Some Results on Operator Theory Based on Unbounded Convergence

Abstract

Main topic of the this work is to use the notion of unbounded convergences on vector lattices to derive properties of various classes of operators and nets of operators defined between vector lattices, Banach lattices, and more generally, between locally solid vector lattices. The classes formed by $uaw$-Dunford-Pettis operators, u\tau-compact operators and $\cc$-Lotz-R{\"a}biger nets are among those classes investigated in this work. We present several properties of these classes with the help of new perspectives provided by unbounded convergences. In addition, various examples with completely new origins are given.

First main chapter deals with uaw-Dunford-Pettis operators. As a result of the theory of classical Dunford-Pettis operators, it is expected that uaw-Dunford-Pettis operators have connections with certain classical classes of operators acting on Banach lattices. Hence, one of the aims to study uaw-Dunford-Pettis operators is to determine their relations with other types of operators. Further, we study domination and iteration properties of $uaw$-Dunford-Pettis operators.

The second class of operators that we investigate is the class of u\tau-compact operators defined between locally solid vector lattices. In this general setting, various notions related to boundedness of operators play a central role. Hence, one of the aims to study u\tau-compact operators is to determine the effect of boundedness on compact operators.

In the last chapter of this work, we study a generalization of norm ergodic operators. The main method is to use various versions asymptotic equivalences to study properties of Lotz-R{\"a}biger nets defined between convergence vector lattices. Because Lotz-R{\"a}biger nets are closely related to the classical notion of ergodic nets and ergodic operators, some of our results further apply to the particular case of classical ergodic operators.