Thesis supervisor: Eve Oja, professor, University of Tartu
Oponendid: Hans-Olav Johannes Tylli, Associate Professor, University of Helsinki, Finland. Dirk Werner, professor, Freie Universität Berlin, Germany
The compactness of sets and operators plays an important role in the baseline studies of functional analysis and its applications. According to Grothendieck’s criterion of compactness, a subset of a Banach space is relatively compact if and only if it is contained in the closed convex hull of a sequence converging to zero. This criterion has inspired several other forms of relative compactness, which in turn define different classes of compact operators. In the current thesis, a unified study of compactnesses and corresponding classes of compact operators is developed. A stronger and more general notion of (p,r)-compactness for sets and for operators is introduced, so that when p=∞, it coincides with the classical compactness and for r=1, it is exactly the p-compactness in the sense of Bougain-Reinov. In the special case when r=p/(p-1), it is the extensively studied p-compactness in the sense of Sinha-Karn. The alternative theory of the thesis encompasses among others the p-compactness in the sense of Sinha-Karn, and the results improve several known ones.