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Nitin Tomar

Publications and source records attributed to Nitin Tomar.

27 records · Page 2Linked to original sources
arXiv2025

Joint reducing subspaces and orthogonal decompositions of operators in an annulus

A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to be an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus \[ \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n \right\} \subseteq \mathbb{C}^n \qquad \quad (0<r<1) \] is a spectral set for $(T_1, \dotsc, T_n)$. We find characterizations for the $\mathbb A_r^n$-unitaries and $\mathbb A_r^n$-isometries and decipher their structures. We find Wold type decompositions for any number of commuting and doubly commuting $\mathbb A_r$-isometries. Then we generalize these results to any family of commuting and doubly commuting $\mathbb A_r$-contractions.

math.FA↗
arXiv2024

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

A tuple $\underline{T}=(T_1, \dotsc, T_k)$ of operators on a Hilbert space $\mathcal H$ is said to be \textit{$q$-commuting with} $\|q\|=1$ or simply $q$-\textit{commuting} if there is a family of scalars $q=\{q_{ij} \in \mathbb C : |q_{ij}|=1, \ q_{ij}=q_{ji}^{-1}, \ 1 \leq i < j \leq k \}$ such that $T_i T_j =q_{ij}T_j T_i$ for $1 \leq i < j \leq k$. Moreover, if each $q_{ij}=-1$, then $\underline{T}$ is called an \textit{anti-commuting tuple}. A well-known result due to Holbrook \cite{Holbrook} states that a commuting $k$-tuple consisting of $2 \times 2$ scalar matrix contractions always dilates to a commuting $k$-tuple of unitaries for any $k\geq 1$. To find a generalization of this result for a $q$-commuting $k$-tuple of $2\times 2$ scalar matrix contractions, we first classify such tuples into three types upto similarity. Then we prove that a $q$-commuting tuple which is unitarily equivalent to any of these three types, admits a $\widetilde{q}$-unitary dilation, where $\widetilde q \subseteq q \cup \{1\}$. A special emphasis is given to the dilation of an anti-commuting tuple of $2 \times 2$ scalar matrix contractions.

math.FA↗
arXiv2024

Characterizations and models for the $C_{1,r}$ class and quantum annulus

For fixed $0<r<1$, let $A_r=\{z \in \mathbb{C} : r<|z|<1\}$ be the annulus with boundary $\partial \overline{A}_r=\mathbb{T} \cup r\mathbb{T}$, where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$. An operator having $\ov{A}_r$ as a spectral set is called an $A_r$-\textit{contraction}. Also, a normal operator with its spectrum lying in the boundary $\partial \overline{A}_r$ is called an \textit{$A_r$-unitary}. The \textit{$C_{1,r}$ class} was introduced by Bello and Yakubovich in the following way: \[ C_{1, r}=\{T: T \ \mbox{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \] McCullough and Pascoe defined the \textit{quantum annulus} $\mathbb Q \mathbb A_r$ by \[ \mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \] If $\mathcal A_r$ denotes the set of all $A_r$-contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$. We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$.

math.FA↗
arXiv2024

Remarks on positive definite functions on a group

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 contractions. We classify all commuting unitary representations over a finite group. We show by examples that the power of a positive-definite function may not be positive definite and also the power of a unitary representation may not be a unitary representation. We also characterize all unitary representations whose powers are also unitary representations.

math.FA↗
arXiv2023

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

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 $\mathbb A_r$-contraction. A celebrated theorem due to Douglas, Muhly and Pearcy gives a necessary and sufficient condition such that a $2 \times 2$ block matrix of operators $ \begin{bmatrix} T_1 & X 0 & T_2 \end{bmatrix} $ is a contraction. We seek an answer to the same question in the setting of annulus, i.e., under what conditions $\widetilde{T}_Y=\begin{bmatrix} T_1 & Y 0 & T_2 \end{bmatrix} $ becomes an $\mathbb A_r$-contraction. For a pair of $\mathbb A_r$-contractions $T_1,T_2$ and an operator $X$ that commutes with $T_1,T_2$, here we find a necessary and sufficient condition such that each of the block matrices \[ T_X= \begin{bmatrix} T & X 0 & T \end{bmatrix} \,, \quad \widehat{T}_X=\begin{bmatrix} T_1 & X(T_1-T_2) 0 & T_2 \end{bmatrix} \] becomes an $\mathbb A_r$-contraction. Thus, the general block matrix $\widetilde{T}_Y$ is an $\mathbb A_r$-contraction if $T_1-T_2$ (as in $\widehat{T}_X$) is invertible.

math.FA↗
arXiv2023

Dilation of normal operators associated with an annulus

For $0<r<1$, let us consider the following annulus: \[ \mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}. \] A Hilbert space operator $T$ for which $\overline{\mathbb A}_r$ is a spectral set is called an $\mathbb A_r$-\textit{contraction}. Also, a normal operator $U$ whose spectrum lies on the boundary $\partial \mathbb A_r$ of $\mathbb A_r$ is called an $\mathbb A_r$-\textit{unitary}. We prove that any $m$ number of commuting normal $\mathbb A_r$-contractions $N_1, \dots , N_m$ can be simultaneously dilated to commuting $\mathbb A_r$-unitaries $U_1, \dots , U_m$. To construct such a dilation, we solve a Dirichlet problem for the polyannulus $\mathbb A_r^m$. Also, we show that any finitely many doubly commuting subnormal $\mathbb A_r$-contractions simultaneously dilate to commuting $\mathbb A_r$-unitaries. Finally, we show that such a simultaneous $\mathbb A_r$-unitary dilation holds for any finite number of doubly commuting $2 \times 2$ scalar $\mathbb A_r$-contractions.

math.FA↗