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Unbounded Translation Invariant Operators and the Derivation Property
Author(s) -
Alboth Dirk
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005218
Subject(s) - mathematics , constant coefficients , spectral theorem , operator theory , abelian group , invariant (physics) , operator (biology) , pure mathematics , linear operators , differential operator , perturbation (astronomy) , linear map , unbounded operator , algebra over a field , discrete mathematics , approximation property , mathematical analysis , banach space , mathematical physics , quantum mechanics , physics , biochemistry , chemistry , gene , bounded function , repressor , transcription factor
We are going to investigate translation invariant derivations on L p spaces of locally compact abelian groups, 1⩽ p <∞. 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 ℝ n .