On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions

Aparna Gupta; Avijit Pal; Bhaskar Paul

Home/Archive/arXiv

arXiv · 2608.03574

On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions

Aparna Gupta·Avijit Pal·Bhaskar Paul

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies the domain $\mathbf{\Theta}_n$ from the perspective of operator theory. We obtain several characterizations of $\mathbf{\Theta}_n$-contractions (respectively, $\mathbf{\Theta}_n$-unitaries and $\mathbf{\Theta}_n$-isometries) and establish their relationships with $\Gamma_n$-contractions (respectively, $\Gamma_n$-unitaries and $\Gamma_n$-isometries), tetrablock contractions (respectively, tetrablock unitaries and tetrablock isometries), and $\mathbf{\Theta}_{n+1}$-contractions (respectively, $\mathbf{\Theta}_{n+1}$-unitaries and $\mathbf{\Theta}_{n+1}$-isometries). We prove that every $\mathbf{\Theta}_n$-contraction admits a canonical decomposition into the direct sum of a $\mathbf{\Theta}_n$-unitary and a completely non-unitary $\mathbf{\Theta}_n$-contraction. We further develop a dilation theory for $\mathbf{\Theta}_n$-contractions by obtaining necessary and sufficient conditions for the existence of minimal $\mathbf{\Theta}_n$-isometric dilations. As an application, we show that the minimal $\Gamma_n$-isometric dilation arises as a special case of the minimal $\mathbf{\Theta}_n$-isometric dilation. Finally, we identify a class of $\mathbf{\Theta}_2$-contractions that always admit $\mathbf{\Theta}_2$-isometric extensions.

Explore related subjects

math.FA math.CV

Keep this discovery

BibTeX RIS

Aparna Gupta, Avijit Pal, Bhaskar Paul. 2026-08-04. On the Dilation Theory and Canonical Decomposition of $\mathbf{\Theta}_n$-Contractions. https://arxiv.org/abs/2608.03574

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Typical dynamical properties of operators on $\ell_p$

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also study the typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

math.FA↗

arXiv2026

A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps

We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.

math.FA↗

arXiv2026

On weak dominance of t-conorms over t-norms

The weak dominance of aggregation operators, particularly between triangular norms (t-norms) and triangular conorms (t-conorms), has attracted considerable attention in aggregation operator theory. While several characterizations have been obtained for Archimedean and continuous cases, a general criterion for continuous t-conorms over continuous t-norms remains to be fully clarified. In this paper, we provide a complete characterization of a continuous t-conorm weakly dominating a continuous t-norm. We first reduce the problem for ordinal sum operators to that for their single Archimedean components, and then express the weak dominance condition entirely in terms of the additive generators of these components. Our approach covers both strict and nilpotent cases uniformly, and recovers the known results for Archimedean operators as a special case.

math.FA↗