Several Researches And Generalizations Of Ideal Convergence

Ideal convergence is one of the most important research contents in the study of statistical convergence, the main topics of this thesis is to investigate the equivalent characterizations of the additive property (called AP for short) of an ideal I,the equivalent representations of I-A-statistical convergence and the decompositions of vector measures.Kostyrko, Saldt and Wilczynski have presented the relationships and the differences between I-convergence and I*-convergence, meanwhile they proved the equivalence between I-convergence and I*-convergence for those ideals with the additive property. Basing on that, first in this thesis, we obtain some equivalent characterizations of AP in terms of I-convergence and I*-convergence of sequences in a Banach space X; we also investigate the relationships among w-I-convergence, w-I*-convergence and uniformly w-I*-convergence, as well as the relationship between w-I-convergence and convergence(Chapter2). Secondly, we define a new type of statistical convergence called I-A-statistical convergence as a generalization of I-convergence; and we establish the equivalent descriptions on I-A-statistical convergence via subdifferential mapping of continuous convex functions defined on l∞; we also investigate the differences between I-A-statistical convergence and I-A-summability(Chapter3). Finally, by employing the one-to-one correspondence between vector-valued measures and operators, we provide a new representation to countably additive vector measures from the angle of operators; and by making use of the extended Yosida-Hewitt theorem, we prove that an operator, which is defined on B(Ω,Σ)=span{XA,A∈Σ} and takes values in a reflexive Banach space X, can be uniquely decomposed into the sum of a w*-norm sequentially continuous operator and a purely continuous operator(Chapter4).