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Type of Document	Dissertation
Author	Ford, Lawrence Charles
URN	etd-10122005-082659
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-10122005-082659
Title	Generalized multipliers on locally compact Abelian groups
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title C. R. DePrima Committee Member W. A. J. Luxemburg Committee Member
Keywords	none
Date of Defense	1974-02-07
Availability	unrestricted
Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let G be a locally compact Abelian group with dual [...], [...], and [...] supp [...] is compact}. Then for [...], the containments are proper if G is noncompact, and [...] is a dense, translation invariant subspace of [...] for [...]. Let [...] be a complex valued function defined on [...], and [...] = [...]. Suppose [...]. Define the operator, [...] by the equation [...] for each [...]. Then [...] is a module over M(G), [...] is a module homomorphism, and [...] is (p, q) closed. We call [...] a generalized (p, q) multiplier. The main results include: (1) Suppose T is an operator satisfying: (a) The domain D(T) is a translation invariant subspace of [...], and the range R(T) [...]; (b) D(T) [...]; (c) T is (p, q) closed, linear, and commutes with all translations; (d) C X T(C) is dense in [...]. Then T = [...] for some [...]. (2) The set of all generalized (p, q) multipliers, denoted [...], is a linear space, and the set of all generalized (p, p) multipliers, denoted [...], is an algebra containing [...] and contained in [...]. (3) If [...], then [...] is locally the transform of a bounded (p, q) multiplier. Further sections include a deeper study of [...], [...], and special results obtainable for compact G.
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