Strongly bounded operators on Crc(X,E)Crc(X,E) with the strict topology βσβσ

Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let Crc(X,E) be a space of all continuous functions f:X→E such that f(X) is a relatively compact set in E, equipped with the strict topology βσ. We study (βσ,‖⋅‖F)-continuous strongly bounded operators T:Crc(X,E)→F. In particular, we establish the relationships between (βσ,‖⋅‖F)-continuous strongly bounded operators and weakly compact (resp. weakly precompact; unconditionally converging; completely continuous; weakly completely continuous) operators T:Crc(X,E)→F. In particular, it is shown that if E is a Schur space, then the space (Crc(X,E),βσ) has the strict Dunford–Pettis property.