Operators associated with the pentablock and their relations with biball and symmetrized bidisc

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a $\mathbb{P}\mathit{\text{-contraction}}$ if the closed pentablock $\bar{\mathbb{P}}$ is a spectral set for $(A,S,P)$, where $$\mathbb{P}:=\{(a_{21},tr(A_0),det(A_0)):A_0=[a_{ij}{]}_{2\times 2}\mathrm{\&}\mathrm{\parallel }{A}_0\mathrm{\parallel }<1\}\subseteq {\mathbb{C}}^3\mathrm{.}$$ A commuting triple of normal operators $(A,S,P)$ acting on a Hilbert space is said to be a $\mathbb{P}\mathit{\text{-unitary}}$ if the Taylor-joint spectrum ${\sigma }_T(A,S,P)$ of $(A,S,P)$ is contained in the distinguished boundary $b\mathbb{P}$ of $\text{PC}$. Also, $(A,S,P)$ is called a $\mathbb{P}\mathit{\text{-isometry}}$ if it is the restriction of a $\mathbb{P}$-unitary $(\widehat{A},\widehat{S},\widehat{P})$ to a joint invariant subspace of $\widehat{A},\widehat{S},\widehat{P}$. 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.