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Maximal subspaces of strong continuity for composition semigroups

Nikolaos Chalmoukis, Álvaro Miguel Moreno

Abstract

Let $(\varphi_t)_{t\geq 0} $ a semigroup of holomorphic self-maps of the unit disk and $C_t f = f \circ \varphi_t $ the semigroup of composition operators which corresponds to $\varphi_t. $ Given a non-separable Banach space of analytic functions $X $ we study the properties of the maximal subspace of $X $ on which the semigroup $C_t $ is strongly continuous. In particular when $X $ contains the polynomials an interesting question is for which semigroups the maximal subspace of strong continuity coincides with the norm closure of the polynomials. This problem has been investigated in several function spaces including $BMOA$, $BMOA_p $, the Bloch space, $Q_s $ space and analytic Morrey spaces. However, in most cases only partial results are available. We offer a unified approach to this problem which encompasses all of the above spaces as particular examples. Moreover, we completely characterize the semigroups for which the maximal subspace of strong continuity coincides with the norm-closure of the polynomials in the space, giving therefore sharp versions of a number of results in the literature.

Maximal subspaces of strong continuity for composition semigroups

Abstract

Let a semigroup of holomorphic self-maps of the unit disk and the semigroup of composition operators which corresponds to Given a non-separable Banach space of analytic functions we study the properties of the maximal subspace of on which the semigroup is strongly continuous. In particular when contains the polynomials an interesting question is for which semigroups the maximal subspace of strong continuity coincides with the norm closure of the polynomials. This problem has been investigated in several function spaces including , , the Bloch space, space and analytic Morrey spaces. However, in most cases only partial results are available. We offer a unified approach to this problem which encompasses all of the above spaces as particular examples. Moreover, we completely characterize the semigroups for which the maximal subspace of strong continuity coincides with the norm-closure of the polynomials in the space, giving therefore sharp versions of a number of results in the literature.
Paper Structure (10 sections, 34 theorems, 157 equations, 2 figures)

This paper contains 10 sections, 34 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Basics on $M_{\alpha}(D^{p}_{s})$ and Composition operators
  4. Univalent functions
  5. Integration operator and related results
  6. Boundedness and compactness
  7. A technical Lemma
  8. Main results on semigroups
  9. First properties of $[\varphi_t,M_\alpha(D^p_s)]$
  10. Characterize $[\varphi_t,M_{\alpha}(D^{p}_{s})]=m_{\alpha}(D^{p}_{s})$

Key Result

Theorem A

ChalDask Let $(\varphi_t)$ and $G$ as above. The following are equivalent

Figures (2)

  • Figure 1: The family of spaces $D^p_s$. The blue region is the admissible region of the exponents $p, s.$
  • Figure 2: The family of spaces $M_0(D^p_s)$. The blue region is the admissible region for the parameters $p, s$.

Theorems & Definitions (52)

  • Theorem A
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma B
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • ...and 42 more