Closedness and invertibility for the sum of two closed operators

Nikolaos Roidos

Published 2016-02-14Version 1

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 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 employ this result in combination with a resolvent construction technique, and recover a classical result on the existence, uniqueness and maximal $L^{p}$-regularity of solution for the abstract non-autonomous linear parabolic problem.