On the Dilation Theory and Canonical Decomposition of Θn-Contractions
Abstract
This paper investigates the domain Θn from an operator-theoretic perspective. We establish several characterizations of Θn-contractions, Θn-unitaries, and Θn-isometries, and explore their connections with Γn- and tetrablock operator tuples, as well as with the corresponding operator classes associated with Θn+1. We prove that every Θn-contraction admits a canonical decomposition as the direct sum of a Θn-unitary and a completely non-unitary Θn-contraction. We further develop the conditional dilation theory for Θn-contractions and establish necessary conditions for the existence of Θn-isometric dilations. We show that the conditions in Theorem~\ref{Conditional Dilation} are not, in general, sufficient by constructing a Θ3-contraction that admits a Θ3-isometric dilation although condition~(2) of Theorem~\ref{Conditional Dilation} fails. We further study pure Θn-isometric dilations of Θn-contractions and establish a Hardy space characterization of such dilations. We show that the resulting Θn-isometric dilation is minimal. We also study isometric dilations of doubly commuting Θn-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of Θ2-contractions that admits a Θ2-isometric dilation.
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BibTeX
@misc{gupta2026dilation,
title = {{On the Dilation Theory and Canonical Decomposition of $\mathbfΘ_n$-Contractions}},
author = {Aparna Gupta and Avijit Pal and Bhaskar Paul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03574},
eprint = {2608.03574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03574},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Dilation Theory and Canonical Decomposition of Θ_n-Contractions
AU - Aparna Gupta
AU - Avijit Pal
AU - Bhaskar Paul
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2608.03574
UR - https://arxiv.org/abs/2608.03574
AB - This paper investigates the domain Θ_n from an operator-theoretic perspective. We establish several characterizations of Θ_n-contractions, Θ_n-unitaries, and Θ_n-isometries, and explore their connections with Γ_n- and tetrablock operator tuples, as well as with the corresponding operator classes associated with Θ_n+1. We prove that every Θ_n-contraction admits a canonical decomposition as the direct sum of a Θ_n-unitary and a completely non-unitary Θ_n-contraction. We further develop the conditional dilation theory for Θ_n-contractions and establish necessary conditions for the existence of Θ_n-isometric dilations. We show that the conditions in Theorem~\ref{Conditional Dilation} are not, in general, sufficient by constructing a Θ_3-contraction that admits a Θ_3-isometric dilation although condition~(2) of Theorem~\ref{Conditional Dilation} fails. We further study pure Θ_n-isometric dilations of Θ_n-contractions and establish a Hardy space characterization of such dilations. We show that the resulting Θ_n-isometric dilation is minimal. We also study isometric dilations of doubly commuting Θ_n-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of Θ_2-contractions that admits a Θ_2-isometric dilation.
ER -