Sur les algèbres $S$-régulières et la $S$-décomposabilité des opérateurs de multiplication

Let A be a commutative Banach algebra and A(A) its maximal ideal space. For given S C A(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.