  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
  \[
  \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \
A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq
\mathbb{C}^3.
  \]
  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. We also 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. A commuting tuple of
Hilbert space operators $(T_1, \dots , T_n)$ having the closed unit ball
$\overline{\mathbb B}_n$ as a spectral set is called a $\mathbb
B_n$-\textit{contraction} and a commuting pair $(S,P)$ having the closed
symmetrized bidisc $\overline{\mathbb G_2} \,(=\Gamma)$ as a spectral set is
called a $\Gamma$-\textit{contraction}. We characterize isometries and
unitaries associated with $\mathbb B_n$. Then we present an analogous canonical
decomposition for a $\mathbb B_n$-contraction. We show interplay between
operator theory on the three domains $\mathbb P, \mathbb B_2$ and $\mathbb
G_2$.
