, Volume 9, Issue 3, pp 357–367 | Cite as

Local Operators and Forms

  • Wolfgang ArendtEmail author
  • Sonja Thomaschewski


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


Hilbert Space Linear Mapping Fourier Analysis Operator Theory Multiplication Operator 
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© Springer 2005

Authors and Affiliations

  1. 1.Abteilung Angewandte AnalysisUniversität UlmUlmGermany

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