Abstract
LetX be a complex Banach space andA: D(A)→X a densely defined closed linear operator whose resolvent set contains the real line and for which ‖λ(λ−A)−1‖ is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and −A, for the existence of a decompositionX=X +⊕X −, whereX ± are closed subspaces, invariant forA, the spectra of the reduced operatorsA ± are {λ∈σ(A);Imλ>0} and {λ∈σ(A);Imλ<0} respectively, and ‖λ(λ−A ±)−1‖ is bounded forImλ≶0.
Finally we give an example of an operator in anL p-type space for which the decomposition exists if 1<p<+∞ and does not exist ifp=1.
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References
H. Bart, I. Gohberg, M. A. Kaashoek: Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators. J. Funct. Anal.68 (1986), 1–42.
T. Burak: On semigroups generated by restrictions of elliptic operators to invariant subspaces, Israel J. Math.12 (1972), 79–93.
G. Dore, A. Venni: On the closedness of the sum of two closed operators. Math. Z.196 (1987), 189–201.
G. Greiner: A spectral decomposition of strongly continuous groups of positive operators; Quart. J. Math. Oxford Ser. (2)35 (1984), 37–47.
P. Grisvard: An approach to the singular solutions of elliptic problems via the theory of differential equations in Banach spaces; in:Differential equations in Banach spaces (Proceedings, Bologna 1985) Lect. Notes Math.1223, Springer, Berlin Heidelberg New York, 1986, 130–155.
K. Hoffman:Banach spaces of analytic functions; Prentice-Hall, Englewood Cliffs, 1962.
T. Kato:Perturbation theory for linear operators, 2nd ed.; Springer, Berlin Heidelberg New York Tokyo, 1984.
J. Marcinkiewicz: Sur les multiplicateurs des séries de Fourier; Studia Math.8 (1939), 78–91.
V. Nollau: Über den Logarithmus abgeschlossener Operatoren in Banachschen Räumen; Acta Sci. Math. (Szeged)30 (1969), 161–174.
J. Prüss: On the spectrum ofC 0-semigroups; Trans. Amer. Math. Soc.284 (1984), 847–857.
R. Seeley: Norms and domains of the complex powersA B z Amer. J. Math.93 (1971), 299–309.
H. Triebel:Interpolation theory, function spaces, differential operators; North Holland, Amsterdam New York Oxford, 1978.
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Dore, G., Venni, A. Separation of two (possibly unbounded) components of the spectrum of a linear operator. Integr equ oper theory 12, 470–485 (1989). https://doi.org/10.1007/BF01199455
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DOI: https://doi.org/10.1007/BF01199455