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Composition operators between Toeplitz kernels

Yuxia Liang, Jonathan R. Partington

TL;DR

This work determines the minimal Toeplitz kernel that contains the image of a Toeplitz kernel under a composition operator with an inner symbol, proving it is $\ker T_{(F\circ\psi)\psi/z}$ and describing how maximal vectors transform under composition. It then extends the analysis to weighted composition operators, establishing explicit minimal kernels for images like $C_\psi(u\ker T_F)$ and $uC_\psi(\ker T_F)$ with symbols $\psi(F\circ\psi)\overline{u}/(z u_o)$, and provides criteria when such images coincide with the minimal kernel. The results also connect to model spaces, giving a concise proof of known containment theorems and showing how maximal vectors are transferred through composition, inner/outer factorizations, and multipliers. Finally, the paper characterizes maximal vectors for kernels with symbols formed via composition, offering explicit constructions for model spaces and highlighting that sums of Toeplitz kernels can fail to be nearly $S^*$-invariant. These contributions deepen the understanding of Toeplitz kernels under composition and weighted composition operators, with implications for near-invariance and model-space theory.

Abstract

Recently, it was shown that the image of a Toeplitz kernel of dimension greater than $1$ under composition by an inner function is nearly $S^*$-invariant if and only if the inner function is an automorphism. Building on this, we determine the minimal Toeplitz kernel containing the image of a Toeplitz kernel under a composition operator with a general inner symbol, and extend this to weighted composition operators. Specifically, the corresponding cases for minimal model spaces are also given, thereby extending known work on the action of composition operators on model spaces. Finally, we use the equivalences between Toeplitz kernels to derive the explicit maximal vectors for several Toeplitz kernels, with symbols expressed in terms of composition operators and inner functions.

Composition operators between Toeplitz kernels

TL;DR

This work determines the minimal Toeplitz kernel that contains the image of a Toeplitz kernel under a composition operator with an inner symbol, proving it is and describing how maximal vectors transform under composition. It then extends the analysis to weighted composition operators, establishing explicit minimal kernels for images like and with symbols , and provides criteria when such images coincide with the minimal kernel. The results also connect to model spaces, giving a concise proof of known containment theorems and showing how maximal vectors are transferred through composition, inner/outer factorizations, and multipliers. Finally, the paper characterizes maximal vectors for kernels with symbols formed via composition, offering explicit constructions for model spaces and highlighting that sums of Toeplitz kernels can fail to be nearly -invariant. These contributions deepen the understanding of Toeplitz kernels under composition and weighted composition operators, with implications for near-invariance and model-space theory.

Abstract

Recently, it was shown that the image of a Toeplitz kernel of dimension greater than under composition by an inner function is nearly -invariant if and only if the inner function is an automorphism. Building on this, we determine the minimal Toeplitz kernel containing the image of a Toeplitz kernel under a composition operator with a general inner symbol, and extend this to weighted composition operators. Specifically, the corresponding cases for minimal model spaces are also given, thereby extending known work on the action of composition operators on model spaces. Finally, we use the equivalences between Toeplitz kernels to derive the explicit maximal vectors for several Toeplitz kernels, with symbols expressed in terms of composition operators and inner functions.
Paper Structure (8 sections, 39 theorems, 83 equations)

This paper contains 8 sections, 39 theorems, 83 equations.

Table of Contents

  1. Preliminaries
  2. Minimal Toeplitz kernel containing $C_\psi(\mathop{\rm ker}\nolimits T_F)$ for an inner $\psi$
  3. The minimal Toeplitz kernel containg $C_\psi (\mathop{\rm ker}\nolimits T_F)$.
  4. Minimal Toeplitz kernel containing $C_\psi(u\mathop{\rm ker}\nolimits T_F)$ and $uC_\psi(\mathop{\rm ker}\nolimits T_F)$
  5. The conditions for $u\mathop{\rm ker}\nolimits T_F =\mathop{\rm ker}\nolimits T_{F\overline{u}/u_o}$
  6. The minimal Toeplitz kernel containing $C_\psi(u\mathop{\rm ker}\nolimits T_F)$
  7. The minimal Toeplitz kernel containing $uC_\psi(\mathop{\rm ker}\nolimits T_F)$
  8. Maximal vectors for Toeplitz kernels with compositional inner symbols

Key Result

Theorem 1.1

hitt The nearly $S^*$-invariant subspaces of $H^2$ have the form $\mathcal{M}=uK$, with $u\in \mathcal{M}$ of unit norm, $u(0)>0,$ and $u$ orthogonal to all elements of $\mathcal{M}$ vanishing at the origin, $K$ an $S^*$-invariant subspace, and the operator of multiplication by $u$ isometric from $K

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • ...and 57 more