Atlas: Matrix Transformations and Measures of Noncompactness by Eberhard Malkowsky

Atlas home || Conferences | Abstracts | about Atlas

18th International Conference on Operator Theory
June 27 - July 1, 2000
University of the West
Timisoara, Romania

Organizers
Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin, Flavius Turcu

View Abstracts
Conference Homepage

Matrix Transformations and Measures of Noncompactness
by
Eberhard Malkowsky
Mathematisches Institut, Justus--Liebig Universität Giessen,Arndtstrasse 2
Coauthors: Vladimir Rakocevic (University of Nis)

We present a survey of recent joint results as well as new research results in the theories of sequence spaces, their dual spaces, matrix transformations and measures of noncompactness.

For the first time, methods from the fields of summability, in particular of sequence spaces and matrix transformations on one hand, and of measures of noncompactness on the other are successfully linked on a large scale to obtain necessary and sufficient conditions for matrix maps between certain sequence spaces of a general class to be compact operators. The original idea for research in this field dates back to the classical paper of L. W. Cohen and N. Dunford [2]. In this paper they gave necessary and sufficient conditions for matrix transformations from l₁ to l_p, l_p to c₀ and l_p to l₁, and found the norm of these transformations. Furthermore they established necessary and sufficient conditions for these operators to be compact. Although the concept of measure of noncompactness is not explicitly mentioned in their paper, their studies and techniques are very closely related to our research.

We apply concepts and results of the theory of FK, BK, and AK spaces, their \beta- and continuous duals, matrix transformations and measures of noncompactness to give necessary and sufficient conditions for the entries of matrices to be compact operators between certain sequences spaces of a general class. The sequence spaces we deal with are closely related to various concepts of summability, such as ordinary and strong summability, spaces of difference sequences of higher order, spaces of strongly convergent and bounded sequences and spaces of generalized weighted means.

Date received: April 14, 2000