On the closedness of the sum of two closed operators. (English) Zbl 0615.47002

Dore, Giovanni; Venni, Alberto

Math. Z. 196, 189-201 (1987).

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

Keywords:

boundedly invertible operator; maximal regularity; linear parabolic Cauchy problem; \(L^ p(L^ q)\)-regularity; parabolic system of PDEs; initial-boundary conditions

Citations:

Zbl 0621.47039

References:

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