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Absorption semigroups and Dirichlet boundary conditions. (English) Zbl 0788.47031
Given a positive $$C_ 0$$-semigroup $$T$$ on $$L^ p(X)$$ and an arbitrary non-negative potential $$V$$, an absorption semigroup $$T_ V$$ is constructed as a $$C_ 0$$-semigroup on $$L^ p(X_ V)$$ for a certain subset $$X_ V$$ of $$X$$. Information is obtained about $$X_ V$$. When $$p=1$$ and $$T$$ is contractive and holomorphic, $$T_ V$$ is also holomorphic. It follows that Schrödinger semigroups for arbitrary non- negative potentials are holomorphic on the appropriate part of $$L^ 1({\mathbf R}^ N)$$, and the semigroup on $$L^ 1(\Omega)$$ generated by the Laplacian with Dirichlet boundary conditions is holomorphic, for arbitrary open sets $$\Omega$$.

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