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We study the interaction between unitary $\rho$-dilations of a pair of Hilbert space operators and Birkhoff-James orthogonality. We prove that for two orthogonal operators $T,A$ if $\|T\|=\rho$, then $U_T \perp_B U_A$ for any unitary $\rho$-dilations $U_T$ of $T$ and $U_A$ of $A$ acting on a common space. We characterize $\varepsilon$-approximate Birkhoff-James orthogonality for complex Hilbert space operators. Then find a sharp bound on $\varepsilon$ such that $T \perp_B A$ implies that $U_T \perp_B^{\varepsilon} U_A$ for any unitary $\rho$-dilations $U_T, U_A$ of $T$ and $A$ respectively. The Sch\"{a}ffer unitary dilations of a pair of contractions $T,A$ are not orthogonal in general. We construct Sch\"{a}ffer-type unitary dilations for contractions which are pairwise orthogonal., Comment: 31 pages, Submitted to journal

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In the literature, we have several results associated with canonical decomposition of commuting contractions. In this paper, we generalize a few of these results to q-commuting contractions. Here we mainly deal with doubly q-commuting contractions when $|q|=1$ or when $q$ is a unitary operator., Contains 25 pages. This is just a first draft. It will be revised soon and some more results will be added to it. arXiv admin note: substantial text overlap with arXiv:2202.01301

A bounded Hilbert space operator $T$ for which the closure of an annulus \[ \mathbb{A}_r=\{z \ : \ r, Comment: 12 pages, Submitted to Journal

A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex algebraic variety that intersects $\mathbb D^n$ and exits the domain through the $n$-torus $\mathbb T^n$ without intersecting any other part of the topological boundary of $\mathbb D^n$. We find two different characterizations for a distinguished variety in the polydisc $\mathbb D^n$ in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy [Acta Math., 2005] on distinguished varieties in $\mathbb D^2$. We show that a distinguished variety in $\mathbb D^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb D^n$., Modified. new results added. 20 Pages

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\bar{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal M_2(\mathbb C), \; \|A\, Revised, 16 pages

There are several proofs of the classical commutant lifting and intertwining lifting theorems in the literature. In this article, we present analogous proofs to a few $Q$-commuting lifting and $Q$-intertwining lifting theorems. We provide several proofs and show explicit constructions of Ando-type dilations for a pair of $Q$-commuting contractions $(T_1,T_2)$ when $Q$ is a bounded operator. We establish a few connections of these results with graph theory., Comment: Contains 52 pages. This is just a first draft and will be revised soon. This is a part of the PhD thesis of the second named author submitted to IIT Bombay on May, 2022

The symmetrized bidisc, which is the symmetrization of the bidisc $\mathbb D^2$, is defined by \[ \mathbb{G}=\{(z_1+z_2, z_1z_2) \ : \ |z_1, This is just a first draft and it will be revised soon. (34 Pages)

For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we show that $(T_1, \dots, T_n)$ dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of $T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$. Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots , T_n)$ and its adjoint. We find a necessary and sufficient condition such that $(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the minimal unitary dilation of $T$. We show an explicit construction of such a unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is necessary and sufficient for the existence of such a unitary dilation when $T$ is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. We construct a different unitary dilation for $(T_1, \dots , T_n)$ when $T$ is a $C._0$ contraction., Comment: Revised, 33 pages. arXiv admin note: text overlap with arXiv:2204.11391

For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer and Sz. Nagy-Foias type isometric dilations for $(T_1,\dots ,T_n)$ on the minimal dilation spaces of $T$. Also, a different dilation is constructed when the product $T$ is a $C._0$ contraction, that is ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\dots ,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed., Comment: Revised, 31 Pages

We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i, Comment: 31 Pages. arXiv admin note: substantial text overlap with arXiv:1904.03745

We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author and Shalit. We show that a distinguished variety $\Lambda$ in $\mathbb G_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which $\Gamma_n=\overline{\mathbb G_n}$ is a spectral set is called a $\Gamma_n$-contraction. To every $\Gamma_n$-contraction $(S_1, \dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, called the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with $\Gamma_n$ and by an application of that model we establish that the $\Gamma_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots , S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilations for a unique distinguished variety $\Lambda_{\Sigma}$ in $\mathbb G_n$, when $\Lambda_{\Sigma}$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots ,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of $P^*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$., Comment: See the previous version for the "Distinguished varieties in the polydisc" part. arXiv admin note: substantial text overlap with arXiv:1708.06015

The symmetrization map $��:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ ��(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $��$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[ ��= ��(\overline{\mathbb D^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\leq 1, i=1,2 \}. \] A pair of commuting Hilbert space operators $(S,P)$ for which $��$ is a spectral set is called a $��$-contraction. Unlike the scalars in $��$, a $��$-contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded operators. We characterize all $��$-contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a $��$-contraction $(S,P)=(T_1+T_2,T_1T_2)$ for a pair of commuting bounded operators $T_1,T_2$, no real number less than $2$ can be a bound for the set $\{ \|T_1\|,\|T_2\| \}$ in general. Then we prove that every $��$-contraction $(S,P)$ is the restriction of a $��$-contraction $(\widetilde S, \widetilde P)$ to a common reducing subspace of $\widetilde S, \widetilde P$ and that $(\widetilde S, \widetilde P)=(A_1+A_2,A_1A_2)$ for a pair of commuting operators $A_1,A_2$ with $\max \{\|A_1\|, \|A_2\|\} \leq 2$. We find new characterizations for the $��$-unitaries and describe the distinguished boundary of $��$ in a different way. We also show some interplay between the fundamental operators of two $��$-contractions $(S,P)$ and $(S_1,P)$., A few typos got fixed. 16 pages

