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Compactness of matrix operators on some new difference sequence spaces. (English) Zbl 1231.47029

The authors establish some identities or estimates for the Hausdorff measures of noncompactness of certain matrix operators on the difference sequence spaces

c o λ (Δ)=(x k ) : lim n 1 λ n k=0 n (λ k -λ k-1 ) (x k -x k-1 ) = 0

and

λ (Δ)=(x k ) : sup n 1 λ n k=0 n (λ k -λ k-1 )(x k -x k-1 ) < + ,

where λ=λ k is a strictly increasing sequence of positive real numbers tending to infinity; see [M. Mursaleen and A. K. Noman, Math. Comput. Modelling 52, No.  3–4, 603–617 (2010; Zbl 1201.40003)]. Furthermore, they characterize some classes of compact operators on these spaces.

MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
46B45Banach sequence spaces
46B50Compactness in Banach (or normed) spaces
46B15Summability and bases in normed spaces
References:
[1]Başar, F.; Malkowsky, E.; Altay, B.: Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequences, Publ. math. Debrecen 73, No. 1 – 2, 193-213 (2008) · Zbl 1164.46003
[2]Djolović, I.: Compact operators on the spaces a0r(Δ) and acr(Δ), J. math. Anal. appl. 318, No. 2, 658-666 (2006) · Zbl 1099.47021 · doi:10.1016/j.jmaa.2005.05.085
[3]Djolović, I.; Malkowsky, E.: A note on compact operators on matrix domains, J. math. Anal. appl. 340, No. 1, 291-303 (2008) · Zbl 1147.47002 · doi:10.1016/j.jmaa.2007.08.021
[4]Djolović, I.; Malkowsky, E.: Matrix transformations and compact operators on some new mth-order difference sequences, Appl. math. Comput. 198, No. 2, 700-714 (2008) · Zbl 1148.46007 · doi:10.1016/j.amc.2007.09.008
[5]Djolović, I.; Malkowsky, E.: A note on Fredholm operators on (c0)T, Appl. math. Lett. 22, No. 11, 1734-1739 (2009) · Zbl 1188.47012 · doi:10.1016/j.aml.2009.06.012
[6]Malkowsky, E.: Compact matrix operators between some BK spaces, Modern methods of analysis and its applications, 86-120 (2010)
[7]E. Malkowsky, V. Rakočević, An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik radova 9 (17), Mat. institut SANU (Beograd), 2000, pp. 143 – 234. · Zbl 0996.46006
[8]Malkowsky, E.; Rakočević, V.: On matrix domains of triangles, Appl. math. Comput. 189, No. 2, 1146-1163 (2007) · Zbl 1132.46011 · doi:10.1016/j.amc.2006.12.024
[9]Malkowsky, E.; Rakočević, V.; Živković, S.: Matrix transformations between the sequence spaces w0p(Λ),v0p(Λ),c0p(Λ) (1<p<) and certain BK spaces, Appl. math. Comput. 147, No. 2, 377-396 (2004) · Zbl 1035.46001 · doi:10.1016/S0096-3003(02)00674-4
[10]Mursaleen, M.; Noman, A. K.: On some new difference sequence spaces of non-absolute type, Math. comput. Modelling 52, No. 3 – 4, 603-617 (2010) · Zbl 1201.40003 · doi:10.1016/j.mcm.2010.04.006
[11]Mursaleen, M.; Noman, A. K.: Compactness by the Hausdorff measure of noncompactness, Nonlinear anal.: TMA 73, No. 8, 2541-2557 (2010) · Zbl 1211.47061 · doi:10.1016/j.na.2010.06.030
[12]Stieglitz, M.; Tietz, H.: Matrixtransformationen von folgenräumen eine ergebnisübersicht, Math. Z. 154, 1-16 (1977) · Zbl 0331.40005 · doi:10.1007/BF01215107