Research Projects

An algebraic variety $V$, defined by a two variable complex polynomial, is said to be a distinguished variety in the bidisk if it intersects the open bidisk non-trivially and its intersection with the topological boundary of the bidisc is the same as that of with the distinguished boundary (the 2-torus). A polynomial in two variables is said to be toral if it defines a distinnguished variety in the bidisk. The famous von Neumann inequality states that, the norm of $p(T)$ for a polynomial $p$ and a contractive operator $T$ is less than or equal to the sup-norm of $p$ on the unit disk. This was later generalized for a pair commuting contractions $(T_1, T_2)$ by Ando. In a seminal work, Agler and McCarthy obtained a sharper version of Ando inequality: the norm of $p(T_1, T_2)$ is less than equal to the sup-norm of $p$ on a distinguished variety in the bidisk provided $T_1$ and $T_2$ are strictly contractive matrices. Thus distinguished varieties in the bidisk (also in the polydisk) became an interesting object to study in function theoretic operator theory. A compact subset $K$ of $\mathbb{C}^2$ is said to be a complete spectral set for a pair of commuting bounded operators $(T_1, T_2)$ on a Hilbert space $\mathcal{H}$, if the Taylor joint spectrum $\sigma(T_1, T_2) $ is a subset of $K$ and for every $n \times n$ matrix-valued rational function $f$ with poless off $K$ such that the norm of $f(T_1, T_2)$ is less than or equal to the sup-norm of $f$ on the compact set $K$ for every $n \geq 1$. If the above inequality holds for every scalar-valued rational functions then we say that $K$ is a spectral set for $(T_1, T_2)$. Dritschel, Jury and McCullough, showed that every pair commuting contractions having the distinguished variety $\{ (z, w ) \in \mathbb{D}^2: z^2-w^2=0\}$ as a spectral set must have the same distinguished variety as a complete spectral set. In contrast, they proved that there is a pair of commuting contractions that has the Neil parabola, $\mathcal{N}=\{ (z, w ) \in \mathbb{D}^2: z^2-w^3=0 $ as a spectral set but not a complete spectral set. Recently, Das and Sau have established that if a pair of commuting contractions is annihilated by a toral polynomial and the defect spaces are finite dimensional then the pair has a distinguished variety (not necessarily given by the annihilating toral polynomial) as a complete spectral set. Thus the above phenomenon motivates us to explore the following problem. When does a commuting contractive pair $(T_1, T_2)$, having a distinguished variety in the bidisk as a spectral set, also possess a distinguished variety (possibly different) as a complete spectral set? The aforementioned result of Das and Sau is obtained via dilating the pair contractions to a pair of commuting isometries annihilated by a toral polynomial. We would like to go beyond the bidisk and explore tuple of commuting tuples of bounded operators that have distinguished varieties in the polydisk as complete spectral set.