Weak and strong approximation of semigroups on Hilbert spaces

TL;DR

This paper investigates different modes of convergence for sequences of bounded semigroups on Hilbert spaces, establishing equivalences among various topological convergence types under common assumptions.

Contribution

It provides a comprehensive comparison of weak and strong convergence modes for semigroups and their generators, clarifying their relationships in Hilbert space settings.

Findings

Convergence of semigroups in weak operator topology is equivalent to convergence in strong operator topology.

Resolvent convergence in various topologies is equivalent to semigroup convergence.

Results apply under natural, frequently met assumptions in applications.

Abstract

For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators, in the weak operator topology, in the strong operator topology or in certain integral norms are equivalent under certain natural assumptions which are frequently met in applications.