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.


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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