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Analytic Gevrey class semigroups generated by \(-A+iB\), and applications. (English) Zbl 0859.47029

Dore, Giovanni (ed.) et al., Differential equations in Banach spaces. Proceedings of the 2nd conference on differential equations in Banach spaces held at Bologna, Italy, 1991. New York: Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. 148, 93-114 (1993).
Let \(A,B\) be selfadjoint operators acting on a Hilbert space \(X\) and let \(A\geq 0\), with dense common domain \(D(A)\cap D(B)\). Then \({\mathcal G}:=-A+iB\) is dissipative, with bounded inverse and generates a \(C_0\)-contraction semigroup on \(X\). The authors investigate \({\mathcal G}\) under additional assumptions such that \(e^{t{\mathcal G}}\) is analytic resp. belongs to a Gevrey class.
These additional assumptions are expressed in terms of fractional powers \((B^+)^\alpha\), where \(B^+:= (B^2)^{1/2}\). The main results are:
(1) If \({\mathcal D}(A^{1/2})\subset {\mathcal D}((B^+)^{1/2})\) then \({\mathcal G}\) is \(m\)-dissipative and \((e^{t{\mathcal G}})\) is analytic.
(2) If \({\mathcal D}(A^{1/2})= {\mathcal D}((B^+)^{\theta/2})\), \(0<\theta<1\), then \((e^{t{\mathcal G}})\) extends to a semigroup of Gevrey class \(>\alpha= 1/\theta\), on \(X\) and on a family of intermediate spaces \(X_{\theta,s}\).
In sections 5 and 6 these results are applied to problems of optimal control with quadratic cost and unbounded control functions, resp. to inhomogeneous Cauchy problems.
For the entire collection see [Zbl 0785.00028].
Reviewer: W.Hazod (Dortmund)

MSC:

47D06 One-parameter semigroups and linear evolution equations
47B44 Linear accretive operators, dissipative operators, etc.
47A60 Functional calculus for linear operators
49J27 Existence theories for problems in abstract spaces
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