Boundary values of holomorphic semigroups

Suppose AA generates a bounded strongly continuous holomorphic semigroup of angle π/2\pi /2. We show that iAiA generates a (1−A)−r{(1 - A)^{ - r}} regularized group, which is O(1+|s|r)∀r>γ⩾0O(1 + |s{|^r})\;\forall r > \gamma \geqslant 0, if and only if ||ezA|||{e^{zA}}|| is O(((1+|z|)/Re⁡(z))r...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1993, Vol.118 (1), p.113-118
Main Authors: Boyadzhiev, Khristo, deLaubenfels, Ralph
Format: Article
Language:English
Subjects:
Boundary value problems
Cauchy problem
Exact sciences and technology
Mathematical analysis
Mathematical theorems
Mathematics
Operator theory
Research article
Sciences and techniques of general use
Semigroups
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Summary:Suppose AA generates a bounded strongly continuous holomorphic semigroup of angle π/2\pi /2. We show that iAiA generates a (1−A)−r{(1 - A)^{ - r}} regularized group, which is O(1+|s|r)∀r>γ⩾0O(1 + |s{|^r})\;\forall r > \gamma \geqslant 0, if and only if ||ezA|||{e^{zA}}|| is O(((1+|z|)/Re⁡(z))r)∀r>γO({((1 + |z|)/\operatorname {Re} (z))^r})\forall r > \gamma and iAiA generates a bounded (1−A)−r{(1 - A)^{ - r}} regularized group ∀r>γ⩾0\forall r > \gamma \geqslant 0 if and only if ||ezA|||{e^{zA}}|| is O((1/Re⁡(z))r)∀r>γO({(1/\operatorname {Re} (z))^r})\;\forall r > \gamma. We apply this to the Schrödinger operator i(Δ−V)i(\Delta - V).
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1993-1128725-X