Operator-valued operators that are associated to vector-valued operators

This paper is motivated by a long-standing conjecture of Dinculeanu from 1967. Let X and Y be Banach spaces and let Ω be a compact Hausdorff space. Dinculeanu conjectured that there exist operators S∈L(C(Ω),L(X,Y)) which are not associated to any U∈L(C(Ω,X),Y). We study this existence problem systematically on three possible levels of generality: the classical case C(Ω,X) of continuous vector-valued functions, p-continuous vector-valued functions, and tensor products. On each level, we establish necessary and sufficient conditions for an L(X,Y)-valued operator to be associated to a Y-valued operator. Among others, we see that examples, proving Dinculeanu's conjecture, come out on the all three levels of generality.