Extension of operators on spaces of vector-valued continuous functions

Let XX be a completely regular Hausdorff space and let BaBa stand for the Baire σσ-algebra of sets in XX. For a Banach space EE let Crc(X,E)Crc(X,E) be the space of all continuous functions f:X→Ef:X→E such that f(X)f(X) is a relatively compact set in EE, and let B(Ba,E)B(Ba,E) be the space of all totally BaBa-measurable functions f:X→Ef:X→E. For a Banach space YY we study the problem of extension of some natural classes of bounded linear operators T:Crc(X,E)→YT:Crc(X,E)→Y to operators T¯:B(Ba,E)→Y.