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Quantitative cyclicity, stability, and geometric analysis in weighted Besov spaces

Saeed Hashemi Sababe, Amir Baghban

Abstract

We introduce new quantitative measures for cyclicity in radially weighted Besov spaces, including the Drury-Arveson space, by defining cyclicity indices based on potential theory and capacity. Extensions to non-commutative settings are developed, yielding analogues of cyclicity in free function spaces. We also study the stability of cyclic functions under perturbations of both the functions and the underlying weight, and we establish geometric criteria linking the structure of zero sets on the boundary to the failure or persistence of cyclicity. These results provide novel invariants and conditions that characterize cyclicity and the structure of multiplier invariant subspaces in a variety of function spaces.

Quantitative cyclicity, stability, and geometric analysis in weighted Besov spaces

Abstract

We introduce new quantitative measures for cyclicity in radially weighted Besov spaces, including the Drury-Arveson space, by defining cyclicity indices based on potential theory and capacity. Extensions to non-commutative settings are developed, yielding analogues of cyclicity in free function spaces. We also study the stability of cyclic functions under perturbations of both the functions and the underlying weight, and we establish geometric criteria linking the structure of zero sets on the boundary to the failure or persistence of cyclicity. These results provide novel invariants and conditions that characterize cyclicity and the structure of multiplier invariant subspaces in a variety of function spaces.

Paper Structure

This paper contains 8 sections, 21 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantitative Measures of Cyclicity
  4. Extensions to Non-Commutative and Free Function Settings
  5. Stability and Perturbation of Cyclic Functions
  6. Geometric Analysis of Zero Sets and Cyclicity
  7. Mixed Norm and Variable Exponent Extensions
  8. Concluding Remarks and Open Problems

Key Result

Lemma 2.8

Let $B^k_\omega$ denote a radially weighted Besov space. Then, for each $k\in\mathbb{N}$, and the inclusion is contractive: for all $\varphi\in \operatorname{Mult}(B^k_\omega)$. (See AlemanPerfRichSund2023.)

Theorems & Definitions (62)

  • Definition 2.1: Hilbert Function Space
  • Definition 2.2: Admissible Radial Measure and Weighted Besov Spaces
  • Definition 2.3: Drury–Arveson Space
  • Definition 2.4: Multiplier Algebra
  • Definition 2.5: Cyclic Function and Invariant Subspace
  • Definition 2.6: Stable Polynomials
  • Definition 2.7: Classes $C_n(H)$
  • Lemma 2.8: Multiplier Inclusion Condition
  • Lemma 2.9: Basic Properties of Cyclic Multipliers
  • Theorem 2.10: One-Function Corona Theorem for $B^N_\omega$
  • ...and 52 more