Abstract. Let A be a commutative Banach algebra and ∆(A) its maximal ideal space. For given S ⊂ ∆(A), we establish necessary and sufficient conditions so that A becomes S-regular. We derive some characterizations of decomposable multiplication operators and a description of the Apostol algebra of A. This provides a class of algebras(including Douglas algebras) for which the Apostol algebra is regular.

No abstract

Abstract. To get hyperinvariant subspaces, we establish a relation between the growth of the resolvent and the geometry of the spectrum. Our approach is based on a resolution of the generalized Dirichlet problem.

No abstract