Compact operators in the class $\left( {bv_k^\theta ,bv} \right)$

Abstract

The space $bv$ of bounded variation sequence plays an important role in the summability. More recently this space has been generalized to the space $bv_k^\theta $ and the class $\left( bv_{k}^{\theta },bv\right) $ of infinite matrices has been characterized by Hazar and Sarıgöl [2]. In the present paper, for $1<k<\infty ,$ we give necessary and sufficient conditions for a matrix in the same class to be compact, where $ \theta $ is a sequence of positive numbers.

Keywords

• Matrix transformations,Sequence spaces,$bv_{k}^{\theta }$ spaces

References

1. [1] E. Malkowsky, V. Rakocevic, S. Živkovic, Matrix transformations between the sequence space bvk and certain BK spaces, Bull. Cl. Sci. Math. Nat. Sci. Math., 123(27) (2002), 33–46.
2. [2] G. C. Hazar, M. A. Sarıgöl, The space bv k and matrix transformations, 8th International Eurasian Converence on Mathematical Sciences and Applications (IECMSA 2019), 2019 (in press).
3. [3] G. C. Hazar, M. A. Sarıgöl, On absolute Nörlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.) 34(5) (2018), 812-826.
4. [4] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad. (Beogr) 9(17) (2000), 143-234.
5. [5] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12 (1998), 87-120.
6. [6] M. A. Sarıgöl, Extension of Mazhar’s theorem on summability factors, Kuwait Jour. Sci., 42(2) (2015), 28-35.
7. [7] M. Stieglitz, H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnisüberischt, Math Z., 154 (1977), 1-16.