Un’osservazione su un criterio di compattezza per le funzioni vettoriali quasi periodiche. (Italian) Zbl 0249.43016

Bratti, Giuliano

Rend. Sem. Mat. Univ. Padova 46(1971), 101-105 (1972).

For locally compact abelian groups \( G \), complex locally convex \( E \), bounded and weakly continuous \( f: G \rightarrow E \) it is shown: Such \( f \) are weakly almost periodic if and only if there is a weakly continuous \( \varphi: \bar{G} \rightarrow E \) with \( f=\varphi \circ \beta \). Here \( \bar{G}:= \) Bohrcompactification of \( G, \beta: G \rightarrow \bar{G} \) corresponding embedding; weakly continuous is defined with respect to the weak topology of \( E \), weakly a.p. means relative compactness of the translates \( \left\{\mathrm{T}_{\mathrm{s}} f: s \in G\right\} \) with respect to uniform convergence on \( G \) in the weak topology of \( E \) (equivalent is: for each linear continuous \( \alpha: E \rightarrow \) complex numbers, \( \alpha \circ f \) is a.p., and the range \( f(G) \) is relatively weakly compact). Also, weak uniform convergence of subsets of equicontinuous and ’equinormal’ ( = weakly a-p. uniformly in \( m \) ) sequences \( \left(f_{m}\right. \) ) is shown, if the ranges of the corresponding \( \varphi_{\mathrm{m}} \) are relatively weakly compact, yielding a characterization of sets of weakly a.p. functions which are compact with respect to weak uniform convergence. (The conditions a), b), c) of the author are probably not independent; compare also with. N. G h e \( \mathrm{rgh} \mathrm{i} \mathrm{u} \) and \( \mathrm{A} \). C o s t in e s c u [Revue Roumaine Math. pur. appl. 11, 341-344 (1966, this ZbI. 163,373\( )] \). )

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MSC:

43A60

Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions

References:

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