Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.Comment: 13 pages, original research articl

Joint spectra of representations of Lie algebras by compact operators

Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $\rho\colon L\to L(X)$, a representation of $L$ in $X$ such that $\rho (l)\in K(X)$ for all $l\in L$, the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $\rho$ are computed.Comment: 8 pages, original research articl

Drazin spectra of Banach space operators and Banach algebra elements

Given a Banach Algebra $A$ and $a\in A$, 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.Comment: 16 pages; original research articl

Further results on regular Fredholm pairs and chains

The main objective of the present article is to characterize regular Fredholm pairs and chains in terms of Fredholm operators.Comment: 8 pages, original research articl

Analytically Riesz operators and Weyl and Browder type theorems

Several spectra of analytically Riesz operators will be characterized. These results will led to prove Weyl and Browder type theorems for the aforementioned class of operators.Comment: 10 pages, original research articl

Moore-Penrose inverse and doubly commuting elements in C∗C^*-algebras

In this work it is proved that the Moore-Penrose inverse of the product of $n$-doubly commuting regular $C^*$-algebra elements obeys the so-called reverse order law. Conversely, conditions regarding the reverse order law of the Moore-Penrose inverse are given in order to characterize when $n$-regular elements doubly commute. Furthermore, applications of the main results of this article to normal $C^*$-algebra elements, to Hilbert space operators and to Calkin algebras will be considered.Comment: 12 pages, original research articl

On Cartan Joint Spectra

In this work several results regarding the Cartan version of the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra will be studied.Comment: 11 pages, original research articl

Characterizations of Fredholm pairs and chains in Hilbert spaces

In this work characterizations of Fredholm pairs and chains of Hilbert space operators are given. Following a well-known idea of several variable operator theory in Hilbert spaces, the aforementioned objects are characterized in terms of Fredholm linear and bounded maps. Furthermore, as an application of the main results of this work, direct proofs of the stability properties of Fredholm pairs and chains in Hilbert spaces are obtained.Comment: 15 pages, original research articl

The Drazin spectrum of tensor product of Banach algebra elements and elementary operators

Given unital Banach algebras $A$ and $B$ and elements $a\in A$ and $b\in B$, the Drazin spectrun of $a\otimes b\in A\overline{\otimes} B$ will be fully characterized, where $A\overline{\otimes} B$ is a Banach algebra that is the completion of $A\otimes B$ with respect to a uniform crossnorm. To this end, however, first the isolated points of the spectrum of $a\otimes b\in A\overline{\otimes} B$ need to be characterized. On the other hand, given Banach spaces $X$ and $Y$ and Banach space operators $S\in L(X)$ and $T\in L(Y)$, using similar arguments the Drazin spectrum of $\tau_{ST}\in L(L(Y,X))$, the elementary operator defined by $S$ and $T$, will be fully characterized.Comment: 12 pages, original research articl

On the Moore-Penrose inverse in C∗C^*-algebras

In this article, two results regarding the Moore-Penrose inverse in the frame of $C^*$-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is $C^*$-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact,these elements will be fully characterized both in the Hilbert space and in the $C^*$-algebra setting. Furthermore, it will be proved that an element is normal and Moore-Penrose hermitian if and only if it is a hermitian partial isometry.Comment: 14 pages, original research articl