## Abstract and Applied Analysis

### Applications of Measure of Noncompactness in Matrix Operators on Some Sequence Spaces

#### Abstract

We determine the conditions for some matrix transformations from $n(\varphi )$, where the sequence space $n(\varphi )$, which is related to the ${\ell }_{p}$ spaces, was introduced by Sargent (1960). We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 378250, 10 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495660

Digital Object Identifier
doi:10.1155/2012/378250

Mathematical Reviews number (MathSciNet)
MR2898049

Zentralblatt MATH identifier
1237.47039

#### Citation

Mursaleen, M.; Latif, A. Applications of Measure of Noncompactness in Matrix Operators on Some Sequence Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 378250, 10 pages. doi:10.1155/2012/378250. https://projecteuclid.org/euclid.aaa/1355495660

#### References

• A. Wilansky, Summability Through Functional Analysis, vol. 85, North-Holland, Amsterdam, The Netherlands, 1984.
• E. Malkowsky and V. Rakočević, “An introduction into the theory of sequence spaces and measures of noncompactness,” Zbornik Radova, vol. 9, no. 17, pp. 143–234, 2000.
• W. L. C. Sargent, “Some sequence spaces related to the ${\ell }_{p}$ spaces,” Journal of the London Mathematical Society, vol. 35, pp. 161–171, 1960.
• E. Malkowsky and M. Mursaleen, “Matrix transformations between FK-spaces and the sequence spaces $m(\phi )$ and $n(\phi )$,” Journal of Mathematical Analysis and Applications, vol. 196, no. 2, pp. 659–665, 1995.
• E. Malkowsky and M. Mursaleen, “Compact matrix operators between the spaces $m(\phi )$, $n(\phi )$ and ${\ell }_{p}$,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 1093–1103, 2011.
• I. T. Gohberg, L. S. Goldenstein, and A. S. Markus, “Investigations of some properties of bounded linear operators with their q-norms,” Ucen. Zap. Kishinevsk.Univ, vol. 29, pp. 29–36, 1957 (Russian).
• R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, vol. 55, Birkhäuser, Basel, Switzerland, 1992.
• J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60, Marcel Dekker Inc., New York, NY. USA, 1980.
• B. de Malafosse and V. Rakočević, “Applications of measure of noncompactness in operators on the spaces ${s}_{\alpha }$, ${s}_{\alpha }^{0}$, ${s}_{\alpha }^{(c)}$, ${\ell }_{\alpha }^{p}$,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 131–145, 2006.
• F. Başar and E. Malkowsky, “The characterization of compact operators on spaces of strongly summable and bounded sequences,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5199–5207, 2011.
• E. E. Kara and M. Başarir, “On compact operators and some Euler ${B}^{(m)}$-difference sequence spaces,” Journal of Mathematical Analysis and Applications, vol. 379, no. 2, pp. 499–511, 2011.
• M. Mursaleen, V. Karakaya, H. Polat, and N. Simşek, “Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 814–820, 2011.
• M. Mursaleen and S. A. Mohiuddine, “Applications of measures of noncompactness to the infinite system of differential equations in ${\ell }_{p}$ spaces,” Nonlinear Analysis Theory, Methods & Applications, vol. 75, no. 4, pp. 2111–2115, 2012.
• M. Mursaleen and A. K. Noman, “Compactness by the Hausdorff measure of noncompactness,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 8, pp. 2541–2557, 2010.
• M. Mursaleen and A. K. Noman, “Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1245–1258, 2010.
• M. Mursaleen and A. K. Noman, “On $\sigma$-conservative matrices and compact operators on the space ${V}_{\sigma }$,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1554–1560, 2011.
• M. Mursaleen and A. K. Noman, “The hausdorff measure of noncompactness of matrix operators on some BK spaces,” Operators and Matrices, vol. 5, no. 3, pp. 473–486, 2011.
• M. Mursaleen and A. K. Noman, “Compactness of matrix operators on some new difference sequence spaces,” Linear Algebra and Its Applications, vol. 436, no. 1, pp. 41–52, 2012.
• M. Mursaleen, “Application of čommentComment on ref. [19?]: Please update the information of this reference, if possible.measure of noncompactness to infinite systems of differential equations,” Canadian Mathematical Society. In press.