On the completion problem for algebra H∞H∞

We study the completion problem to an invertible operator-valued function for the class of bounded holomorphic functions on the unit disk D⊂SD⊂S with relatively compact images in the space of bounded linear operators between complex Banach spaces. In particular, we prove that in this class of functions the operator-valued corona problem and the completion problem are not equivalent, and establish an Oka-type principle asserting that the completion problem is solvable if and only if it is solvable in the class of continuous operator-valued functions on DD with relatively compact images.