Related papers: Multivariable $\rho$-contractions

Let $\mathcal{H}$ be a complex Hilbert space and let $\big\{A_{n}\big\}_{n\geq 1}$ be a sequence of bounded linear operators on $\mathcal{H}$. Then a bounded operator $B$ on a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ is said to be…

We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_\delta \subseteq \mathbb{C}$ given by \[ K_\delta {\stackrel{\rm…

Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A \otimes I$ whenever the numerical range inclusion relation $W(B) \subseteq W(A)$ holds.…

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}.…

Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times…

We explore aspects of dilation theory in the finite dimensional case and show that for a commuting $n$-tuple of operators $T=(T_1,...,T_n) $ acting on some finite dimensional Hilbert space $H$ and a compact set $X\subset \mathbb{C}^n$ the…

We define a function on the $C^{\ast}$-algebra of all bounded linear Hilbert space operators, which generalizes the operator radii, and we present some basic properties of this function. Our results extend several results in the literature.

Let $\sigma(A)$, $\rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$\rho(AB)\le r(A)r(B)…

The purpose of this paper is to describe the Harnack parts for the operators of class C $\rho$ ($\rho$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of…

Let $ \mathbb{B}(\mathscr{H})$ represent the $C^*$-algebra, which consists of all bounded linear operators on $\mathscr{H},$ and let $N ( .) $ be a norm on $ \mathbb{B}(\mathscr{H})$. We define a norm $w_{(N,e)} (. , . )$ on $…

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with…

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain…

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$…

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…

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…

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ be the C*-algebra of all bounded linear operators on $\mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction,…

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<j\leq n}z_iz_j,\dots,…

Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model…

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…