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.

- All seminars.
- Seminars for 2023

Last updated: 11 Nov 2024