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Joint reducing subspaces and orthogonal decompositions of operators in an annulus

Sourav Pal, Nitin Tomar

TL;DR

The paper develops a comprehensive framework for joint reducing subspaces and orthogonal decompositions of commuting tuples of $\mathbb{A}_r$-contractions on Hilbert spaces. It charters $\mathbb{A}_r$-unitaries and $\mathbb{A}_r$-isometries, proves a Wold type decomposition for $\mathbb{A}_r$-isometries, and establishes canonical decompositions for $\mathbb{A}_r$-contractions, extending these results to finite and infinite (doubly) commuting families. A key theme is translating classical operator theory concepts, such as unitary parts and completely non-unitary parts, to the annulus domain via $\mathbb{A}_r$-positivity and spectral-set considerations, including Brehmer-type positivity conditions. The work provides both a polyannulus-specific analogue of Levan and Burdak decompositions and a general combinatorial approach for infinite families, with implications for structure theory and spectral analysis of operator tuples on multiply connected domains.

Abstract

A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to be an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus \[ \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n \right\} \subseteq \mathbb{C}^n \qquad \quad (0<r<1) \] is a spectral set for $(T_1, \dotsc, T_n)$. We find characterizations for the $\mathbb A_r^n$-unitaries and $\mathbb A_r^n$-isometries and decipher their structures. We find Wold type decompositions for any number of commuting and doubly commuting $\mathbb A_r$-isometries. Then we generalize these results to any family of commuting and doubly commuting $\mathbb A_r$-contractions.

Joint reducing subspaces and orthogonal decompositions of operators in an annulus

TL;DR

The paper develops a comprehensive framework for joint reducing subspaces and orthogonal decompositions of commuting tuples of -contractions on Hilbert spaces. It charters -unitaries and -isometries, proves a Wold type decomposition for -isometries, and establishes canonical decompositions for -contractions, extending these results to finite and infinite (doubly) commuting families. A key theme is translating classical operator theory concepts, such as unitary parts and completely non-unitary parts, to the annulus domain via -positivity and spectral-set considerations, including Brehmer-type positivity conditions. The work provides both a polyannulus-specific analogue of Levan and Burdak decompositions and a general combinatorial approach for infinite families, with implications for structure theory and spectral analysis of operator tuples on multiply connected domains.

Abstract

A commuting tuple of Hilbert space operators is said to be an \textit{-contraction} if the closure of the polyannulus is a spectral set for . We find characterizations for the -unitaries and -isometries and decipher their structures. We find Wold type decompositions for any number of commuting and doubly commuting -isometries. Then we generalize these results to any family of commuting and doubly commuting -contractions.
Paper Structure (12 sections, 41 theorems, 103 equations)

This paper contains 12 sections, 41 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. A brief outline of the main results
  4. The Polyannulus unitaries
  5. The Polyannulus isometries
  6. Canonical decomposition of an $\mathbb{A}_r$-contraction
  7. Canonical decomposition of finitely many doubly commuting $\mathbb{A}_r$-contractions
  8. A different canonical decomposition for commuting $\mathbb{A}_r$-contractions
  9. Generalization
  10. Decomposition of an infinite family of doubly commuting $\mathbb{A}_r$-contractions
  11. Decomposition of an infinite family of doubly commuting c.n.u. $\mathbb{A}_r$-contractions
  12. Declarations

Key Result

Theorem 1.3

Let $V$ be an isometry on a Hilbert space $\mathcal{H}$. Then there is an orthogonal decomposition of $\mathcal{H}$ into an orthogonal sum of two subspaces reducing $V$, say $\mathcal{H}=\mathcal{H}_0 \oplus \mathcal{H}_1$ such that $V|_{\mathcal{H}_{0}}$ is a unitary and $V|_{\mathcal{H}_1}$ is a p

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: von Neumann-Wold, Wold
  • Theorem 1.4: Nagy-Foias-001 & Langer-1
  • Theorem 1.5: Levan, Theorem 1
  • Theorem 1.6: Sł ociński, Slocinski
  • Theorem 2.1: N-S1, Theorem 3.1
  • Proposition 2.2: N-S2, Lemma 3.4
  • Remark 2.3
  • Proposition 2.4
  • ...and 63 more