A cheap way to closed operator sums

TL;DR

This work develops a concise, unified method to establish the closedness of sums of sectorial operators in Banach spaces when resolvents commute and $\omega_A+\omega_B<\pi$. Central to the approach is a Littlewood–Paley type norm framework and an auxiliary operator $S$ that reduces the problem to proving boundedness of $AS$ (or $BS$); this yields streamlined proofs of the Da Prato–Grisvard and Kalton–Weis results and extends to $\ell^q$-interpolation via generalized Triebel–Lizorkin spaces, with applications to maximal regularity for abstract parabolic equations. The paper also provides a variant proof of the Dore–Venni result by aligning complex interpolation with the real-type square-function viewpoint and discusses duality aspects. Together, these results illuminate a common interpolation-norm structure underlying operator-sum closedness across real, $\ell^q$, and complex interpolation theories. The practical upshot includes new maximal-regularity insights and a broader, cohesive toolbox for analyzing linear parabolic problems in diverse Banach spaces.

Abstract

Let $A$ and $B$ be sectorial operators in a Banach space $X$ of angles $ω_A$ and $ω_B$, respectively, where $ω_A+ω_B<π$. 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.

Theorems & Definitions (50)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 2.6