# Local Operators and Forms

Local Operators and Forms Let $$a:\,V\times V \rightarrow \mathbb{R}$$ be a continuous, coercive form where V is a Hilbert space, densely and continuously embedded into L 2(Ω). Denote by T the associated semigroup on L 2(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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Local Operators and Forms

, Volume 9 (3) – Mar 11, 2005
11 pages
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2005 by Springer
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-3558-1
Publisher site
See Article on Publisher Site

### 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 L 2(Ω). Denote by T the associated semigroup on L 2(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

### Journal

PositivitySpringer Journals

Published: Mar 11, 2005

### References

• On operators preserving disjointness
Abramovich, Yu.A.; Veksler, A.I.; Kaldunov, A.V.

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