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 ℓ-contraction if the closed pentablock ℓ is a spectral set for (A,S,P), where (Formula presented). A commuting triple of normal operators (A,S,P) acting on a Hilbert space is said to be a ℓ-unitary if the joint spectrum σ_T(A,S,P) of (A,S,P) is contained in the distinguished boundary bℓ of ℓ. Also, (A,S,P) is called a ℓ-isometry if it is the restriction of a ℓ-unitary (Ã,Š,P̃) to a joint invariant subspace of Ã,Š,P̃. We find several characterizations for the ℓ-unitaries and ℓ-isometries. We show that every ℓ-isometry admits a Wold type decomposition that splits it into a direct sum of a ℓ-unitary and a pure ℓ-isometry. Moving one step ahead we show that every ℓ-contraction (A,S,P) possesses a canonical decomposition that orthogonally decomposes (A,S,P) into a ℓ-unitary and a completely non-unitary ℓ-contraction. We find a necessary and sufficient condition such that a ℓ-contraction (A,S,P) dilates to a ℓ-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.