Entire solutions of the abstract Cauchy problem in a Hilbert space

deLaubenfels, Ralph; Yao, Fuyuan

doi:10.1090/S0002-9939-1995-1273486-9

Entire solutions of the abstract Cauchy problem in a Hilbert space
HTML articles powered by AMS MathViewer

by Ralph deLaubenfels and Fuyuan Yao PDF

Proc. Amer. Math. Soc. 123 (1995), 3351-3356 Request permission

Abstract:

We show that, whenever the linear operator A is symmetric and densely defined, on a Hilbert space, then the abstract Cauchy problem \[ \frac {d}{{dz}}u(z) = {A^ \ast }(u(z))\quad (z \in {\mathbf {C}}),\qquad u(0) = x\] has an entire solution, for all initial data x in the image of ${e^{ - \bar A{A^ \ast }}}$, which is a dense set.

References

Laurent Autret, Entire vectors and time reversible Cauchy problems, Semigroup Forum 46 (1993), no. 3, 347–351. MR 1206212, DOI 10.1007/BF02573577
Ralph deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Mathematics, vol. 1570, Springer-Verlag, Berlin, 1994. MR 1290783, DOI 10.1007/BFb0073401
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
Roe Goodman, Complex Fourier analysis on a nilpotent Lie group, Trans. Amer. Math. Soc. 160 (1971), 373–391. MR 417334, DOI 10.1090/S0002-9947-1971-0417334-3
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 107176, DOI 10.2307/1970331
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727