Operators associated with the pentablock and their relations with biball and symmetrized bidisc
Sourav Pal, Nitin Tomar
TL;DR
This work develops a comprehensive operator-theoretic framework for the pentablock domain by introducing and analyzing $\mathbb P$-contractions, $\mathbb P$-unitaries, and $\mathbb P$-isometries. It establishes deep structural results, including a Wold-type decomposition for $\mathbb P$-isometries and a canonical decomposition for $\mathbb P$-contractions, while linking these objects to the well-studied $\mathbb B_2$-contractions and $\Gamma$-contractions. A central contribution is a dilation theory for $\mathbb P$-contractions: a necessary-and-sufficient condition for dilating to a $\mathbb P$-isometry with the last component $P$ dilated minimally, together with an explicit construction using an auxiliary operator equation $I - A^*A - \frac14 S^*S = D_P X D_P$ and its unique solution with $\omega(X)\le 1$. The paper also clarifies the interplay among operator theory on the pentablock, the biball, and the symmetrized bidisc, and discusses the implications for rational dilation on the pentablock, drawing on $\Gamma$-dilation techniques. Overall, it advances understanding of multivariable spectral sets and their dilations in domains beyond the bidisc and ball.
Abstract
A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq \mathbb{C}^3. \] A commuting triple of normal operators $(A, S, P)$ acting on a Hilbert space is said to be a \textit{$\mathbb P$-unitary} if the Taylor-joint spectrum $σ_T(A, S, P)$ of $(A, S, P)$ is contained in the distinguished boundary $b\mathbb{P}$ of $\PC$. Also, $(A, S , P)$ is called a \textit{$\mathbb P$-isometry} if it is the restriction of a $\mathbb P$-unitary $(\hat A, \hat S, \hat P)$ to a joint invariant subspace of $\hat A, \hat S, \hat P$. We find several characterizations for the $\mathbb P$-unitaries and $\mathbb P$-isometries. We show that every $\mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $\mathbb P$-unitary and a pure $\mathbb P$-isometry. Moving one step ahead we show that every $\mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $\mathbb P$-unitary and a completely non-unitary $\mathbb P$-contraction. We find a necessary and sufficient condition such that a $\mathbb P$-contraction $(A, S, P)$ dilates to a $\mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. We show interplay between operator theory on the following three domains: the pentablock, the biball and the symmetrized bidisc.