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Characteristic function of a power partial isometry

Kritika Babbar, Amit Maji

Abstract

The celebrated Sz.-Nagy-Foiaş model theory says that there is a bijection between the class of purely contractive analytic functions and the class of completely non-unitary (c.n.u.) contractions modulo unitary equivalence. In this paper we provide a complete classification of the purely contractive analytic functions such that the associated contraction is a c.n.u. power partial isometry. As an application of our findings, we determine a class of contractive polynomials such that the associated c.n.u. power partial isometry is of the explicit diagonal form $S \oplus N \oplus C$, where $S$ and $C^*$ are unilateral shifts and $N$ is nilpotent. Finally, we obtain a characterization of operator-valued symbols for which the corresponding Toeplitz operator on vector-valued Hardy space is a partial isometry.

Characteristic function of a power partial isometry

Abstract

The celebrated Sz.-Nagy-Foiaş model theory says that there is a bijection between the class of purely contractive analytic functions and the class of completely non-unitary (c.n.u.) contractions modulo unitary equivalence. In this paper we provide a complete classification of the purely contractive analytic functions such that the associated contraction is a c.n.u. power partial isometry. As an application of our findings, we determine a class of contractive polynomials such that the associated c.n.u. power partial isometry is of the explicit diagonal form , where and are unilateral shifts and is nilpotent. Finally, we obtain a characterization of operator-valued symbols for which the corresponding Toeplitz operator on vector-valued Hardy space is a partial isometry.
Paper Structure (4 sections, 16 theorems, 160 equations)

This paper contains 4 sections, 16 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partially isometric Toeplitz operators
  4. Examples

Key Result

Theorem 1.1

Let $\Theta : \mathbb{D} \rightarrow \mathcal{B}(\mathcal{E},\mathcal{E}_*)$ be a purely contractive analytic function such that where $\theta_m \in \mathcal{B}(\mathcal{E},\mathcal{E}_*)$ are partial isometries for all $m \geq 1$. Then there exist a Hilbert space where $\Delta_\Theta$ is a constant projection, and a c.n.u. power partial isometry $T$ on $\mathcal{H}$ defined by such that the ch

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: cf. halmospowers
  • Theorem 2.2
  • Definition 2.3: Contractive analytic function
  • Definition 2.4: Inner function
  • Definition 2.5: Characteristic function
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 26 more