Properties of a subalgebra of H{infty}(D) and stabilization

Let D denote the open unit disc in C. Let T denote the unit circle and let S sub T. We denote by AS(D) the set of all functions f : D {cup} S -> C that are holomorphic in D and are bounded and continuous in D {cup} S. Equipped with the supremum norm, AS(D) is a Banach algebra, and it lies between the extreme cases of the disc algebra A(D) and the Hardy space H{infty}(D). We show that AS(D) has the following properties: P1. The corona theorem holds for AS(D). P2. The integral domain AS(D) is not a Bézout domain, but it is a Hermite ring. P3. The stable rank of AS(D) is 1. P4. The Banach algebra AS(D) has topological stable rank 2. The classes AS(D) serve as appropriate transfer function classes for infinite-dimensional systems that are not exponentially stable, but stable only in some weaker sense. Consequences of the above properties to stabilizing controller synthesis using a coprime factorization approach are discussed