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The Hille-Yosida space of an arbitrary operator. (English) Zbl 0675.47033
Summary: Let A be an arbitrary Banach space operator with resolvent defined for all $\lambda >0$. We define a linear manifold Z in the given space and norm $\Vert\vert \cdot \Vert\vert$ on Z majorizing the given norm, such that (Z,$\Vert\vert \cdot \Vert\vert)$ is a Banach space, and the restriction of A to Z generates a strongly continuous semigroup of contractions in Z. This so-called Hille-Yosida space (Z,$\Vert\vert \cdot \Vert\vert)$ is “maximal- unique” in a suitable sense.

47D03(Semi)groups of linear operators
Full Text: DOI
[1] Hille, E.; Phillips, R. S.: Functional analysis and semigroups. Publ. 31 (1957) · Zbl 0078.10004
[2] Kantorovitz, S.: The semi-simplicity manifold of arbitrary operators. Trans. amer. Math. soc. 123, 241-252 (1966) · Zbl 0154.16001
[3] Kantorovitz, S.: Spectral theory of Banach space operators. ”Lecture notes in mathematics,” no. 1012 (1983) · Zbl 0527.47001
[4] S. Kantorovitz and R. Hughes, Spectral representations for unbounded operators with real spectrum, Math. Ann., in press. · Zbl 0468.47026