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