Teresa Álvarez

Some examples of Tauberian operators

We study some examples of tauberian operators and show that the second conjugate of a tauberian operators is not always tauberian, answering a question of Kalton and Wilansky. Also we show that the class of tauberian operators is not open, although tauberian operators in the boundary of the class must have nonclosed range

QUANTITIES RELATED TO UPPER AND LOWER SEMI-FREDHOLM TYPE LINEAR RELATIONS

Several authors ([11, 15, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]) have in-troduced operational quantities in order to obtain characterisations and perturbationresults for various classes of operators of Predholm theory. We remark that all theabove authors considered only the case of bounded linear operators in Banach spaces.It is the purpose of this paper to consider these quantities in the more general settingof linear relations between normed spaces.In Section 2 we define some quantities associated with an arbitrary quantity /, andin particular, with the measures of nonprecompactness p and K and the measure of nonstrict singularity 55. These quantities are related to certain quantities generated by thenorm and these relations will be applied to obtain characterisations and perturbationresults for F

PERTURBATION THEORY OF MULTIVALUED ATKINSON OPERATORS IN NORMED SPACES

We prove several stability results for Atkinson linear relations under additive perturbation by small norm, strictly singular and strictly cosingular multivalued linear operators satisfying some additional conditions.

Paracomplete normed spaces and Fredholm theory

A normed space is paracomplete if it admits a new norm, stronger than the initial one, that makes it complete. Here we give a characterization of paracomplete normed spaces. As a consequence, we show that operators on paracomplete spaces have compact spectrum in the algebra of all operators, and that the class of paracomplete spaces is not stable under ℓ2-sums. Moreover, we give characterizations for the closed Fredholm operators on paracomplete spaces and for the almost semi-Fredholm operators of Harte on normed spaces.

Coperturbation function and lower semi-Browder multivalued linear operators

In this article, we investigate the perturbation theory of lower semi-Browder and Browder linear relations. Our approach is based on the concept of a coperturbation function for linear relations in order to establish some perturbation theorems and deduce the stability under strictly cosingular operator perturbations. Furthermore, we apply the obtained results to study the invariance and the characterization of Browders essential defect spectrum and Browders essential spectrum.

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract Let X and Y be normed spaces and T: D(T) ⊂ X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily lp. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.

Generalized Fredholm operators

0. Introduction. The classical Fredholm theory in Banach spaces studies normally solvable operators with null space or conull space in F, the ideal of all finite dimensional Banach spaces. The aim of this paper is to study normally solvable operators with null space or conull space in an arbitrary space ideal A. We use the operator ideal Op(A) to replace the operator ideal Fi := Op (F) of the classical theory in a natural way. Operators invertible modulo Op (A) are studied as well. Since F c A and Fi ~Op (A), Fredholm operators are particular A-Fredholm operators. Yang [10], [11] developed Fredholm theory relative to the ideal R of all reflexive Banach spaces with a functorial approach which does not allow to develop the theory in an arbitrary space ideal; theorems of [9], [ 10] and [ 11] are particular cases of some results presented here.

Adjoint characterisations of quasi-weakly compact linear relations

In this paper, the class of all quasi-weakly compact linear relations is introduced and described in terms of their first and second adjoints. Complete characterisations are obtained in the case when the adjoint is continuous. We investigate the connection between a quasi-weakly compact linear relation and its adjoint. We also characterise the quasi-reflexive spaces in terms of quasi-weak compactness of operators. Examples of linear relations belonging to this class are exhibited.

Quantities Characterizing Lower Semi-Fredholm Linear Relations

Certain operational quantities derived from the norm and from the injection modulus in the context of multivalued linear operators are considered in order to obtain characterizations of lower semi-Fredholm multivalued linear operators.

Left-right Browder linear relations and Riesz perturbations

Abstract A closed linear relation T in a Banach space X is called left (resp. right) Fredholm if it is upper (resp. lower) semiFredholm and its range (resp. null space) is topologically complemented in X . We say that T is left (resp. right) Browder if it is left (resp. right) Fredholm and has a finite ascent (resp. descent). In this paper, we analyze the stability of the left (resp. right) Fredholm and the left (resp. right) Browder linear relations under commuting Riesz operator perturbations. Recent results of Zivkovic et al. to the case of bounded operators are covered.