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Embedding of some classes of operators into strongly continuous semigroups

Isabelle Chalendar, Romain Lebreton

TL;DR

The paper investigates when bounded operators can embed into $C_0$-semigroups, using Eisner's necessary condition to focus on cases with rich structure. It develops embedding criteria for isometric operators on the Hardy space $H^2$, with detailed results for composition operators and analytic Toeplitz operators, including explicit semigroup realizations in several inner-function and Blaschke product scenarios. Key contributions include conditions under which $C_\varphi$ and $T_\varphi$ are embeddable, constructions of semigroups that realize these embeddings (including weighted and non-composition semigroups), and structural consequences such as the preservation of isometry across the semigroup and the impossibility of compactness for $t>0$. These results clarify when natural operator classes on $H^2$ admit semigroup embeddings and delineate obstructions arising from injectivity or finite Blaschke structure, advancing the practical realization of semigroups for operator theory on Hardy spaces.

Abstract

In this paper we study the embedding problem of an operator into a strongly continuous semigroup. We obtain characterizations for some classes of operators, namely composition operators and analytic Toeplitz operators on the Hardy space H^2. In particular, we focus on the isometric ones using the necessary and sufficient condition observed by T. Eisner.

Embedding of some classes of operators into strongly continuous semigroups

TL;DR

The paper investigates when bounded operators can embed into -semigroups, using Eisner's necessary condition to focus on cases with rich structure. It develops embedding criteria for isometric operators on the Hardy space , with detailed results for composition operators and analytic Toeplitz operators, including explicit semigroup realizations in several inner-function and Blaschke product scenarios. Key contributions include conditions under which and are embeddable, constructions of semigroups that realize these embeddings (including weighted and non-composition semigroups), and structural consequences such as the preservation of isometry across the semigroup and the impossibility of compactness for . These results clarify when natural operator classes on admit semigroup embeddings and delineate obstructions arising from injectivity or finite Blaschke structure, advancing the practical realization of semigroups for operator theory on Hardy spaces.

Abstract

In this paper we study the embedding problem of an operator into a strongly continuous semigroup. We obtain characterizations for some classes of operators, namely composition operators and analytic Toeplitz operators on the Hardy space H^2. In particular, we focus on the isometric ones using the necessary and sufficient condition observed by T. Eisner.

Paper Structure

This paper contains 9 sections, 13 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Strongly continuous semigroups of operators
  4. Analytic semiflows on D
  5. Hardy spaces and Blaschke products
  6. Embedding results
  7. Composition operators on H2
  8. Analytic Toeplitz operators on H2
  9. Isometric operators and properties of semigroups

Key Result

Theorem 1.1

Let $V : H \rightarrow H$ be an isometry on a Hilbert space $H$. Then $V$ is embeddable into a $C_0$-semigroup on $H$ if and only if $V$ is unitary or $\text{codim}(VH) = \infty$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 18 more