Closedness and invertibility for the sum of two closed operators

TL;DR

This paper extends a classical theorem to non-commuting operators in Banach spaces, showing conditions under which their sum is closed, invertible, and sectorial, with applications to linear parabolic problems.

Contribution

It generalizes the Kalton-Weis theorem to non-commuting operators, establishing closedness and invertibility criteria under weaker resolvent conditions.

Findings

Sum of operators is closed under specified conditions.

Sum becomes invertible and sectorial after a large shift.

Application to maximal regularity in parabolic problems.

Abstract

We show a Kalton-Weis type theorem for the general case of non-commuting operators. More precisely, we consider sums of two possibly non-commuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible, and moreover sectorial. As an application we recover a classical result on the existence, uniqueness and maximal $L^{p}$-regularity for solutions of the abstract linear non-autonomous parabolic problem.