Grauert- and Lax–Halmos-type theorems and extension of matrices with entries in H∞

In the paper we prove an extension theorem for matrices with entries in H∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.