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Nitin Tomar

Publications and source records attributed to Nitin Tomar.

At least 19 recordsLinked to original sources
arXiv2026

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

The realization, interpolation, extension and Toeplitz corona problems are amongst the central themes in function theory of a domain in $\mathbb{C}^d$. The present work addresses these four problems for the hexablock $\mathbb{H}$, a domain in $\mathbb{C}^4$ arising in connection with a special case of $\mu$-synthesis in $H^{\infty}$ control theory. We determine Schur-Agler class $SA(\mathbb{H})$ for $\mathbb{H}$ and find a realization formula for $\mathbb H$. With the help of this realization formula we state and prove interpolation, extension and Toeplitz corona theorems for the hexablock. As an application, we obtain analogous theorems for the Euclidean unit ball in $\mathbb{C}^2$. Moreover, we recover the existing same results for the tetrablock $\mathbb E$, another domain associated with the $\mu$-synthesis, as consequences of the hexablock theory and in terms of the Schur-Agler class $SA(\mathbb{H})$ and admissible kernels on $\mathbb{H}$.

math.CV↗
arXiv2026

Function theoretic aspects of the symmetrized polydisc and generalization

We introduce Schur-Agler type class for the symmetrized polydisc $\mathbb{G}_d$ and establish a realization theorem for functions in this class. We state and prove an interpolation theorem on $\mathbb{G}_d$ with interpolating functions belonging to the associated Schur-Agler type class. Moreover, Toeplitz corona and extension theorems are established for $\mathbb{G}_d$. We also extend the realization, interpolation, Toeplitz corona and extension theorems to a more general symmetrized family of domains $\Theta_d$.

math.CV↗
arXiv2026

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

A tuple $\underline{T}=(T_1,\dots,T_k)$ of contractions on a Hilbert space $\mathcal H$ is said to be $q$-commuting with $\|q\|=1$ if there exists a family of scalars $q=\{q_{ij}\in\mathbb C : |q_{ij}|=1,\ q_{ij}=q_{ji}^{-1},\ 1\le i<j\le k\}$ such that $T_iT_j=q_{ij}T_jT_i$ for $1\le i<j\le k$. In this article, we characterize $q$-commuting pairs of contractions with $\|q\|=1$ that admit a minimal $*$-regular $q$-isometric dilation. We present sufficient conditions for the commutant lifting theorem for such pairs. Moreover, sufficient conditions are obtained for a $q$-commuting triple of contractions with $\|q\|=1$ to admit a $q$-isometric dilation.

math.FA↗
arXiv2026

Regular quantum annulus unitary dilation and applications

Consider the annulus $\mathbb{A}_r=\{z\in\mathbb{C}:r^{-1}<|z| 1$ and the quantum annulus \[ Q\mathbb{A}_r=\{T:\ T \text{ is an invertible operator and} \ \|T\|, \|T^{-1}\|\leq r\}. \] McCullough and Pascoe proved that $T\in Q\mathbb{A}_r$ if and only if $\beta(T^*,T)=(r^2+r^{-2})-T^*T-(T^*T)^{-1}\ge0$. We call an invertible operator $T$ a quantum annulus unitary if $\beta(T^*,T)=0$. In this article, we construct an explicit doubly commuting $d$-tuple of quantum annulus unitaries that simultaneously extends a given doubly commuting $d$-tuple of operators in $Q\mathbb{A}_r$. We introduce the notion of a regular quantum annulus unitary dilation and show that the dilation arising from our construction is regular. As an application of the dilation theorem, we show that $\overline{\mathbb A}_r$ is a complete $K_t$-spectral set for operators in $Q\mathbb{A}_r$ and $\overline{\mathbb{A}}_r^d$ is a complete $K_{dc}^{(d)}$-spectral set for doubly commuting $d$-tuples of operators in $Q\mathbb{A}_r$, where \[ K_t=2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right) \quad \text{and} \quad K_{dc}^{(d)}=\left[2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right)\right]^d. \] We further prove that every doubly commuting tuple of operators in $Q\mathbb{A}_r$ is similar to a commuting tuple having $\overline{\mathbb{A}}_r^{d}$ as a complete spectral set. In addition, we establish bounds for the optimal spectral constants and show that they converge to $2^d$ as $r\to\infty$. We also obtain an alternative characterization of operators in $Q\mathbb{A}_r$ and quantum annulus unitaries, and prove that $\overline{\mathbb{A}}_r^d$ is a $K$-spectral set for a subclass of commuting $d$-tuples in $Q\mathbb{A}_r$.

math.FA↗
arXiv2026

Realization, interpolation, extension on the pentablock and applications to $\mathbb D^2$, $\mathbb G_2$

