Collectively compact and collectively strictly singular sets of linear operators

Abstract

In this thesis the concept of collectively compact sets of operators is studied. As a reason for the study of such operators it is shown how collectively compact sets of operators are applicable to an approximation theory for Fredholm integral equations of the second kind where the kernel is continuous. In this case the integral operator mapping C[a, b] into C[a, b] is compact and the set of numerical-integral operators approximating the integral operator is collectively compact. Convergence theorems and error bounds are given for this type of situation. Once the importance of the concept of collective compactness has been established, properties of such sets of operators are studied. A characterisation of collectively compact sets of operators in terms of countable subsets is given. In addition, a comparison between totally bounded sets and collectively compact sets of compact operators is done since the approximation theory mentioned above is applicable to sets of operators that are collectively compact but not totally bounded. Perturbation theorems involving perturbations of semi-Fredholm operators with collectively compact sets of operators are also studied. The concept of collectively strictly singular sequences of operators is defined and perturbation theorems for perturbations of semi-Fredholm operators with collectively strictly singular sequences of operators are given. It is probable that the concept of collective strict singularity might be applicable in establishing an approximation theory for Fredholm integral equations of the second kind with measurable, discontinuous kernel where the integral operator maps the Lebesgue space £ 1 into £ 1• The concept of collectively strictly cosingular sequences of operators naturally arises and is therefore defined. It is noted that analogous perturbation theorems to the ones proved for collectively strictly singular sequences of operators could easily be proven by suitably dualising the proofs for the above-mentioned theorems.