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On the closedness of the sum of two closed operators. (English) Zbl 0615.47002
The main result of the paper is the following. Let $$X$$ be a $$\zeta$$- convex, complex Banach space and $$A: {\mathcal D}(A)\to X$$, $$B: {\mathcal D}(B)\to X$$ be densely defined closed linear operators in $$X$$ such that:
(i) the resolvent sets of $$A$$ and $$B$$ contain $${\mathbb{R}}^ -\cup \{0\}$$ and there exists $$M\geq 1$$ such that $$\max \{\| A+t)^{-1}\|,\| (B+t)^{-1}\| \}\leq M/(1+t)$$ $$\forall t\in {\mathbb{R}}^ +\cup \{0\};$$
(ii) the resolvent operators of $$A$$ and $$B$$ commute;
(iii) $$\forall s\in {\mathbb{R}}$$, $$A^{is}$$ and $$B^{is}$$ are bounded, the groups $$s\mapsto A^{is}$$, $$s\mapsto B^{is}$$ are strongly continuous and the following estimates hold: $\| A^{is}\| \leq Ke^{\theta _ A| s|},\quad \| B^{is}\| \leq Ke^{\theta _ B| s|},\text{ with } \theta _ A+\theta _ B<\pi.$ Then $$A+B$$ is a closed, boundedly invertible operator.
This result is applied to get maximal regularity in the sense of $$L^ p$$ $$(1<p<+\infty)$$ for the solution of the linear parabolic Cauchy problem $u'(t)+Au(t)=f(t)\text{ for } 0\leq t\leq T,\quad u(0)=0,$ under the same assumptions on A as above, provided that $$\theta _ A<\pi /2$$. In particular we obtain the $$L^ p(L^ q)$$-regularity $$(1<p,q<+\infty)$$ for the solution of a parabolic system of PDEs, with suitable initial- boundary conditions.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47D03 Groups and semigroups of linear operators 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 47A50 Equations and inequalities involving linear operators, with vector unknowns 34G10 Linear differential equations in abstract spaces
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