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The spectral analysis of the difference quotient operator on model spaces

Carlo Bellavita, Eugenio Alberto Dellepiane, Javad Mashreghi

Abstract

We conduct a spectral analysis of the difference quotient operator $Q^u_ζ$, associated with a boundary point $ζ\in \partial \mathbb{D}$, on the model space $K_u$. We describe the operator's spectrum and provide both upper and lower estimates for its norm, and furthermore discussing the sharpness of these bounds. Notably, the upper estimate offers a new characterization of the one-component property for inner function.

The spectral analysis of the difference quotient operator on model spaces

Abstract

We conduct a spectral analysis of the difference quotient operator , associated with a boundary point , on the model space . We describe the operator's spectrum and provide both upper and lower estimates for its norm, and furthermore discussing the sharpness of these bounds. Notably, the upper estimate offers a new characterization of the one-component property for inner function.

Paper Structure

This paper contains 9 sections, 11 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Some profound results about model spaces
  4. Spectrum of $\|\text{Q}_\zeta^u\|$
  5. A lower estimate for $\|\text{Q}_\zeta^u\|$
  6. Upper estimates for $\|\text{Q}_\zeta^u\|$
  7. Proof of Theorem \ref{['T:characterization']}
  8. An example
  9. Concluding remarks

Key Result

Theorem 2.1

Let $u$ be an inner function and $\zeta \in \mathbb{T} \setminus \sigma(u)$. Then Moreover, the lower estimate holds.

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.6
  • Theorem 2.8
  • Theorem 2.10
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.4: Aleksandrov Aleksandrov2000
  • Proposition 3.5: Bessonov bessonov2015duality
  • proof : Proof of Theorem \ref{['T:spectrumQ']}
  • Lemma 5.1
  • ...and 8 more