  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.
