THE GENERALIZED $B^r-$ DIFFERENCE RIESZ $X^2$SEQUENCE SPACES AND UNIFORM OPIAL PROPERTY

Abstract

We define the new generalized difference Riesz sequence spaces $\Lambda_r^2q(p,B^r)$ and $X_r^2q(p,B^r)$ which consist of all the sequences whose $B^r-$transforms are in the Riesz sequence spaces $r_\infty^q(p), r_c^q(p)$ and$r_0^q(p),$ respectively, introduced by Altay and Basar(2006).We examine some topological properties and compute the $\alpha-, \beta-,$ and $\gamma-$ duals of the spaces$ \Lambda_r^2q(p,B^r)$ and $X_r^2q(p,B^r)$.Fianlly, we determine the necessary and sufficient conditions on the matrix transform from the spaces $\Lambda_r^2q(p,B^r)$ and $X_r^2q(p,B^r)$ to the spaces$\Lambda^2$ and $X^2$ and prove that sequence space $X_r^2q(p,B^r)$ have the uniform Opial property for $p_{mn} \geq 1$ for all $m,n \in N.$