2013

Tartu Ülikool

## Abstract

This master thesis aims to study the notions of operator ideals, generating systems of sets and generating systems of sequences and connections between them. The master thesis consists of seven chapters. In the first chapter the necessary notions and general lemmas are introduced. In the second chapter the definition and historical background of the notion of relatively (p; r)-compact sets are given, where 1 p 1; 1 r p and p is the conjugate index of p. It is shown that the relatively (1; 1)-compact sets are exactly the relatively compact sets. In the third chapter we look at the notion of an operator ideal. Let the class of all operator ideals be denoted by OI. An operator is said to be (p; r)-compact if it maps every bounded set to a relatively (p; r)-compact set. We denote the class of all (p; r)-compact operators by K(p;r) and show that K(p;r) 2 OI. The fourth chapter starts with the notion of generating system of sets that was introduced in  by I. Stephani in 1980s. This notion is of importance because it gives a possibility to generate a new operator ideal from two given generating systems of sets. We denote the class of all generating systems of sets by GHS and define a partial order on GHS. Let the class of all relatively (p; r)-compact sets be denoted by K(p;r). The class K(1;1) coincides with the class K of the relatively compact sets. The fifth chapter is devoted to the notion of generating system of sequences that was also introduced in  by Stephani. Let the class of all generating systems of sequences be denoted by GJS. Stephani showed that from a given generating system of sequences it is possible to generate a new generating system of sets. Let g 2 GJS. We denote the system of sets generated from g by ->g . Denote the system of convergent sequences by c. It is easily obtained that ->c = K. We define a system G 2 GHS to be generatable if there exists a system g 2 GJS such that ->g = G. We ask the question: is the system K(p;r) generatable? More generally, given a system G 2 GHS, how to decide whether this system is generatable? We start the sixth chapter by introducing a pre-order on the class GJS. With the help of this pre-order, we define an equivalence relation on GJS and find 57 the corresponding quotient class GJS= . On this class we now introduce a partial order. We then define operations ! and between the classes GJS= and GHS. We show that these operations (!; ) form a Galois connection. Using the Galois connection we give a criterion that allows to decide whether a given generating system of sets is generatable. We also give an answer to the question whether the system of sets K(p;r) is generatable. In the seventh chapter we show that the classes of operator ideals and generating systems of sets and sequences are all examples of lattices.