Year: 2026
Type: preprint
Abstract: This paper investigates the domain $\mathbfΘ_n$ from an operator-theoretic perspective. We establish several characterizations of $\mathbfΘ_n$-contractions, $\mathbfΘ_n$-unitaries, and $\mathbfΘ_n$-isometries, and explore their connections with $Γ_n$- and tetrablock operator tuples, as well as with the corresponding operator classes associated with $\mathbfΘ_{n+1}$. We prove that every $\mathbfΘ_n$-contraction admits a canonical decomposition as the direct sum of a $\mathbfΘ_n$-unitary and a completely non-unitary $\mathbfΘ_n$-contraction. We further develop the conditional dilation theory for $\mathbfΘ_n$-contractions and establish necessary conditions for the existence of $\mathbfΘ_n$-isometric dilations. We show that the conditions in Theorem~\ref{Conditional Dilation} are not, in general, sufficient by constructing a $\mathbfΘ_3$-contraction that admits a $\mathbfΘ_3$-isometric dilation although condition~(2) of Theorem~\ref{Conditional Dilation} fails. We further study pure $\mathbfΘ_n$-isometric dilations of $\mathbfΘ_n$-contractions and establish a Hardy space characterization of such dilations. We show that the resulting $\mathbfΘ_n$-isometric dilation is minimal. We also study isometric dilations of doubly commuting $\mathbfΘ_n$-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of $\mathbfΘ_2$-contractions that admits a $\mathbfΘ_2$-isometric dilation.
Source: arXiv (Cornell University)
Authors: Gupta, Aparna, Pal, Avijit, Paul, Bhaskar
Institution: Indian Institute of Technology Bhilai
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172554382