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Characterizations of vector-valued weakly almost periodic functions. (English) Zbl 1014.43005

For \(f:S\to X\) \((X\) a Banach space) it is shown that “right translation orbit of \(f\) relatively weakly compact in \(C_b(S,X)\)” is equivalent with “left translation orbit relatively weakly compact”, provided the range \(f(S)\) is relatively weakly compact in \(X\), with \(C_b(S,X):=\) Banach space of continuous bounded \(g:S\to X\) with the sup-norm, and \(S\) a semitopological semigroup. This and some related results extend characterizations of Eberlein weakly almost periodic functions \(f\) for the scalar case resp. \(f\) with relatively compact range.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
43A07 Means on groups, semigroups, etc.; amenable groups
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