We introduce Schur-Agler class for the pentablock $\mathbb P$ and establish a realization theorem for functions in this class. Then we prove an interpolation theorem for the pentablock with interpolating functions belonging to the corresponding Schur-Agler class. Also, we obtain an extension theorem for $\mathbb P$. Applying these results, we add a few new characterizations in the existing realization and interpolation theorems for the bidisc $\mathbb D^2$ and the symmetrized bidisc $\mathbb G_2$. Also, we give alternative proofs to the existing extension theorems for $\mathbb D^2, \, \mathbb G_2$.

math.CV↗
arXiv2026

A Toeplitz corona theorem for the pentablock and applications

We state and prove a Toeplitz corona theorem for the pentablock $\mathbb{P}$, a domain in $\mathbb{C}^3$ given by \[ \mathbb{P}=\{(a_{21}, \text{tr}(A), \det(A)) \in \mathbb C^3 : A=[a_{ij}] \in M_2(\mathbb C), \|A\|<1\}. \] By two different applications of this theorem, we obtain a few new characterizations in the Toeplitz corona theorems for the bidisc and the symmetrized bidisc.

math.CV↗
arXiv2026

Rigidity of the structured singular value and applications

The structured singular value $\mu_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by \[ \mu_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})), \] and $\mu_E(A)=0$ if there is no $X \in E$ with $\det(I_n-AX)=0$. It is well-known that $\mu_E(A)$ coincides with the spectral radius $r(A)$ when $E=\{cI_n: c \in \mathbb C \}$ and $\mu_E(A)=\|A\|$ when $E=M_n(\mathbb C)$, for all $A\in M_n(\mathbb C)$. Also, for any linear subspace $E$ satisfying $\{cI_n: c \in \mathbb C \} \subseteq E \subseteq M_n(\mathbb C)$, we have $r(A)\leq \mu_E(A) \leq \|A\|$. We prove that if $E=\{cI_n: c \in \mathbb C \}$ and $F$ is any linear subspace of $M_n(\mathbb C)$ containing $E$, then $\mu_E=\mu_F$ if and only if $E=F$. We prove the exact same rigidity theorem for the linear subspace consisting of the diagonal matrices of order $n$. On the contrary, when $E=M_n(\mathbb C)$, we show that there is a proper subspace $F$ of $M_n(\mathbb C)$, viz. the space of symmetric matrices such that $\mu_E=\mu_F=$ operator norm. Further, we characterize all linear subspaces $F\subseteq M_n(\mathbb C)$ such that $\mu_F$ coincides with the operator norm. Next, we show that in general there is no subspace $E$ of $M_n(\mathbb C)$ such that $\mu_E=$ the numerical radius, not even for $M_2(\mathbb C)$. We establish the rigidity of the structured singular value for each of the subspaces $E$ of $M_2(\mathbb C)$ such that the corresponding $\mu_E$-unit ball induces the domains -- symmetrized bidisc, tetrablock, pentablock, hexablock.

math.FA↗
arXiv2026

A domain in $\mathbb C^4$ and its connection with $\mu$-synthesis problem

We explore a domain $\mathbb F$ in $\mathbb C^4$ that has structural similarities with the hexablock $\mathbb{H} \subset \mathbb C^4$. It leads to the question if these two domains are biholomorphic. In this paper, we answer this question in negative. We provide alternative characterizations of the domain $\mathbb{F}$ and find its connection with the domains associated with $\mu$-synthesis problem such as the symmetrized bidisc, the tetrablock, the pentablock and the hexablock. We also address the following question: does $\mathbb{F}$ (which is biholomorphic with $\mathbb L_4$) arise from a $\mu$-synthesis problem in the same manner as $\mathbb{G}_2$ and $\mathbb{E}$ ?

math.CV↗
arXiv2026

Spectral constants for the quantum annulus

We find several new estimates for the spectral constants $K(\mathbb A_r)$ for which a closed annulus $\overline{\mathbb A}_r$ or closed polyannulus $\overline{\mathbb A}^n_r$ is a $K$-spectral set for operators in the quantum annulus $\mathbb Q \mathbb A_r$. We give two alternative proofs to an existing estimate of spectral constant. The first proof capitalizes a dilation theorem due to McCullough and Pascoe, while the second proof involves a certain variety in the Euclidean biball. For commuting and doubly commuting operators in $\mathbb Q \mathbb A_r$, we find upper and lower bounds for the smallest spectral constants.

