Alladi Ramakrishnan Hall

Certain affine varieties as a spectral set

Haripada Sau

IISER Pune

An affine variety is considered 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 can be dilated (or lifted) to a commuting isometric pair. Inspired by the intrinsic connection of distinguished varieties with the bidisk operator theory, we proposed a question that can be seen as a Constrained Ando Dilation Problem:

When does a commuting contractive pair have a commuting isometric lift that `lives’ in a distinguished variety?

In this talk, we shall see why one may want to find an answer to this problem. While the problem remains unsolved in its full generality, it is solved affirmatively for operators with finite dimensional defects extending earlier work of Agler-McCarthy (Acta Math., 2005) and of
Das-Sarkar (JFA 2017). We shall also see two representations of these varieties which play a key role in the quest for an answer to the above
problem.