Let T be a bounded operator with (SVEP) on its localizable spectrum \(\sigma _\mathrm{loc}(T)\). We show that for every open subset U of \(\sigma _\mathrm{loc}(T)\), there exists a unit vector x whose local spectrum coincides with the closure of U, and such that its local resolvent function is bounded. This result answers positively to an open question stated by several authors, and extends the both cases of operators with trivial divisible subspace and operators whose point spectrum has empty interior.

We investigate common spectral properties between operators A and B satisfying the operator equations A2 = ABA and B2 = BAS. We show that A and B shareall their spectral and local spectral behavior. In this paper, X is a Banach space and .c(X) denotes the space of aH bounded linear operators on X. For a bounded linear operator T E.c(X), let ker(T), R(T), O"(T), O"p(T) , O"ap(T) and O"r(T) denote the kernel of T, the range space, the spectrum, the pointspectrum, the approximate point spectrum :<J,nd the residual spectrum of T respectively. The adjoint operator oí T will be denbted by T*. Recall that a bounded operator is semi Fredholm provided that R(T) is closed and either a(T) = dim(ker(T)) or j3(T) = dim(X/R(T)) is finite. The index of a semi Fredhom operator is ind(T) = a(T) - (3(T) and a semi Fredholm operator with afinite index is said to be Fredholm. The semi Fredholm spectrum O"e(T) and the Fredholm spectrum O"s¡(T) stand respectively for the set of complex numbers A such that T - A is not Fredhom (respectively not semi Fredholm). In the sequel, we will consider A, B E L(X) given bounded operators satisfying

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

L’insiemeP degli elementi idempotenti di un’algebra di BanachA é in generale non connesso. J. Zemanek [7] e B. Aupetit [1] hanno dimostrato che le componenti connesse di tale insieme sono connesse per archi. Inoltre J. Esterle [4] ha dimostrato che due elementi diP appartenti alla stessa componente connessa possono essere collegati attraverso un cammino polinomiale. In questo lavoro si studia il minimo grado di tali polinomi seA è un’algebra di Banach di dimensione finita oppure seA è l’algebra degli operatori limitati su uno spazio di Banach.

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.