math.FA↗
arXiv2025

Constrained dilation and $\Gamma$-contractions

A commuting pair of Hilbert space operators having the closed symmetrized bidisc \[ \Gamma=\{(z_1+z_2, z_1z_2) \in \mathbb C^2 \ : \ |z_1| \leq 1, |z_2| \leq 1\} \] as a spectral set is called a \textit{$\Gamma$-contraction}. A $\Gamma$-contraction $(S,P)$ is called \textit{$\Gamma$-distinguished} if $(S,P)$ is annihilated by a polynomial $q \in \mathbb C[z_1,z_2]$ whose zero set $Z(q)$ defines a distinguished variety in the symmetrized bidisc $\mathbb G$. There is Schaffer-type minimal $\Gamma$-isometric dilation of a $\Gamma$-contraction $(S,P)$ in the literature. In this article, we study when such a minimal $\Gamma$-isometric dilation is $\Gamma$-distinguished provided that $(S,P)$ is a $\Gamma$-distinguished $\Gamma$-contraction. We show that a pure $\Gamma$-isometry $(T,V)$ with defect space $\dim \mathcal D_{V^*}< \infty$, is $\Gamma$-distinguished if and only if the fundamental operator of $(T^*,V^*)$ has numerical radius less than $1$. Further, it is proved that a $\Gamma$-contraction acting on a finite-dimensional Hilbert space dilates to a $\Gamma$-distinguished $\Gamma$-isometry if its fundamental operator has numerical radius less than $1$. We also provide sufficient conditions for a pure $\Gamma$-contraction to be $\Gamma$-distinguished. Wold decomposition splits an isometry into two orthogonal parts of which one is a unitary and the other is a completely non-unitary contraction. In this direction, we find a few decomposition results for the $\Gamma$-distinguished $\Gamma$-unitaries and $\Gamma$-distinguished pure $\Gamma$-isometries.

math.FA↗
arXiv2025

Operators associated with a domain in $\mathbb C^4$ and applications

The hexablock is a domain arising from a special case of the $\mu$-synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is called a hexablock-contraction or simply $\mathbb H$-contraction. We characterize the unitaries and isometries associated with $\mathbb H$-contractions. Two different types of dilation results for $\mathbb H$-contractions are obtained. We find connection of this theory with the operators associated with the symmetrized bidisc and tetrablock, two other domains related to the $\mu$-synthesis problem.

math.FA↗
arXiv2025

The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$

We introduce a domain named \textit{hexablock} in $\mathbb C^4$ and show that its origin is a special case of $\mu$-synthesis in $M_2(\mathbb C)$, more precisely the $\mu_E$-unit ball with respect to the linear subspace $E$ consisting of $2 \times 2$ upper triangular matrices. The hexablock is denoted by $\mathbb H$ and is defined by \[ \mathbb{H}=\left\{(a, x_1, x_2, x_3) \,\in\, \mathbb{C} \times \mathbb{E}\,\,\big\vert\,\, \sup_{z_1,\, z_2 \,\in\, \mathbb D}\left|\frac{a\sqrt{(1-|z_1|^2)(1-|z_2|^2)}}{1-x_1z_1-x_2z_2+x_3z_1z_2}\right| <1\right\}, \] where $\mathbb{E}$ is the \textit{tetrablock}, another domain in $\mathbb C^3$ associated with a different case of $\mu$-synthesis, and is given by \[ \mathbb{E}=\{(x_1, x_2, x_3) \in \mathbb{C}^3 : 1-x_1z_1-x_2z_2+x_3z_1z_2 \ne 0 \ \text{for all } \, z_1, z_2 \in \overline{\mathbb D}\}. \] We show that two other objects in $\mathbb C^4$ namely, the $\mu$-hexablock $\mathbb H_{\mu}$ and the normed hexablock $\mathbb H_N$ naturally arise in the $\mu_E$-unit ball and the norm unit ball of $M_2(\mathbb C)$, respectively and pave the way to reach the domain $\mathbb H$. A set of independent characterizations for the points in $\mathbb H_{\mu}, \mathbb H_N$ and $\mathbb H$ are obtained. Geometric and function theoretic aspects of $\mathbb H$ are studied and its connections with the popular domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$ are explored.

math.CV↗
arXiv2025

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

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 between operators associated with an annulus, $C_{1,r}$ or quantum annulus and operator pairs living on a certain variety in $\mathbb C^2$ and its intersection with the biball. It is shown that the minimal spectral sets and von Neumann's inequality for these classes $C_{1,r}$, quantum annulus can also be studied via appropriate operator pairs associated with the biball.

math.FA↗
arXiv2025

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

For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples.

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

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

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 any family $\mathcal T$ of $q$-commuting contractions with $\|q\|=1$ by showing the equivalence of the following three statements: $(i)$ $\mathcal T$ admits a regular $q$-unitary dilation; $(ii)$ $\mathcal T$ satisfies Brehmer's positivity condition; $(iii)$ $\mathcal T$ admits a $Q$-unitary dilation for a family of $Q$-commuting unitaries. We achieve the first part of the result by an application of Stinespring's dilation theorem on a particular completely positive map acting on a quotient algebra of a group $C^*$-algebra, where the underlying group is a free group, and the second part is obtained by an application of Naimark's theorem. Next, we find several cases when $\mathcal{T}$ admits a regular $q$-unitary dilation and establish a von Neumann type inequality for such a $q$-commuting family.

math.FA↗