发表机构
IIT Bhilai（印度理工学院比莱分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从算子理论视角研究域Θₙ，刻画了Θₙ压缩算子等的性质及与相关算子的关系，证明其典范分解，建立其扩张理论，应用中推导极小Γₙ等距扩张的特例，确定了一类Θ₂压缩算子的等距扩张存在性。
AI 中文摘要
本文从算子理论的视角研究域Θₙ，得到了Θₙ压缩算子（对应Θₙ酉算子、Θₙ等距算子）的若干刻画，并建立了它们与Γₙ压缩算子（对应Γₙ酉算子、Γₙ等距算子）、四面体压缩算子（对应四面体酉算子、四面体等距算子）及Θₙ₊₁压缩算子（对应Θₙ₊₁酉算子、Θₙ₊₁等距算子）之间的关系。我们证明每个Θₙ压缩算子都可典范分解为Θₙ酉算子与完全非酉Θₙ压缩算子的直和，还通过推导极小Θₙ等距扩张存在的充要条件，建立了Θₙ压缩算子的扩张理论；作为应用，证明极小Γₙ等距扩张是极小Θₙ等距扩张的特例，最终确定了一类总能容许Θ₂等距扩张的Θ₂压缩算子。
英文摘要
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.
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