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Journal of the London Mathematical Society 1997 56(1):1-15; doi:10.1112/S0024610797005218
© 1997 by London Mathematical Society
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© The London Mathematical Society

Unbounded Translation Invariant Operators and the Derivation Property

Dirk Alboth

Fachbereich 17–Mathematik, University-GH Paderborn 33095 Paderborn, Germany. E-mail: dirka{at}uni-paderborn.de

Received 20 February 1995. Revision received 22 June 1995.

We are going to investigate translation invariant derivations on Lp spaces of locally compact abelian groups, 1≤p<{infty}. By these we mean densely defined closed linear operators which commute with translations and obey a Leibniz rule, that is, T(fg)=(Tf)·g+f·(Tg); see Definition 1 for details.

The original motivation for studying these operators was to find an abstract description of constant coefficient partial differential operators as a link to perturbation theory which is usually formulated in terms of abstract operator theoretic notions. This fits, for example, Schrödinger operator theory (as in [1]). Here, in view of the applications to perturbation theory, the point is that this identification is not just a formal one but includes assertions about domains (Theorem 1).

In this paper, however, we shall concentrate on groups other than Rn.


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