Generalized Multipliers on Locally Compact Abelian Groups

Author: Ford, Lawrence Charles

Year: 1974

Degree: Dissertation (Ph.D.)

Advisors: De Prima, Charles R.; Luxemburg, W. A. J.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/NB31-1Y34

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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