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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

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

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq \mathbb{C}^3. \] We find several characterizations for the $\mathbb P$-unitaries and $\mathbb P$-isometries. We show that every $\mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $\mathbb P$-unitary and a pure $\mathbb P$-isometry. We also show that every $\mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $\mathbb P$-unitary and a completely non-unitary $\mathbb P$-contraction. We find a necessary and sufficient condition such that a $\mathbb P$-contraction $(A, S, P)$ dilates to a $\mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. A commuting tuple of Hilbert space operators $(T_1, \dots , T_n)$ having the closed unit ball $\overline{\mathbb B}_n$ as a spectral set is called a $\mathbb B_n$-\textit{contraction} and a commuting pair $(S,P)$ having the closed symmetrized bidisc $\overline{\mathbb G_2} \,(=\Gamma)$ as a spectral set is called a $\Gamma$-\textit{contraction}. We characterize isometries and unitaries associated with $\mathbb B_n$. Then we present an analogous canonical decomposition for a $\mathbb B_n$-contraction. We show interplay between operator theory on the three domains $\mathbb P, \mathbb B_2$ and $\mathbb G_2$., Comment: 40 pages. This article will be revised soon

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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 $\mathbb D^n$., Comment: Modified. new results added. 20 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 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, An Appendix added, 33 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 \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal M_2(\mathbb C), \; \|A\| <1 \} \subset \mathbb C^3 \] is a polynomially convex domain which is naturally associated with the $\mu$-synthesis problem. We introduce triangular $\mathbb E$-contractions and prove that every pure triangular $\mathbb E$-contraction dilates to a pure triangular $\mathbb E$-isometry. We construct a functional model for a pure triangular $\mathbb E$-isometry and apply that model to find a new proof for the famous Berger-Coburn-Lebow Model Theorem for commuting isometries. Next we give an alternative proof to the more generalized version of Berger-Coburn-Lebow Model, namely the factorization of a pure contraction due to Das, Sarkar and Sarkar (\textit{Adv. Math.} 322 (2017), 186 -- 200). We find a necessary and sufficient condition for the existence of $\mathbb E$-unitary dilation of an $\mathbb E$-contraction $(A,B,P)$ on the smallest dilation space and show that it is equivalent to the existence of a distinguished variety in $\mathbb E$ when the defect space $D_{P^*}$ is finite dimensional., Comment: Revised, 16 pages

We consider distinguished varieties in the bidisc $\mathbb D^2$ and the symmetrized bidisc $\mathbb{G}_2$. A commuting pair of Hilbert space operators $(S,P)$ for which $\Gamma=\overline{\mathbb{G}}_2$ is a spectral set is called a \textit{$\Gamma$-contraction}. A commuting pair of contractions $(T_1,T_2)$ is said to be \textit{toral} if there is a toral polynomial $p\in \mathbb C[z_1,z_2]$ such that $p(T_1,T_2)=0$. Similarly, a $\Gamma$-contraction $(S,P)$ is called $\Gamma$-\textit{distinguished} if $(S,P)$ is annihilated by a $\Gamma$-distinguished polynomial in $\mathbb C[z_1,z_2]$. The main results af this article are the following: we find a necessary and sufficient condition such that a toral pair of contractions dilates to a toral pair of isometries. Also, we characterize all $\Gamma$-distinguished $\Gamma$-contractions that admit $\Gamma$-distinguished $\Gamma$-isometric dilations. We determine the distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$. Then we prove that a pair of commuting contractions $(T_1,T_2)$ (or a $\Gamma$-contraction $(S,P)$) dilates to a toral pair of isometries (or a $\Gamma$-distinguished $\Gamma$-isometry) if and only if there is a toral polynomial (or $\Gamma$-distinguished polynomial) $p\in \mathbb C[z_1,z_2]$ such that $Z(p) \cap \overline{\mathbb D}^2$ (or $Z(p) \cap \overline{\mathbb G}_2$) is a complete spectral set for $(T_1,T_2)$ (or $(S,P)$). We provide several examples at places to show the contrasts between the theory of toral contractions and $\Gamma$-distinguished $\Gamma$-contractions., Comment: Revised, typos fixed, new results added. (55 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

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

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetilde{\Gamma}_n$., Comment: Revised, 31 pages

We find new characterizations for the points in the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i

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

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 \[ \Gamma_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i

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 $\mathbb G_2$.

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., Comment: Error in one of the main results

The symmetrized polydisc of dimension three is the set \[ \Gamma_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. We show that every $\Gamma_3$-contraction admits a decomposition into a $\Gamma_3$-unitary and a completely non-unitary $\Gamma_3$-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 $\Gamma_3$ and $\Gamma_3$-contractions., Comment: To appear in Operators and Matrices, 2017

We show that every distinguished variety in the symmetrized tridisc $\mathbb G_3$ is one-dimensional and can be represented as \begin{equation}\label{eqn:1} \Lambda=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in \sigma_T(F_1^*+pF_2\,,\, F_2^*+pF_1) \}, \end{equation} where $F_1,F_2$ are commuting square matrices of the same order satisfying $[F_1^*,F_1]=[F_2^*,F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb G_3$. We show that for a triple of commuting operators $\Sigma = (S_1,S_2,P)$ having $\Gamma_3$ as a spectral set, there is a one-dimensional subvariety $\Lambda_{\Sigma}$ of $\Gamma_3$ depending on $\Sigma$ such that von-Neumann's inequality holds, i.e, \[ f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\in\Lambda_{\Sigma}}\, |f(s_1,s_2,p)|, \] for any holomorphic polynomial $f$ in three variables, provided that $P^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $\Lambda_\Sigma$ has been shown to have representation like (\ref{eqn:1}), where $F_1,F_2$ are the unique solutions of the operator equations \begin{gather*} S_1-S_2^*P=(I-P^*P)^{\frac{1}{2}}X_1(I-P^*P)^{\frac{1}{2}} \text{ and } \\ S_2-S_1^*P=(I-P^*P)^{\frac{1}{2}}X_2(I-P^*P)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $\Lambda_\Sigma$ is a distinguished variety in $\mathbb G_3$. We produce an explicit dilation and a concrete functional model for such a triple $(S_1,S_2,P)$ in which the unique operators $F_1,F_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the symmetrized bidisc and in the unit bidisc $\mathbb D^2$., Comment: We withdraw this article because we could resolve the n-variable case

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