Matrix-valued corona theorem for multiply connected domains. (Q2733847)

Let \(D\) be a bounded domain in the complex plane whose boundary consists of \( k\) simple closed continuous curves and let \(H^{\infty }(D) \) be the algebra of all bounded analytic functions on \(D.\) Under this assumption, the author proves a matrix-valued corona theorem generalizing a result of Sz.-Nagy (for the case of the open unit disk, hence \(k=1\)): let \(n\) be natural number \(\geq k\) and let \(f\) be a \(k\times n\)-matrix whose entries belong to \(H^{\infty }(D)\). Assume that there exists \(\delta >0\) such that NEWLINE\[NEWLINE \sum_{s\in S}| F_{s}(z)|\geq\delta\text{ for all }z\in DNEWLINE\]NEWLINE where \((F_{s}) _{s\in S}\) is the family of determinants of submatrices of order \(k\) of \(f\). Then there exists a unimodular \(n\times n\)-matrix \(F\) with entries in \(H^{\infty }(D)\) which extends the matrix \(f\). The proof is based on a result of \textit{F.~D. Suaréz} [J. Funct. Anal. 123, 233--263 (1994; Zbl 0808.46076)] about the topological dimension of the maximal ideal space of \(H^{\infty}(D)\).