2022/07/07 by Joseph A. Ball, Ball, Joseph A., Haripada Sau +1 · 3 citations
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2207.03229
openalex publication_date 2022/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator T can be dilated to a unitary \cU. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain Ω contained in \mathbb Cd, (ii) the contraction operator T is replaced by a commuting tuple \bfT = (T1, …, Td) such that ‖ r(T1, …, Td) ‖\cL(\cH) ≤ sup\lam ∈ Ω | r(\lam) | for all rational functions with no singularities in Ω and the unitary operator \cU is replaced by an Ω-unitary operator tuple, i.e., a commutative operator d-tuple \bfU = (U1, …, Ud) of commuting normal operators with joint spectrum contained in the distinguished boundary bΩ of Ω. For a given domain Ω⊂ \mathbb Cd, the \em rational dilation question asks: given an Ω-contraction \bfT on \cH, is it always possible to find an Ω-unitary \bfU on a larger Hilbert space \cK ⊃ \cH so that, for any d-variable rational function without singularities in Ω, one can recover r(T) as r(T) = P_\cH r(\bfU)|_\cH. We focus here on the case where Ω is the \em tetrablock. (i) We identify a complete set of unitary invariants for a \mathbb 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 \em pseudo-commutative \mathbb E-isometries (a priori slightly larger than the class of \mathbb E-isometries) to which any \mathbb 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-isometric lift (V1, V2, V3) of a special type for a \mathbb E-contraction (A,B,T).