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We study Banach-valued holomorphic functions defined on open subsets of themaximal ideal space of the Banach algebra H^\infty of bounded holomorphicfunctions on the unit disk D\subset C with pointwise multiplication andsupremum norm. In particular, we establish vanishing cohomology for sheaves ofgerms of such functions and, solving a Banach-valued corona problem forH^\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 acommutative unital complex Banach algebra A is homeomorphic to the directproduct of maximal ideal spaces of H^\infty and A.

Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^\infty