Entire solutions of the abstract Cauchy problem in a Hilbert space

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

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1995-11, Vol.123 (11), p.3351-3356
Main Authors: deLaubenfels, Ralph, Yao, Fuyuan
Format: Article
Language:English
Subjects:
Adjoints
Boundary conditions
Cauchy problem
Hilbert spaces
Linear transformations
Mathematical functions
Mathematical theorems
Mathematical vectors
Range searching
Research article
Semigroups
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Summary:We show that, whenever the linear operator A is symmetric and densely defined, on a Hilbert space, then the abstract Cauchy problem \[ ddzu(z)=A∗(u(z))(z∈C),u(0)=x\frac {d}{{dz}}u(z) = {A^ \ast }(u(z))\quad (z \in {\mathbf {C}}),\qquad u(0) = x \] has an entire solution, for all initial data x in the image of e−A¯A∗{e^{ - \bar A{A^ \ast }}}, which is a dense set.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1995-1273486-9