On sequences of sectorial forms converging `from above'

TL;DR

This paper proves a convergence theorem for sequences of sectorial forms and their semigroups in complex Hilbert spaces without requiring monotonicity or closedness, broadening the scope of previous results.

Contribution

It introduces a new form convergence theorem for sectorial forms that do not need to be monotonic, closed, or densely defined, expanding the applicability of form convergence analysis.

Findings

Strong resolvent convergence of linear relations

Convergence of associated semigroups

Applications to numerical analysis methods

Abstract

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.