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PhD Thesis colloquium

Title: Interaction of distinguished varieties and the Nevanlinna-Pick interpolation problem in some domains
Speaker: Poornendu Kumar (IISc Mathematics)
Date: 31 March 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department

A distinguished variety in $\mathbb C^2$ has been the focus of much research in recent years because of good reasons. One of the most important results in operator theory is Ando’s inequality which states that for any pair of commuting contractions $(T_1, T_2)$ and two variables polynomial $p$, the operator norm of of the operator $p(T_1, T_2)$ does not exceed the sup norm of $p$ over the bidisc, i.e., \begin{equation} |p(T_1, T_2)|\leq \sup_{(z_1,z_2)\in\mathbb{D}^2}|p(z_1, z_2)|. \end{equation} A quest for an improvement of Ando’s inequality led to the study of distinguished varieties. Since then, distinguished varieties are a fertile field for function theoretic operator theory and connection to algebraic geometry. This talk is divided into two parts.

In the first part of the talk, we shall see a new description of distinguished varieties with respect to the bidisc. It is in terms of the joint eigenvalue of a pair of commuting linear pencils. There is a characterization known of $\mathbb{D}^2$ due to a seminal work of Agler–McCarthy. We shall see how the Agler–McCarthy characterization can be obtained from the new one and vice versa. Using the new characterization of distinguished varieties, we improved the known description by Pal–Shalit of distinguished varieties over the symmetrized bidisc: \begin{equation} \mathbb {G}=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: (z_1,z_2)\in\mathbb{D}^2\}. \end{equation} Moreover, we will see complete algebraic and geometric characterizations of distinguished varieties with respect to $\mathbb G$. In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc.

In the second part of the talk, we shall discuss the uniqueness of the solutions of a solvable Nevanlinna–Pick interpolation problem in $\mathbb G$. The uniqueness set is the largest set in $\mathbb G$ where all the solutions to a solvable Nevanlinna–Pick problem coincide. For a solvable Nevanlinna–Pick problem in $\mathbb G$, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in $\mathbb G$. The algebraic variety is called the uniqueness variety. We shall see if an $N$-point solvable Nevanlinna–Pick problem is such that it has no solutions of supremum norm less than one and that each of the $(N-1)$-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the $N$-point problem contains a distinguished variety containing all the initial nodes, this is called the Sandwich Theorem. Finally, we shall see the converse of the Sandwich Theorem.


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Last updated: 19 Jul 2024