×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.17, 35-92 (1964) · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[2] Bourgain, J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat.21, 163-168 (1983) · Zbl 0533.46008 · doi:10.1007/BF02384306
[3] Burkholder, D.L.: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab.9, 997-1011 (1981) · Zbl 0474.60036 · doi:10.1214/aop/1176994270
[4] Burkholder, D.L.: A geometric condition that implies the existence of certain singular intergrals of Banach-space-valued functions. In: Conference on harmonic analysis in honor of Antoni Zygmund (Chicago 1981), pp. 270-286. Belmont: Wadsworth 1983
[5] Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. In: Probability and analysis (Varenna 1985), pp. 61-108; Lect. Notes Math. 1206. Berlin Heidelberg New York: Springer 1986
[6] Cior?nescu, I., Zsidó, L.: Analytic generators for one-parameter groups. Tôhoku Math. J. II Ser.28, 327-362 (1976) · Zbl 0361.47014 · doi:10.2748/tmj/1178240775
[7] Cobos, F.: Some spaces in which martingale difference sequences are unconditional. Bull. Pol. Acad. Sci., Math.34, 695-703 (1986) · Zbl 0617.46079
[8] Da Prato, G., Grisvard, P.: Sommes d’opérateurs linéaires et équations différentielles opérationnelles J. Math. Pures Appl., IX Ser.54, 305-387 (1975) · Zbl 0315.47009
[9] Dunford, N., Schwartz, J.T.: Linear operators, Part I. New York: Interscience 1958 · Zbl 0084.10402
[10] Marschall, E.: On the analytical generator of a group of operators. Indiana Univ. Math. J.35, 289-309 (1986) · Zbl 0608.47048 · doi:10.1512/iumj.1986.35.35017
[11] McConnell, T.R.: On Fourier multiplier transformations of Banach-valued functions. Trans. Am. Math. Soc.285, 739-757 (1984) · Zbl 0566.42009 · doi:10.1090/S0002-9947-1984-0752501-X
[12] Peetre, J.: Sur la transformation de Fourier des fonctions à valeurs vectorielles. Rend. Sem. Mat. Univ. Padova.42, 15-26 (1969) · Zbl 0241.46033
[13] Rubio de Francia, J.L.: Martingale and integral transforms of Banach space valued functions. In: Probability and Banach spaces (Proceedings, Zaragoza 1985), pp. 195-222; Lect. Notes Math.1221. Berlin Heidelberg New York: Springer 1986 · Zbl 0615.60041
[14] Seeley, R.: Norms and domains of the complex powersA B r . Am. J. Math.93, 299-309 (1971) · Zbl 0218.35034 · doi:10.2307/2373377
[15] Triebel, H.: Interpolation theory, function spaces, differential operators. Amsterdam, New York, Oxford: North Holland 1978 · Zbl 0387.46032
[16] von Wahl, W.: The equationú+A(t)u=f in a Hilbert space andL p-estimates for parabolic equations. J. Lond. Math. Soc., II Ser.25, 483-497 (1982) · Zbl 0493.35050 · doi:10.1112/jlms/s2-25.3.483
[17] Yagi, A.: Coincidence entre des espaces d’interpolations et des domains de puissances fractionnaires d’opérateurs J.R. Acad. Sci. Paris, Ser I299, 173-176 (1984). · Zbl 0563.46042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.