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Local Operators and Forms

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Abstract

Let \(a:\,V\times V \rightarrow \mathbb{R}\)be a continuous, coercive form where V is a Hilbert space, densely and continuously embedded into L2(Ω). Denote by T the associated semigroup on L2(Ω). We show that T consists of multiplication operators if and only if V is a sublattice with normal cone and

$$a(u^+, \, u^-)\,=\,0 \quad (u \in V)$$

We also prove a vector-valued version of this result. For this we characterize multiplication operators \(M:\, L^p(\Omega,E) \rightarrow L^p(\Omega,E)\) by locality. If Ω has no atoms, we show that each local, linear mapping is automatically continuous

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  1. Abteilung Angewandte Analysis, Universität Ulm, 89069, Ulm, Germany

    Wolfgang Arendt & Sonja Thomaschewski

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  1. Wolfgang Arendt
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  2. Sonja Thomaschewski
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Correspondence to Wolfgang Arendt.

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Arendt, W., Thomaschewski, S. Local Operators and Forms. Positivity 9, 357–367 (2005). https://doi.org/10.1007/s11117-005-3558-1

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