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Approximation of degenerate semigroups. (English) Zbl 1025.47023
A continuous degenerate semigroup is defined as a strongly continuous mapping \(T: [0,\infty)\to{\mathcal L}(X)\) having the semigroup property, but where \(T(0)\) may be a projection different from the identity. The main result of the paper is a Trotter-Kato type approximation result establishing the relation between the convergence of continuous degenerate semigroups and pseudoresolvents. A similar result is shown for holomorphic degenerate semigroups. As an application, the author studies, convergence of heat semigroups (with absorption at the boundary) on a sequence of domains.
In a preliminary section, the author gives – among other things – a new proof of a remarkable extension of Pettis’ measurability criterion (Theorem 1.2). A more general version of this result in the context of locally convex spaces is contained in E. F. Thomas [Trans. Am. Math. Soc. 212, 61-81 (1975; Zbl 0312.28014)].

MSC:
47D06 One-parameter semigroups and linear evolution equations
35K05 Heat equation
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