### Operational quantities.

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In this paper we consider maps called operational quantities, which assign a non-negative real number to every operator acting between Banach spaces, and we obtain relations between the kernels of these operational quantities and the classes of operators of the Fredholm theory.

Several authors have defined operational quantities derived from the norm of an operator between Banach spaces. This situation is generalized in this paper and we present a general framework in which we derivate several maps $X\to \mathbb{R}$ from an initial one $X\to \mathbb{R}$, where $X$ is a set endowed with two orders, $\le $ and ${\le}^{*}$, related by certain conditions. We obtain only three different derivated maps, if the initial map is bounded and monotone.

Several operational quantities have appeared in the literature characterizing upper semi-Fredholm operators. Here we show that these quantities can be divided into three classes, in such a way that two of them are equivalent if they belong to the same class, and are comparable and not equivalent if they belong to different classes. Moreover, we give a similar classification for operational quantities characterizing lower semi-Fredholm operators.

Several concepts of incomparability of Banach spaces have been considered in the literature, which allow one to describe some of the properties of the product of two Banach spaces as a juxtaposition of the corresponding properties of the factors. In this paper we study the relations between these concepts of incomparability, survey the main results and applications, and state some open problems.

The problem we are concerned with in this research announcement is the algebraic characterization of chain-finite operators (global case) and of locally chain-finite operators (local case).

Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.

Several operational quantities, defined in terms of the norm and the class of finite dimensional Banach spaces, have been used to characterize the classes of upper and lower semi-Fredholm operators, strictly singular and strictly cosingular operators, and to derive some perturbation results. In this paper we shall introduce and study some operational quantities derived from the norm and associated to a space ideal. By means of these quantities we construct a generalized Fredholm theory...

The notion of a measure of noncompactness turns out to be a very important and useful tool in many branches of mathematical analysis. The current state of this theory and its applications are presented in the books [1,4,11] for example. The notion of a measure of weak noncompactness was introduced by De Blasi [8] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations (cf. [2,3,9,10,11], for instance). In...

Let X and Y be infinite dimensional Banach spaces and let L(X,Y) be the class of all (linear continuous) operators acting between X and Y. Mil'man [5] introduced the isometry spectrum I(T) of T ∈ L(X,Y) in the following way:
I(T) = {α ≥ 0: ∀ ε > 0, ∃M ∈ S_{∞}(X), ∀x ∈ S_{M}, | ||T_{x}|| - α | < ε}},
where S_{∞}(X) is the set of all infinite dimensional closed subspaces of X and S_{M...
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