On the absence of necessary conditions for linear evolution operators

Goldstein, Jerome A.

doi:10.1090/S0002-9939-1977-0500284-2

On the absence of necessary conditions for linear evolution operators

Author: Jerome A. Goldstein
Journal: Proc. Amer. Math. Soc. 64 (1977), 77-80
MSC: Primary 47D05; Secondary 34G05
DOI: https://doi.org/10.1090/S0002-9939-1977-0500284-2
MathSciNet review: 0500284
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Abstract: There exist selfadjoint operators $A(t)(t \geqslant 0)$, whose resolvents depend smoothly on t, such that the initial value (Schrödinger) problem $du(t)/dt = iA(t)u(t),u(0) = f$ has a unique $({C^\infty })$ solution u for each f in a dense set in the underlying Hilbert space, with u depending continuously on f, and yet the intersection over $t \geqslant 0$ of the domain of $A(t)$ is {0}.

G. F. Darmois, Evolution equations in a Banach space, Ph.D. Thesis, Univ. of California, Berkeley, 1975.

J. R. Dorroh, A simplified proof of a theorem of Kato on linear evolution equations, J. Math. Soc. Japan 27 (1975), no. 3, 474–478. MR 380504, DOI https://doi.org/10.2969/jmsj/02730474
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI https://doi.org/10.1016/S0001-8708%2871%2980006-3

J. A. Goldstein, Review of Kōmura’s paper, [8]. MR 44 #4567.

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
Tosio Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241–258. MR 279626
Tosio Kato, Linear evolution equations of “hyperbolic” type. II, J. Math. Soc. Japan 25 (1973), 648–666. MR 326483, DOI https://doi.org/10.2969/jmsj/02540648
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
David Lowell Lovelady, On the generation of linear evolution operators, Duke Math. J. 42 (1975), 57–69. MR 397477
R. S. Phillips, A note on the abstract Cauchy problem, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 244–248. MR 61759, DOI https://doi.org/10.1073/pnas.40.4.244

K. Yosida, Functional analysis, Springer, Berlin, 1965.