  We study Banach-valued holomorphic functions defined on open subsets of the
maximal ideal space of the Banach algebra H^\infty of bounded holomorphic
functions on the unit disk D\subset C with pointwise multiplication and
supremum norm. In particular, we establish vanishing cohomology for sheaves of
germs of such functions and, solving a Banach-valued corona problem for
H^\infty, prove that the maximal ideal space of the algebra H_{\rm comp}^\infty
(A) of holomorphic functions on $\Di$ with relatively compact images in a
commutative unital complex Banach algebra A is homeomorphic to the direct
product of maximal ideal spaces of H^\infty and A.
