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Positivity

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

Local Operators and Forms

  • Wolfgang ArendtEmail author
  • Sonja Thomaschewski
Article

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

Keywords

Hilbert Space Linear Mapping Fourier Analysis Operator Theory Multiplication Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Abteilung Angewandte AnalysisUniversität UlmUlmGermany

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