Abstract: An affine variety in the two dimensional complex space is a distinguished variety if it intersects the open bidisk and exits the domain through the torus. The distinguished varieties are important in the operator theory and the Pick interpolation problem associated with the bidisk. Ando's Dilation Theorem states that every commuting contractive pair has the bidisk as a complete spectral set. Inspired by the intrinsic connection of distinguished varieties with the bivariate matrix theory, we proposed in [Das-Sau, Proc. AMS, 2024] the following question: "When does a commuting contractive pair have a distinguished variety as a complete spectral set?" We shall discuss what we have been able to say about this problem. Our results significantly improves earlier work of Agler-McCarthy (Acta Math., 2005) and Das-Sarkar (JFA 2017). Several interesting results about the classical dilation theory of Ando emerge. We shall also see two representations of distinguished varieties which play a key role in the quest for an answer to the above problem.

Meeting ID: 948 0044 7244
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