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Zbl 1033.46002
Malkowsky, E.; Rakočević, V.; Živković, Snežana
Matrix transformations between the sequence space $bv^p$ and certain $BK$ spaces.
(English)
[J] Bull., Cl. Sci. Math. Nat., Sci. Math. 123, No. 27, 33-46 (2002). ISSN 0561-7332

Let $\omega$ be the set of all complex sequences $x=(x_k)_{k=0}^{\infty}$, $1\le p<\infty$, and $bv^p=\{x\in\omega:\sum_{k=0}^{\infty}\vert x_k-x_{k- 1}\vert ^p<\infty\}$. For $X\subset\omega$, its $\beta$-dual is defined as $X^{\beta}=\{a\in\omega:\sum_{k=0}^{\infty}a_kx_k\text{ converges for all } x\in X\}$. The authors determine the $\beta$-duals of the sets $bv^p$, characterize some matrix transformations between these sets and apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for the entries of an infinite matrix to be a compact operator between the space $bv^p$ and certain $BK$-spaces, where a Banach space is said to be $BK$ if its convergence implies coordinatewise convergence.
MSC 2000:
*46A45 Sequence spaces
40H05 Functional analytic methods in summability
46B45 Banach sequence spaces
47B37 Operators on sequence spaces, etc.

Keywords: matrix transformations; sequence spaces; measure of noncompactness; $BK$-spaces

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