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 \($\mathbb{P}\)-contraction if the closed pentablock \(\overline{\mathbb P}\) is a spectral set for \((A,S,P)\), where \(\mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\colon A_0=[a_{ij}]_{2 \times 2}\) and \(\|A_0\| <1 \right\} \subseteq \mathbb{C}^3.\) A commuting triple of normal operators \((A, S, P)\) acting on a Hilbert space is said to be a \(\mathbb P\)-unitary if the joint spectrum \(\sigma_T(A, S, P)\) of \((A, S, P)\) is contained in the distinguished boundary \(b\mathbb{P}\) of \(\overline{\mathbb{P}}\). Also, \((A, S , P)\) is called a \(\mathbb P\)-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.