Composition Semigroups on Weighted Bergman Spaces Induced by Doubling Weights

Fanglei Wu,Dimitri Mugnai

doi:10.1155/2021/6613928

Abstract

We prove that composition semigroups are strongly continuous on weighted Bergman spaces with doubling weights. Point spectra and compact resolvent operators of infinitesimal generators of composition semigroups are characterized.

Highlights

Let H(D) denote the space of analytic functions in the unit disc D {z ∈ C: |z| < 1}
For 0 < p < ∞ and a radial weight ω, the weighted Bergman space Apω consists of f ∈ H(D) such that
Apα stands for the classical weighted Bergman space induced by the standard radial weight ω(z) (1 − |z|2)α, where − 1 < α < ∞

Summary

Introduction

Let H(D) denote the space of analytic functions in the unit disc D {z ∈ C: |z| < 1}. A semigroup (φt)t ≥ 0 is said to be trivial if each φt is the identity of D. e infinitesimal generator of (φt)t ≥ 0 is defined as the function. Notice that each semigroup (φt)t ≥ 0 gives rise to a semigroup (Ct)t ≥ 0 consisting of composition operators on H(D), the set of analytic functions on D, where. Given a semigroup (φt)t ≥ 0 and a Banach space X of analytic functions on D, we say that (φt)t ≥ 0 generates a strongly continuous composition operator on X if Ct is bounded on X and lim t⟶0+. Journal of Mathematics (Ct)t ≥ 0 is called uniformly continuous on X if limt⟶0+ ‖Ct − I‖ 0, where I is the identity map on X. e infinitesimal generator of a strongly continuous semigroup (Ct)t ≥ 0 on a Banach space X is the operator. We say that A≲B if there exists a constant C such that A ≤ CB

Strongly Continuous Composition Semigroup

Resolvent Operator