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Operational quantities

Antonio Martinón — 1997

Commentationes Mathematicae Universitatis Carolinae

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.

Generating real maps on a biordered set

Antonio Martinón — 1991

Commentationes Mathematicae Universitatis Carolinae

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 from an initial one X , where X is a set endowed with two orders, and * , related by certain conditions. We obtain only three different derivated maps, if the initial map is bounded and monotone.

Operational quantities characterizing semi-Fredholm operators

Manuel GonzálezAntonio Martinón — 1995

Studia Mathematica

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.

On incomparability of Banach spaces

Manuel GonzálezAntonio Martinón — 1994

Banach Center Publications

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.

Operational quantities derived from the norm and generalized Fredholm theory.

Manuel GonzálezAntonio Martinón — 1991

Extracta Mathematicae

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...

Note on measures of noncompactness in Banach sequence spaces.

Jozef BanasAntonio Martinón — 1990

Extracta Mathematicae

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...

Note on operational quantities and Mil'man isometry spectrum.

Manuel GonzálezAntonio Martinón — 1991

Extracta Mathematicae

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 ∈ SM, | ||Tx|| - α | < ε}}, where S(X) is the set of all infinite dimensional closed subspaces of X and SM...

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