Sums of generators of analytic semigroups and multivalued linear operators (Q1817936)

The author proves the following theorem: Let \(A\) and \(B\) be generators of analytic semigroups in a Banach space. Assume that (i) \(A\) and \(B\) are linear operators in \(X\) with domain \(D_A\), \(D_B\) and there exists \(\theta\in\left(0,{\pi\over 2}\right)\), \(c>0\) such that \[ \|(A- z)^{-1}\|_{{\mathcal L}(X)}+\|(B- z)^{-1}\|_{{\mathcal L}(X)}\leq C|z|^{-1} \] for every \(z\in\mathbb{C}\) with \(|\text{arg}(z)|< \pi-\theta\); (ii) \((A-(ii)\) \((A- v)^{-1}D_B\subset D_B\) and there exist \(C>0\), \(\alpha\), \(\beta\) such that \(-1\leq\alpha< \beta\leq 1\), \[ \|[B;(A- v)^{-1}](B- z)^{-1}\|_{{\mathcal L}(X)}\leq C{1\over|v|^{1-\alpha}|z|^\beta} \] for every \(v,z\in\mathbb{C}\) with \(|\text{arg}(v)|\), \(|\text{arg}(z)|<\pi-\theta\), where \([P;Q]\) is the commutator \(PQ-QP\). Then \(A+B\) is closable and its closure \(\overline{A+B}\) generates a strongly continuous analytic semigroup.