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Negative powers of Hilbert-space contractions

Thomas Ransford

Abstract

We show that, given a closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a Hilbert-space contraction such that $σ(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ and rank$(I-T^*T)<\infty$, then $T$ is a unitary operator. We further show that the condition of measure zero is sharp.

Negative powers of Hilbert-space contractions

Abstract

We show that, given a closed subset of the unit circle of Lebesgue measure zero, there exists a positive sequence with the following property: if is a Hilbert-space contraction such that and and rank, then is a unitary operator. We further show that the condition of measure zero is sharp.
Paper Structure (11 sections, 31 theorems, 77 equations)

This paper contains 11 sections, 31 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Compressed shifts
  3. Basic results
  4. Norms of negative powers
  5. Decay of singular inner functions
  6. Return to Theorems \ref{['T:main']} and \ref{['T:converse']}
  7. Functional models
  8. Compressed shifts for operator-valued inner functions
  9. The model theorem
  10. Proof of Theorem \ref{['T:main']}
  11. Conclusion

Key Result

Theorem 1.1

Let $T$ be a Banach-space contraction such that $\sigma(T)$ is a countable subset of the unit circle $\mathbb{T}$ and Then $T$ is an isometry, i.e., $\|T\|=\|T^{-1}\|=1$.

Theorems & Definitions (60)

  • Theorem 1.1: Za93
  • Theorem 1.2: Es23
  • Conjecture 1.3
  • Theorem 1.4: Es94a
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 50 more