## Digitized Theses

1994

Dissertation

#### Degree Name

Doctor of Philosophy

#### Abstract

We investigate matrices and sequences of operators as bounded linear operators on Banach sequence spaces in various situations, and some topics related to these matrices and sequences. This thesis consists of five chapters.;In the first chapter we study whether an infinite matrix, particularly a summability matrix, is a bounded linear operator on {dollar}l\sb{lcub}p{rcub} (p \ge{dollar} 1). Some restrictive conditions for Norlund and weighted mean matrices to be in {dollar}B(l\sb{lcub}p{rcub}){dollar} imposed by earlier authors we eliminated. Some results for weighted mean matrices are proved as consequences of more general results for generalized Hausdorff matrices.;A necessary and sufficient condition for a non-negative matrix to map {dollar}l\sb{lcub}p{rcub}{dollar} to {dollar}l\sb{lcub}q{rcub}{dollar} (1 {dollar}\le q \le p < \infty{dollar}) is refined in the second chapter, and a generalized Vere-Jones conjecture related to this problem is solved.;The third chapter is on the topic of subalgebras {dollar}\Gamma\sb{lcub}\omega{rcub}{dollar} and {dollar}\Omega\sb{lcub}\omega{rcub}{dollar} of B(X), where X is a nonreflexive Banach space and {dollar}\omega \in X\sp{lcub}\*\*{rcub}{dollar}. These algebras arise as generalizations of the classical algebra of 'conservative matrices', i.e., matrices that are in B(c). Algebraic and set-theoretic relationships between these subalgebras are studied. The relationships among such subalgebras of {dollar}B(c\sb0{dollar}) are clarified. For {dollar}B(l\sb1{dollar}), the subalgebras associated with any Dirac measure, i.e., a unit point mass, are found to be isomorphic, but never equal, to those associated with some Banach limit, i.e., a translation invariant extended limit. With the aid of the Stone-Cech compactification of N, the intersections of all {dollar}\Gamma{dollar}'s or all {dollar}\Omega{dollar}'s associated with Banach limits or Dirac measures are characterized.;In chapter 4, we investigate operators on sequence spaces with terms in a Banach space. Some classes of transformations e.g., those between spaces of convergent sequences are characterized. In the course of the work various questions about weak, norm, unconditional and absolute convergence arise and are discussed.;The last chapter is concerned with generalized Kothe-Toeplitz duals of Banach sequence spaces. The relationship between the various duals of {dollar}c\sb0(X),{dollar} c(X), {dollar}l\sb{lcub}\infty{rcub}(X),{dollar} and bv(X) are examined.

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