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Hypercyclicity of Toeplitz operators with smooth symbols

Emmanuel Fricain, Sophie Grivaux, Maëva Ostermann

Abstract

This paper is devoted to the study of the dynamics of Toeplitz operators $T_F$ with smooth symbols $F$ on the Hardy spaces of the unit disk $H^p$, $p>1$. Building on a model theory for Toeplitz operators on $H^2$ developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for $T_F$ to be hypercyclic on $H^p$. In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators.

Hypercyclicity of Toeplitz operators with smooth symbols

Abstract

This paper is devoted to the study of the dynamics of Toeplitz operators with smooth symbols on the Hardy spaces of the unit disk , . Building on a model theory for Toeplitz operators on developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for to be hypercyclic on . In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators.

Paper Structure

This paper contains 67 sections, 90 theorems, 465 equations, 44 figures.

Table of Contents

  1. Introduction and main results
  2. Model theory for Toeplitz operators on $H^p$ spaces with smooth symbols
  3. Some spectral theory
  4. Eigenvectors
  5. Interior and exterior boundary values
  6. Boundary conditions in some Smirnov spaces
  7. The main result of Yakubovich
  8. The functional calculus
  9. Spanning eigenvectors
  10. Necessary conditions for hypercyclicity of Toeplitz operators
  11. A condition on the winding number
  12. Intersection with connected components of the interior of the spectrum
  13. Proving the hypercyclicity of $T_F$: from a connected component to another
  14. Example of two circles
  15. From the interior to the exterior
  16. ...and 52 more sections

Key Result

Theorem 2.2

The operator $U$ defined above is a linear isomorphism from $H^q$ onto $E_F^q$. Moreover, we have In other words, the following diagram commutes:

Figures (44)

  • Figure 6:
  • Figure 7:
  • Figure 8:
  • Figure 11:
  • Figure 12:
  • ...and 39 more figures

Theorems & Definitions (169)

  • Theorem 2.2
  • Corollary 2.3: Yakubovich Yakubovich1991
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • ...and 159 more