A cheap way to closed operator sums

TL;DR

This paper introduces a simple, unified approach using Littlewood-Paley norms and interpolation theory to analyze the closedness of sums of sectorial operators, providing new proofs and results in operator theory and maximal regularity.

Contribution

It offers a novel, straightforward method for proving closedness of operator sums and extends results to new interpolation spaces, simplifying existing proofs and deriving new theorems.

Findings

Unified approach to closedness of operator sums

New result in ll^q-interpolation spaces

Maximal regularity for abstract parabolic equations

Abstract

Let $A$ and $B$ be sectorial operators in a Banach space $X$ of angles $\omega_A$ and $\omega_B$, respectively, where $\omega_A+\omega_B<\pi$. We present a simple and common approach to results on closedness of the operator sum $A+B$, based on Littlewood-Paley type norms and tools from several interpolation theories. This allows us to give short proofs for the well-known results due to Da~Prato-Grisvard and Kalton-Weis. We prove a new result in $\ell^q$-interpolation spaces and illustrate it with a maximal regularity result for abstract parabolic equations. Our approach also yields a new proof for the Dore-Venni result.