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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.

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We study composition operators on Hardy and Dirichlet spaces belonging to Schatten classes. We give some new examples and analyse the size of contact set of the symbol of such operators.The contact set of ϕ is E ϕ := E ϕ (1). Let H be a Hillbert space, a compact operator is said to belong in the Schatten class S p (H ) if its sequence of singular numbers is in the sequence space ℓ p .2000 Mathematics Subject Classification. 47B38, 30H05, 30C85, 47A15.

We give in this paper some asymptotic Von Neumann inequalities for power bounded operators in the class C p Π C\ t and some spacial von Neumann inequalities associated with non zero elements of the point spectrum, when it is non void, of generalized Toeplitz operators.Introducing perturbed kernel, we consider classes CR which extend the classical classes C p . We give results about absolute continuity with respect to the Haar measure for operators in class CR Π (7ι ; . This allows us to give new results on cyclic vectors for such operators and provides invariant subspaces for their powers. Relationships between cyclic vectors for Τ and T* involving generalized Toeplitz operators are given and the commutativity of {T}', the commutant of Τ is discussed.