Interaction of distinguished varieties and the nevanlinna-pick interpolation problem in some domains

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PhD; 2023; Mathematics

This thesis explores the interplay between complex geometry and operator theory, focusing on characterizing certain objects from algebraic geometry. Two concepts that have been of prime importance in recent times in the analysis of Hilbert space operators are distinguished varieties, which are a priori geometric in nature, and joint spectra, which are a priori algebraic in nature. This thesis brings them together to characterize all distinguished varieties with respect to the bidisc, more generally the polydisc and the symmetrized bidisc in terms of the joint spectrum of certain linear pencils. Some of the results are shown to refine earlier work in these directions. The binding force is provided by an operator-theoretic result, the Berger-Coburn-Lebow characterization of a tuple of commuting isometries. The thesis then turns to studying the uniqueness of solutions of the solvable NevanlinnaPick interpolation problems on the symmetrized bidisc and its connection with distinguished varieties. Several sucient conditions have been identified for a given data to have a unique solution. Moreover, for a class of solvable data on the symmetrized bidisc, there exists a distinguished variety where all solutions agree. Additionally, the thesis explores the more general concept of the determining sets.