor Mihai Putinar

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Abstract

© Springer International Publishing AG 2017. By interpreting the exponential orthogonal polynomials as characteristic polynomials of finite central truncations of the underlying hyponormal operator one opens a vast toolbox of Hilbert space geometry methods. In particular we prove in this chapter that trace class modifications of the hyponormal operator attached to a domain will not alter the convex hull of the support of any cluster point of the count in measures of the roots of the orthogonal polynomials. As a sharp departure from the case of complex orthogonal polynomials associated to a Lebesgue space we prove that the convex hull of these supports is not affected by taking the union of an open set with a disjoint quadrature domain. However, similar to the case of Bergman orthogonal polynomials, we prove that the exponential orthogonal polynomials satisfy a three term relation only in the case of an ellipse. Some general perturbation theory arguments are collected in the last section.