Characterizations of vector-valued weakly almost periodic functions (Q1864276)

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.