The main objective of this paper is to develop a notion of joint spectrum for complex solvable Lie algebras of operators acting on a Banach space, which generalizes Taylor joint spectrum (T.J.S.) for several commuting operators.

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C * -algebra elements.

Given a Banach algebra A and a ∈ A, several relations among the Drazin spectrum of a and the Drazin spectra of the multiplication operators L a and R a will be stated. The Banach space operator case will be also examined. Furthermore, a characterization of the Drazin spectrum will be considered.

Properties of the inverse along an element in rings with an involution, Banach algebras and C * -alegbras will be studied unifying known expressions concerning generalized inverses.

EP Banach space operators and EP Banach algebra elements are characterized using different kinds of factorizations. The results obtained generalize well-known characterizations of EP matrices, EP Hilbert space operators and EP C * -algebra elements. Furthermore, new results that hold in these contexts are presented.