N. Tomar

Function theoretic aspects of the symmetrized polydisc and generalization

Sourav Pal, Nitin Tomar

We introduce Schur-Agler type class for the symmetrized polydisc $G_{d}$ and establish a realization theorem for functions in this class. We state and prove an interpolation…

Function theory of the hexablock and applications to the tetrablock and Euclidean biball

Sourav Pal, Nitin Tomar

The realization, interpolation, extension and Toeplitz corona problems are amongst the central themes in function theory of a domain in $C^{d}$ . The present work addresses t…

Intertwining of $*$ -regular $q$ -isometric dilations

Nitin Tomar

A tuple $T = (T_{1}, \dots, T_{k})$ of contractions on a Hilbert space $H$ is said to be $q$ -commuting with $∥ q ∥ = 1$ if there exists a family of scalars $q=\{q_{ij}\i…

Regular quantum annulus unitary dilation and applications

Nitin Tomar

Consider the annulus $A_{r} = {z \in C : r^{- 1} < ∣ z ∣ < r}$ for $r > 1$ and the quantum annulus \[ Q\mathbb{A}_r=\{T:\ T \text{ is an invertible operator and} \ \|T\|, \|T^{-…

A Toeplitz corona theorem for the pentablock and applications

Sourav Pal, Nitin Tomar

We state and prove a Toeplitz corona theorem for the pentablock $P$ , a domain in $C^{3}$ given by \[ \mathbb{P}=\{(a_{21}, \text{tr}(A), \det(A)) \in \mathbb C^3 :…

Realization, interpolation, extension on the pentablock and applications to $D^{2}$ , $G_{2}$

Sourav Pal, Nitin Tomar

We introduce Schur-Agler class for the pentablock $P$ and establish a realization theorem for functions in this class. Then we prove an interpolation theorem for the pentab…

Rigidity of the structured singular value and applications

Sourav Pal, Nitin Tomar

The structured singular value $μ_{E}$ for a linear subspace $E$ of $M_{n} (C)$ is defined by \[ μ_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathb…

A domain in $C^{4}$ and its connection with $μ$ -synthesis problem

Sourav Pal, Nitin Tomar

We explore a domain $F$ in $C^{4}$ that has structural similarities with the hexablock $H \subset C^{4}$ . It leads to the question if these two domains…

Spectral constants for the quantum annulus

Sourav Pal, James E. Pascoe, Nitin Tomar

We find several new estimates for the spectral constants $K (A_{r})$ for which a closed annulus $A_{r}$ or closed polyannulus $A_{r}^{n}$ is a…

Constrained dilation and $Γ$ -contractions

Sourav Pal, Nitin Tomar

A commuting pair of Hilbert space operators having the closed symmetrized bidisc \[ Γ=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\} \] as a spectral set is…

Operators associated with a domain in $C^{4}$ and applications

Sourav Pal, Nitin Tomar

The hexablock is a domain arising from a special case of the $μ$ -synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is c…

math.CV2025

The Hexablock: a domain associated with the $μ$ -synthesis in $M_{2} (C)$

Indranil Biswas, Sourav Pal, Nitin Tomar

We introduce a domain named \textit{hexablock} in $C^{4}$ and show that its origin is a special case of $μ$ -synthesis in $M_{2} (C)$ , more precisely the $μ_{E}$ -unit ball…

Dilation on an annulus and von Neumann's inequality on certain varieties in the biball

Sourav Pal, Nitin Tomar

We give an alternative proof to Agler's famous result on success of rational dilation on an annulus by an application of a result due to Dritschel and McCullough. We show interplay…

On doubly commuting operators in $C_{1, r}$ class and quantum annulus

Nitin Tomar

For $0 < r < 1$ , let $A_{r} = {z \in C : r < ∣ z ∣ < 1}$ be the annulus with boundary $\partial A_{r} = T \cup r T$ , wher…

math.FA2024

Classification and dilation for $q$ -commuting $2 \times 2$ scalar matrices

Sourav Pal, Prajakta Sahasrabuddhe, Nitin Tomar

A tuple $T = (T_{1}, \dots, T_{k})$ of operators on a Hilbert space $H$ is said to be \textit{ $q$ -commuting with} $∥ q ∥ = 1$ or simply $q$ -\textit{commuting} if ther…

math.FA2024

Theory of $q$ -commuting contractions-II: Regular dilation, Brehmer's positivity and von Neumann's inequality

Sourav Pal, Prajakta Sahasrabuddhe, Nitin Tomar

It is well-known that a commuting family of contractions possesses a regular unitary dilation if and only if it satisfies Brehmer's positivity condition. We extend this theorem to…

math.FA2024

Remarks on positive definite functions on a group

Swapan Jana, Sourav Pal, Nitin Tomar

We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contracti…

Joint reducing subspaces and orthogonal decompositions of operators in an annulus

Sourav Pal, Nitin Tomar

A commuting tuple of Hilbert space operators $(T_{1}, \dots, T_{n})$ is said to be an \textit{ $A_{r}^{n}$ -contraction} if the closure of the polyannulus \[ \mathbb A_r^n=\left\{(…

The $2 \times 2$ block matrices associated with an annulus

Sourav Pal, Nitin Tomar

A bounded Hilbert space operator $T$ for which the closure of the annulus \[ \mathbb{A}_r=\{z \ : \ r<|z|<1\} \subseteq \mathbb{C}, \qquad (0<r<1) \] is a spectral set is called an…

Operators associated with the pentablock and their relations with biball and symmetrized bidisc

Sourav Pal, Nitin Tomar

A commuting triple of Hilbert space operators $(A, S, P)$ is said to be a \textit{ $P$ -contraction} if the closed pentablock $P$ is a spectral set for $(A,…