On the Matrix Representations of Operators on the Classical Sequence Spaces

Yıl 2019, , 489 - 498, 24.03.2019

Erdal Bayram

https://doi.org/10.18185/erzifbed.508746

Öz

Sunulan çalışma, Banach örgüsü
yapısına sahip bazı klasik dizi uzayları üzerinde tanımlı olan L-zayıf ve
M-zayıf kompakt operatörlerin matris temsilleri için gerekli ve yeterli
koşullar sağlar. Bu operatör sınıflarının, Banach örgüleri üzerinde tanımlı
zayıf kompakt ve kompakt operatörlerle çakışabildiği bilinmektedir. Böylece
kompakt ve zayıf kompakt operatörler için bilinen bazı sonuçlar L-zayıf ve
M-zayıf kompaktlık açısından farklı bir alternatif olarak sunulmuş oldu.

Anahtar Kelimeler

Matris dönüşümleri, Zayıf kompakt operatör, Kompakt operatör, L-zayıf kompakt operatör, M-zayıf kompakt operatör

On the Matrix Representations of Operators on the Classical Sequence Spaces

Öz

The present study provides the necessary and sufficient conditions for the matrix characterizations of 𝐿- and 𝑀-

weakly compact operators which are defined on certain classical sequence spaces as Banach lattices. It is known

that these operators may coincide with both weakly compact and compact operators on Banach lattices. Our

study offers a different alternative to some known results for the matrix characterizations of compact and

weakly compact operators which are presented in terms of L- and M-weakly compactness.

Anahtar Kelimeler

Matrix transformation, Weakly compact operator, Compact operator, L-weakly compact operator, M-weakly compact operator

Kaynakça

• Aliprantis C. D., Burkinshaw O. (1985). "Positive Operators", Springer, Dordecht.
• Altın Y., Et M. 2005, “Generalized difference sequence spaces defined by a modulus function in a locally convex space”, Soochow J.Math., 31(2), 233-243.
• Aqzzouz B., Elbour A., Wickstead A.W. 2011. "Compactness of L-weakly and M-weakly Compact Operators on Banach Lattices, Rend.Circ. Mat. Palermo, 60, 43-50.
• Basarır M., Kara E.E. 2011, “On some difference sequence spaces of weighted means and compact operators”, Ann.Func.Anal., 2(2),114-129.
• Chen Z.L., Wickstead A.W. 1999. "L-weakly and M-weakly compact operators", Indag. Math. (N.S.), 10(3), 321-336.
• Çolak R., Et M., “On some generalized difference sequence spaces and related matrix transformations”, Hokkaido Mathematical Journal, 26(3), 483-492.
• Djolovic I. 2003, "Two ways to compactness", Filomat, 17, 15-21.
• Djolovic I., Malkowsky E. 2008. “A note on compact operators on matrix domains”, J.Math.Anal.Appl., 340, 291-303.
• İlkhan M., Kara E.E. 2018, “Compactness of matrix operators on the Banach space l_p (T)”, Conference Proceedings of Science and Technology, 1(1),11-15.
• Jarrah A.M., Malkowsky E. 2003. "Ordinary, Absolute and Strong Summability and Matrix Transformations", Filomat, 17, 59-78.
• Maddox I.J. (1971). "Elements of Functional Analysis", Cambridge University Press.
• Maddox I.J. (1980). "Infinite Matrices of Operators, Lecture Notes in Mathematics 780, Springer-Verlag.
• Malkowsky E. 2013, “Characterization of compact operators between certain BK spaces”, Filomat, 27(3), 447-457.
• Meyer-Nieberg P. 1974. "Uber klassen schwach kompakter operatoren in Banachverbanden", Math.Z., 138, 145-159.
• Meyer-Nieberg P. (1991). "Banach Lattices", Springer-Verlag, Berlin.
• Mursaleen M. (2014). "Applied Summability Methods", Springer.
• Sargent W.L.C.1966. "On Compact Matrix Transformations Between Sectionally Bounded BK-spaces", Journal London Math.Soc., 41, 79-87.
• Stieglitz M., Tietz H. 1977. "Matrixtransformationen von Folgenraumen Eine Ergebnisübersicht", Math. Zeitschrift, 154, 1-16.
• Wilansky A. (1984). "Summability through Functional Analysis", North-Holland Mathematical Studies 85, Elsevier Science Publishers.
• Wilansky A. 1985, "What Infinite Matrices Can Do”, Mathematics Magazine, 58(5), 281-283.