Related papers: Constrained dilation and $\Gamma$-contractions

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…

A commuting triple of operators $(A,B,P)$ on a Hilbert space $\mathcal{H}$ is called a tetrablock contraction if the closure of the set $$ E = \{\underline{x}=(x_1,x_2,x_3)\in \mathbb{C}^3: 1-x_1z-x_2w+x_3zw \neq 0 \text{whenever}|z| \leq…

A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. We show that every $\Gamma_n$-contraction admits a decomposition into a…

We prove some estimates for elementary symmetric polynomials on $\mathbb D^n.$ We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc $\Gamma_n.$ Furthermore, we show the existence and…

We find an explicit tetrablock isometric dilation for every member $(A_\alpha, B, P)$ of a family of tetrablock contractions indexed by a parameter $\alpha$ in the closed unit disc (only the first operator of the tetrablock contraction…

A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. In this paper, we study Hilbert space operators having liftings or…

We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] =…

A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator $T$ can be lifted to an isometry $V$. A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced…

A commuting tuple of $n$ 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_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots,…

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…

In this article, we discuss necessary condition of conditional dilation for both completely non-unitary (c.n.u) $\Gamma_{n}$-contractions and c.n.u $\mathbb E$-contractions. Consider two tuples, $(A_1, \dots, A_{n-1})$ and $(B_1, \dots,…

For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the…

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…

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}.…

Let $H_1, H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator from its domain $D(T)$, a dense subspace of $H_1$, into $H_2$. Let $N(T)$ denote the null space of $T$ and $R(T)$ denote the range of $T$. Recall…

Given a contractive tuple of Hilbert space operators satisfying certain $A$-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger…

The most degenerate unitary principal series representations {\pi}_{i{\lambda},{\delta}} (with {\lambda} \in R, \delta \in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary…

The isometric dilation of a pair of commuting contractions due to And\^{o} is not minimal. We modify And\^{o}'s dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a…

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…

The explicit constructions of minimal isometric, and minimal unitary dilations of an arbitrary linear pencil of operators $T(\lambda)=T_0+\lambda T_1$ consisting of contractions on a separable Hilbert space for $|\lambda |=1$, which…