zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Applications of measure of noncompactness in matrix operators on some sequence spaces. (English) Zbl 1237.47039
Summary: We determine the conditions for some matrix transformations from $n(\phi)$, where the sequence space $n(\phi)$, which is related to the $\ell_p$ spaces, was introduced by {\it W. L. C. Sargent} [J. Lond. Math. Soc. 35, 161--171 (1960; Zbl 0090.03703)]. 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.

47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
46A45Sequence spaces
Full Text: DOI
[1] A. Wilansky, Summability Through Functional Analysis, vol. 85, North-Holland, Amsterdam, The Netherlands, 1984. · Zbl 0531.40008
[2] E. Malkowsky and V. Rako\vcević, “An introduction into the theory of sequence spaces and measures of noncompactness,” Zbornik Radova, vol. 9, no. 17, pp. 143-234, 2000. · Zbl 0996.46006
[3] 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. · Zbl 0090.03703 · doi:10.1112/jlms/s1-35.2.161
[4] E. Malkowsky and M. Mursaleen, “Matrix transformations between FK-spaces and the sequence spaces m(\varphi ) and n(\varphi ),” Journal of Mathematical Analysis and Applications, vol. 196, no. 2, pp. 659-665, 1995. · Zbl 0846.40004 · doi:10.1006/jmaa.1995.1432
[5] E. Malkowsky and M. Mursaleen, “Compact matrix operators between the spaces m(\varphi ), n(\varphi ) and \ell p,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 1093-1103, 2011. · Zbl 1231.47028 · doi:10.4134/BKMS.2011.48.5.1093 · http://www.mathnet.or.kr/mathnet/kms_content.php?no=408258
[6] 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).
[7] 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. · Zbl 0748.47045
[8] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60, Marcel Dekker Inc., New York, NY. USA, 1980. · Zbl 0441.47056
[9] B. de Malafosse and V. Rako\vcević, “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. · Zbl 1106.47029 · doi:10.1016/j.jmaa.2005.10.024
[10] F. Ba\csar 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. · Zbl 1213.47019 · doi:10.1016/j.amc.2010.12.007
[11] E. E. Kara and M. Ba\csarir, “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. · Zbl 1236.46015 · doi:10.1016/j.jmaa.2011.01.028
[12] M. Mursaleen, V. Karakaya, H. Polat, and N. Sim\csek, “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. · Zbl 1247.47009 · doi:10.1016/j.camwa.2011.06.011
[13] 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. · Zbl 1256.47060 · doi:10.1016/j.na.2011.10.011
[14] 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. · Zbl 1211.47061 · doi:10.1016/j.na.2010.06.030
[15] 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. · Zbl 1201.40002 · doi:10.1016/j.camwa.2010.06.005
[16] 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. · Zbl 1263.47040 · doi:10.1016/j.aml.2011.03.045
[17] 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. · Zbl 1227.47032 · http://files.ele-math.com/abstracts/oam-05-35-abs.pdf
[18] 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. · Zbl 1231.47029 · doi:10.1016/j.laa.2011.06.014
[19] M. Mursaleen, “Application of measure of noncompactness to infinite systems of differential equations,” Canadian Mathematical Society. In press. · Zbl 1275.47133 · doi:10.4153/CMB-2011-170-7