## 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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