Preserving closedness of operators under summation

TL;DR

This paper establishes conditions under which the sum of two closed operators remains closed, focusing on sectorial operators and their applications to parabolic problems and maximal regularity.

Contribution

It provides a new sufficient condition ensuring the closedness of the sum of sectorial operators, extending previous results and applying to $L^{p}$-maximal regularity.

Findings

Sum of two sectorial operators is closed under specified conditions.

Bounded $H^{ ext{infinity}}$-calculus plays a key role.

Application to abstract parabolic problems and $L^{p}$-maximal regularity.

Abstract

We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than $\pi$. We show that if one of the operators admits bounded $H^{\infty}$-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for $L^{p}$-maximal regularity.