Domain of the double sequential band matrix in the classical sequence spaces

Murat Candan

doi:10.1186/1029-242x-2012-281

Abstract

Let λ denote any one of the classical spaces , c, and of bounded, convergent, null and absolutely p-summable sequences, respectively, and also be the domain of the double sequential band matrix in the sequence space λ, where and are given convergent sequences of positive real numbers and . The present paper is devoted to studying the sequence space . Furthermore, the β- and γ-duals of the space are determined, the Schauder bases for the spaces , and are given, and some topological properties of the spaces , and are examined. Finally, the classes and of infinite matrices are characterized, where and . MSC:46A45, 40C05.

Highlights

Let λ denote any one of the classical spaces ∞, c, c0 and p of bounded, convergent, null and absolutely p-summable sequences, respectively, and λ be the domain of the double sequential band matrix B(r, s) in the sequence space λ, where∞ n=0 and∞ n=0 are given convergent sequences of positive real numbers and 1 ≤ p < ∞
The main purpose of the present paper is to introduce the sequence space λB(r,s) and to determine the β- and γ -duals of the space, where λ denotes any one of the spaces ∞, c, c or p
In Section, we summarize the studies on the difference sequence spaces

Summary

Nq C m

T S Aru B(r, s) S u u2 G(u, v) G(u, v) Az Ar1 Rt Rt m u (m) B(r, s) B(r, s, t) λA cNq. In Section , we introduce the domain λB(r,s) of the generalized difference matrix B(r, s) in the sequence space λ with λ ∈ { ∞, c, c , p} and determine the β- and γ -duals of λB(r,s). In Section , we state and prove a general theorem characterizing the matrix transformations from the domain of a triangle matrix to any sequence space. The inclusion relation λ ⊂ λ( ) strictly holds He determined the α-, β- and γ duals of the difference spaces and characterized the classes (λ( ) : μ) and (μ : λ( )) of infinite matrices, where λ, μ ∈ { ∞, c}. Malkowsky [ ] determined the Köthe-Toeplitz duals of the sets ∞(p, ) and c (p, ) and gave new proofs of the characterization of the matrix transformations considered in [ ].

Suppose that sup sn inf rn

To From

Conclusion