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Dilation and Model Theory for Pairs of Contractions with a Twisted Commutation Relation

Sourav Ghosh

TL;DR

$(T_1,T_2)$ is a $q$-commuting pair of contractions with product $T=T_1T_2=qT_2T_1$. The paper develops two constructive dilation-model schemes (Schäffer-type and Douglas-type) to realize isometric lifts and unitary dilations that respect the $q$-twist, and then builds Sz.-Nagy–Foias–type functional models for the pair. Central to this theory are the fundamental operators $(G_1,G_2)$, the canonical $q$-commuting unitary pair $(W_1,W_2)$, and the characteristic function $ heta_T$, which together form a characteristic triple that serves as a complete unitary invariant for pairs whose product is completely non-unitary. A key outcome is a NF-type functional model that expresses $(T_1,T_2)$ as a compressed model on the Hilbert space $ ext{H}_{ heta_T}$, parameterized by the characteristic triple, thereby unifying dilation, canonical unitary structure, and an explicit analytic model. The results culminate in a robust framework for q-commuting contractions, including the notion of admissible triples and pseudo $q$-commuting lifts, enabling precise classification and functional realization in analogy with the classic Nagy–Foias theory.

Abstract

In this note, we develop a parallel theory of the classical Sz.-Nagy--Foias dilation and model theory for a single contraction operator in the setting of pairs of \em{$q$-commuting} contraction operators for a unimodular complex number $q$.

Dilation and Model Theory for Pairs of Contractions with a Twisted Commutation Relation

TL;DR

is a -commuting pair of contractions with product . The paper develops two constructive dilation-model schemes (Schäffer-type and Douglas-type) to realize isometric lifts and unitary dilations that respect the -twist, and then builds Sz.-Nagy–Foias–type functional models for the pair. Central to this theory are the fundamental operators , the canonical -commuting unitary pair , and the characteristic function , which together form a characteristic triple that serves as a complete unitary invariant for pairs whose product is completely non-unitary. A key outcome is a NF-type functional model that expresses as a compressed model on the Hilbert space , parameterized by the characteristic triple, thereby unifying dilation, canonical unitary structure, and an explicit analytic model. The results culminate in a robust framework for q-commuting contractions, including the notion of admissible triples and pseudo -commuting lifts, enabling precise classification and functional realization in analogy with the classic Nagy–Foias theory.

Abstract

In this note, we develop a parallel theory of the classical Sz.-Nagy--Foias dilation and model theory for a single contraction operator in the setting of pairs of \em{-commuting} contraction operators for a unimodular complex number .

Paper Structure

This paper contains 13 sections, 21 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Models for q-commuting isometries
  4. A Schäffer-type model for isometric lifts
  5. A Douglas-type model for isometric lifts
  6. Canonical $q$-commuting unitary pair
  7. A Douglas-type model for isometric lifts
  8. A Sz.-Nagy--Foias-type model for isometric lifts
  9. A Sz.-Nagy--Foias-type functional model
  10. The fundamental operators
  11. Pseudo $q$-commuting contractive lift
  12. The characteristic triple
  13. Functional model

Key Result

Lemma 2.3

Let $(T_1,T_2)$ be a $q$-commuting pair of contractions on ${\mathcal{H}}$, and $({\mathcal{F}};\Lambda,P,U)$ and $({\mathcal{F}}_*;\Lambda_*,P_*,U_*)$ be the special Andô tuples for $(T_1,T_2)$ and $(T_1^*,T_2^*)$, respectively. Then we have and

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • ...and 37 more