THE GENERALIZED B^r- DIFFERENCE RIESZ X^2SEQUENCE SPACES AND UNIFORM OPIAL PROPERTY - Subramanian, N.; Misra, U. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2011)
We define the new generalized difference Riesz sequence spaces Λ_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_∞^q(p), r_c^q(p) andr_0^q(p), respectively, introduced by Altay and Basar(2006).We examine some topological properties and compute the α-, β-, and γ- duals of the spaces Λ_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 Λ_r^2q(p,B^r) and X_r^2q(p,B^r) to the spacesΛ^2 and X^2 and prove that sequence space X_r^2q(p,B^r) have the uniform Opial property for p_mn ≥ 1 for all m,n N.