  The main result of the paper is the theorem giving a sufficient condition for
the existence of a bounded analytic projection onto a holomorphic family of
(generally infinite-dimensional) subspaces (a holomorphic sub-bundle of a
trivial bundle). This sufficient condition is also necessary in the case of
finite dimension or codimension of the bundle. A simple lemma of N. Nikolski
connects the existence of a bounded analytic projection with the Operator
Corona Problem (existence of a bounded analytic left inverse for an
operator-valued function), so as corollaries of the main result we obtain new
results about the Operator Corona Problem. In particular, we find a new
sufficient condition, a complete solution in the case of finite codimension,
and, a solution of the generalized Corona Problem.
