A faithful Hille-Yosida theorem for finite-dimensional evolutions
Author:
M. A. Freedman
Journal:
Trans. Amer. Math. Soc. 265 (1981), 563-573
MSC:
Primary 47D05; Secondary 47D99
DOI:
https://doi.org/10.1090/S0002-9947-1981-0610966-5
MathSciNet review:
610966
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Abstract | References | Similar Articles | Additional Information
Abstract: As a natural generalization of the classical Hille-Yosida theorem to evolution operators, necessary and sufficient conditions are found for an evolution $U$ acting in ${R^N}$ so that for each $s \geqslant t$, $U(s,t)$ can be uniquely represented as a product integral $\prod _t^s{[I + V]^{ - 1}}$ for some additive, accretive generator $V$. Under these conditions, we further show that $U(\xi ,\zeta )$ is differentiable a.e.
- J. V. Herod, A product integral representation for an evolution system, Proc. Amer. Math. Soc. 27 (1971), 549–556. MR 275244, DOI https://doi.org/10.1090/S0002-9939-1971-0275244-4
- J. V. Herod, A pairing of a class of evolution systems with a class of generators, Trans. Amer. Math. Soc. 157 (1971), 247–260. MR 281059, DOI https://doi.org/10.1090/S0002-9947-1971-0281059-8
- James V. Herod, Generators for evolution systems with quasi continuous trajectories, Pacific J. Math. 53 (1974), 153–161. MR 372700
- J. V. Herod and R. W. McKelvey, A Hille-Yosida theory for evolutions, Israel J. Math. 36 (1980), no. 1, 13–40. MR 589655, DOI https://doi.org/10.1007/BF02761228
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
- Yukio K\B{o}mura, On linear evolution operators in reflexive Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 17 (1970), 529–542. MR 0287361
- J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
- J. S. MacNerney, A linear initial-value problem, Bull. Amer. Math. Soc. 69 (1963), 314–329. MR 146613, DOI https://doi.org/10.1090/S0002-9904-1963-10905-2
- Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. MR 0500055
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Additional Information
Keywords:
Evolution,
generator,
product and sum integral
Article copyright:
© Copyright 1981
American Mathematical Society