Compactness of Matrix Operators on the Banach Space $\ell_p(T)$

Year 2018, Volume: 1 Issue: 1, 11 - 15, 14.12.2018

Merve İlkhan Emrah Evren Kara

Abstract

In this study, by using the Hausdorff measure of non-compactness, we obtain the necessary and sufficient conditions for certain matrix operators on the spaces $\ell_p(T)$ and $\ell_\infty(T)$ to be compact, where $1\leq p<\infty$.

Keywords

Compact operators, Hausdorff measure of non-compactness, Sequence spaces

References

• [1] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik radova, Matematicki Inst. SANU, Belgrade, 9(17) (2000), 143–234.
• [2] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12(2) (1998), 87–120.
• [3] M. Başarır, E. E. Kara, On compact operators on the Riesz B(m)-difference sequence spaces, Iran. J. Sci. Technol., 35(A4) (2011), 279–285.
• [4] M. Başarır, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal., 2 (2011), 114–129.
• [5] M. Başarır, E. E. Kara, On the B-difference sequence space derived by generalized weighted mean and compact operators, J. Math. Anal. Appl., 391 (2012), 67–81.
• [6] M. Mursaleen, V. Karakaya, H. Polat, N. Şimşek, Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Comput. Math. Appl., 62 (2011), 814–820.
• [7] M. Mursaleen, S. A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $\ell_p$ spaces, Nonlinear Anal., 75 (2012), 2111–2115.
• [8] M. Mursaleen, A. K. Noman, Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Inequal. Appl., 16(1) (2013), 207–220.
• [9] M. Mursaleen, A. K. Noman, The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Oper. Matrices, 5(3) (2011), 473–486.
• [10] E. E. Kara, M. İlkhan, On some Banach sequence spaces derived by a new band matrix, Br. J. Math. Comput. Sci. 9(2) (2015), 141–159.
• [11] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73(8) (2010), 2541–2557.