A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ ��_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i, 22 Pages, Submitted to Journal

Given a complex polynomial $p(z)$ of degree $n$ and an ellipse, we find an algorithm of determining the number of zeros of $p$ in the interior and exterior of the ellipse. Our results generalize the previous results of Pt\'ak and Young about the circles., Comment: We withdraw the paper temporarily because of a mistake in Theorem 2.1

The main aims of this article are to characterize a class of operators associated with the symmetrized polydisc that admit rational dilations on the minimal space and to show an interplay between rational dilation and distinguished varieties in the symmetrized polydisc., Comment: Revised, 41 Pages

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock $\mathbb E$. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc $\mathbb G_2$. In either case, our results generalize all previous results in this direction for $\mathbb E$ and $\mathbb G_2$.

A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ ��_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i, 16 pages

The closed symmetrized tridisc $\Gamma_3$ and its distinguished boundary $b\Gamma_3$ are the sets $\Gamma_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $b\Gamma_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq \Gamma_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal H$ for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. In this article we show by a counter example that there are $\Gamma_3$-contractions which do not dilate. It is also shown that under certain conditions a $\Gamma_3$-contraction can have normal $b\Gamma_3$ dilation. We determine several classes of $\Gamma_3$-contractions which dilate and show explicit construction of their dilations. A concrete functional model is provided for the $\Gamma_3$-contractions which dilate. Various characterizations for $\Gamma_3$-unitaries and $\Gamma_3$-isometries are obtained; the classes of $\Gamma_3$-unitaries and $\Gamma_3$-isometries are analogous to the unitaries and isometries in one variable operator theory. Also we find out a model for the class of pure $\Gamma_3$-isometries. En route we study the geometry of the sets $\Gamma_3$ and $b\Gamma_3$ and provide variety of characterizations for them., Comment: Incomplete

We produce a Schwarz lemma for the symmetrized tridisc \[ \mathbb G_3 =\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|< 1, i=1,2,3 \}. \] We show that an interpolating function related to the Schward lemma for $\mathbb G_3$ is not unique and present an explicit description of all such interpolating functions. We also study the complex geometry of $\mathbb G_3$ and present a variety of new characterizations for the open and closed symmetrized tridisc., Error in one of the main results

The symmetrized polydisc of dimension three is the set \[ ��_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $��_3$ is a spectral set is called a $��_3$-contraction. We show that every $��_3$-contraction admits a decomposition into a $��_3$-unitary and a completely non-unitary $��_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $��_3$ and $��_3$-contractions., To appear in Operators and Matrices, 2017

We provide an explicit normal distinguished boundary dilation to a pair of commuting operators $(S,P)$ having the closed symmetrized bidisc $\Gamma$ as a spectral set. This is called Sz.-Nagy dilation of $(S,P)$. The operator pair that dilates $(S,P)$ is obtained by an application of Stinespring dilation of $(S,P)$ given by Agler and Young. We further prove that the dilation is minimal and the dilation space is no bigger than the dilation space of the minimal unitary dilation of the contraction $P$. We also describe model space and model operators for such a pair $(S,P)$.

A pair of commuting operators $(S,P)$ defined on a Hilbert space $\mathcal H$ for which the closed symmetrized bidisc $$ ��= \{(z_1+z_2,z_1z_2):: |z_1|\leq 1,\, |z_2|\leq 1 \}\subseteq \mathbb C^2, $$ is a spectral set is called a $��$-contraction in the literature. A $��$-contraction $(S,P)$ is said to be pure if $P$ is a pure contraction, i.e, ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty $. Here we construct a functional model and produce a complete unitary invariant for a pure $��$-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation $$ S-S^*P=D_PXD_P, \textup{where} X\in \mathcal B(\mathcal D_P), $$and is called the fundamental operator of the $��$-contraction $(S,P)$. We also discuss some important properties of the fundamental operator., Journal of Operator Theory, 71 (2014), 327-339

For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation S-S*P = (I-P*P)^1/2 X(I-P*P) 1/2, where X is a bounded operator on Ran(I-P*P) 1/2 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domain \Gamme = {(z 1 +z 2, z 1z 2) : |z1|{\leq} 1, |z2| {\leq}1} {\subseteq} C2 as a spectral set, is called a \Gamme-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a ��-contraction (S,P). This allows us to construct an explicit ��-isometric dilation of a ��-contraction (S,P). We prove the other way too, i.e, for a commuting pair (S,P) with |P|| {\leq} 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S,P) is a ��-contraction. We show that for a pure ��-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S = C +C*P. Any ��-isometry can be written in this form where P now is an isometry commuting with C and C*. Any ��-unitary is of this form as well with P and C being commuting unitaries. Examples of ��-contractions on reproducing kernel Hilbert spaces and their ��-isometric dilations are discussed., Advances in Mathematics, 230 (2012), 577-606