Dilation Theory and Functional Models for Tetrablock Contractions

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator T can be dilated to a unitary U $${{\mathcal {U}}}$$ , i.e., T n = P H U n | H $$T^n = P_{{\mathcal {H}}}{{\mathcal {U}}}^n|{{\mathcal {H}}}$$ for all n = 0 , 1 , 2 , … $$n =0,1,2,\ldots $$ . A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain Ω $$\Omega $$ contained in C d $${{\mathbb {C}}}^d$$ , (ii) the contraction operator T is replaced by an $$\Omega $$ Ω -contraction, i.e., a commutative operator d-tuple $${{\textbf{T}}}= (T_1, \ldots , T_d)$$ T = ( T 1 , … , T d ) on a Hilbert space $${{\mathcal {H}}}$$ H such that $$\Vert r(T_1, \ldots , T_d) \Vert _{{{\mathcal {L}}}({{\mathcal {H}}})} \le \sup _{\lambda \in \Omega } | r(\lambda ) |$$ ‖ r ( T 1 , … , T d ) ‖ L ( H ) ≤ sup λ ∈ Ω | r ( λ ) | for all rational functions with no singularities in $$\overline{\Omega }$$ Ω ¯ and the unitary operator $${{\mathcal {U}}}$$ U is replaced by an $$\Omega $$ Ω -unitary operator tuple, i.e., a commutative operator d-tuple $${{\textbf{U}}}= (U_1, \ldots , U_d)$$ U = ( U 1 , … , U d ) of commuting normal operators with joint spectrum contained in the distinguished boundary $$b\Omega $$ b Ω of $$\Omega $$ Ω . For a given domain $$\Omega \subset {\mathbb C}^d$$ Ω ⊂ C d , the rational dilation question asks: given an $$\Omega $$ Ω -contraction $${{\textbf{T}}}$$ T on $${{\mathcal {H}}}$$ H , is it always possible to find an $$\Omega $$ Ω -unitary $${{\textbf{U}}}$$ U on a larger Hilbert space $${{\mathcal {K}}}\supset {{\mathcal {H}}}$$ K ⊃ H so that, for any d-variable rational function without singularities in $${\overline{\Omega }}$$ Ω ¯ , one can recover r(T) as $$r(T) = P_{{\mathcal {H}}}r({{\textbf{U}}})|_{{\mathcal {H}}}$$ r ( T ) = P H r ( U ) | H . We focus here on the case where $$\Omega = {{\mathbb {E}}}$$ Ω = E , a domain in $${{\mathbb {C}}}^3$$ C 3 called the tetrablock. (i) We identify a complete set of unitary invariants for a $${{\mathbb {E}}}$$ E -contraction (A, B, T) which can then be used to write down a functional model for (A, B, T), thereby extending earlier results only done for a special case, (ii) we identify the class of pseudo-commutative $${{\mathbb {E}}}$$ E -isometries (a priori slightly larger than the class of $${{\mathbb {E}}}$$ E -isometries) to which any $${{\mathbb {E}}}$$ E -contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a $${{\mathbb {E}}}$$ E -isometric lift $$(V_1, V_2, V_3)$$ ( V 1 , V 2 , V 3 ) of a special type for a $${{\mathbb {E}}}$$ E -contraction (A, B, T).