Abstract
The difference sequence spaces , and were introduced by Kizmaz (Can. Math. Bull. 24:169-176, 1981). In this paper, we introduce the Cesaro summable difference sequence space which strictly includes the spaces and but overlaps with . It is shown that the newly introduced space turns out to be an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Kothe-Toeplitz duals of are computed and as an application, the matrix classes , and are also characterized. MSC:40C05, 40A05, 46A45.
Highlights
We introduce the Cesáro summable difference sequence space C1( ) which strictly includes the spaces c0( ) and c( ) but overlaps with ∞( )
The notion of difference sequence space was introduced by Kizmaz [ ] in as follows: X( ) = x = ∈ s : ∈ X
We introduce a sequence space C ( ), Cesàro summable difference sequence space, as follows: C ( ) = x = ∈ s : ∈ C
Summary
By s we shall denote the linear space of all complex sequences over C (the field of complex numbers). ∞, c and c denote the spaces of all bounded, convergent and null sequences x = (xk) with complex terms, respectively, normed by x ∞ = supk |xk|. A complete metric linear space is called a Frèchet space. C we shall denote the linear space of all (C, ) summable sequences of complex numbers over C, i.e.,. The notion of difference sequence space was introduced by Kizmaz [ ] in as follows: X( ) = x = (xk) ∈ s : ( xk) ∈ X for X = ∞, c, c ; where xk = xk – xk+ for all k ∈ N (the set of natural numbers). For a detailed account of difference sequence spaces, one may refer to [ – ] where many more references can be found
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