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Hyponormal Quantization of Planar Domains pp 57–75Cite as

Finite Central Truncations of Linear Operators

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2199))

Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

    Björn Gustafsson

  2. Mathematics Department, University of California, Santa Barbara, CA, USA

    Mihai Putinar

  3. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle Upon Tyne, UK

    Mihai Putinar

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  1. Björn Gustafsson
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Gustafsson, B., Putinar, M. (2017). Finite Central Truncations of Linear Operators. In: Hyponormal Quantization of Planar Domains. Lecture Notes in Mathematics, vol 2199. Springer, Cham. https://doi.org/10.1007/978-3-319-65810-0_5

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