Convergence of sectorial operators on varying Hilbert space

TL;DR

This paper introduces a new notion of convergence for sectorial operators on varying Hilbert spaces, extending classical resolvent convergence, and demonstrates its compatibility with functional calculus and spectral convergence, with applications to manifold limits.

Contribution

It defines a novel convergence concept for sectorial operators on different Hilbert spaces, linking resolvent convergence to spectral and functional calculus convergence.

Findings

The new convergence generalizes classical resolvent convergence.

It is compatible with the functional calculus of operators.

Applications include convergence in parabolic problems and manifold limits.

Abstract

Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.