The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces

• Autorzy: Fernández Julio C. Ramos , Rivera-Sarmiento María A. , Salas-Brown Margot

Identyfikatory

DOI

10.1515/dema-2023-0263

• Języki publikacji: EN

Abstrakty

EN

We calculate the essential norm of bounded diagonal infinite matrices acting on Köthe sequence spaces. As a consequence of our result, we obtain a recent criteria for the compactness of multiplication operator acting on Köthe sequence spaces.

Słowa kluczowe

EN

diagonal operators; Banach sequence spaces; compact operators; essential norm

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20230263
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Fernández Julio C. Ramos

• Facultad de Ciencias Matemáticas y Naturales, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia

autor

Rivera-Sarmiento María A.

• Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia

autor

Salas-Brown Margot

• Programa de Matemáticas, Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Bogotá, Colombia

Bibliografia

• [1] H. Poincaré, Sur les groupes des équations linéaires (French), Acta Math. 4 (1884), no. 1, 201–312, DOI: https://doi.org/10.1007/BF02418420.
• [2] D. Hilbert, Zur Variationsrechnung (German), Math. Ann. 62 (1906), no. 3, 351–370, DOI: https://doi.org/10.1007/BF01450516.
• [3] R. G. Cooke, Infinite Matrices and Sequence Spaces, Dover Publications, New York, 1955.
• [4] M. Bernkopf, A history of infinite matrices. A study of denumerably infinite linear systems as the first step in the history of operators defined on function spaces, Arch. History Exact Sci. 4 (1968), no. 4, 308–358, DOI: https://doi.org/10.1007/BF00411592.
• [5] P. N. Shivakumar and K. C. Sivakumar, A review of infinite matrices and their applications, Linear Algebra Appl. 430 (2009), no. 4, 976–998, DOI: https://doi.org/10.1016/j.laa.2008.09.032.
• [6] P. N. Shivakumar, K. C. Sivakumar and Y. Zhang, Infinite Matrices and Their Recent Applications, Springer International Publishing, Switzerland, 2016.
• [7] J. J. Williams and Q. Ye, Infinite matrices bounded on weighted ℓ1 spaces, Linear Algebra Appl. 438 (2013), no. 12, 4689–4700, DOI: https://doi.org/10.1016/j.laa.2013.02.023.
• [8] D. Foroutannia and H. Roopaei, Bounds for the norm of lower triangular matrices on the Cesàro weighted sequence space, J. Inequal. Appl. 2017 (2017), no. 67, 1–11, DOI: https://doi.org/10.1186/s13660-017-1339-6.
• [9] H. Hudzik, R. Kumar, and R. Kumar, Matrix multiplication operators on Banach function spaces, Proc. Indian Acad. Sci. Math sci. 116 (2006), no. 1, 71–81, DOI: https://doi.org/10.1007/BF02829740.
• [10] J. C. Ramos-Fernández and M. Salas-Brown, On multiplication operators acting on Köthe sequence spaces, Afr. Mat. 28 (2017), no. 3–4, 661–667, DOI: https://doi.org/10.1007/s13370-016-0475-3.
• [11] S. C. Arora, G. Datt, and S. Verma, Operators on Lorentz sequence spaces, Math. Bohem. 134 (2009), no. 1, 87–89, DOI: https://mb.math.cas.cz/full/134/1/mb134_1_9.pdf.
• [12] P. Bala, A. Gupta, and N. Bhatia, Multiplication operators on Orlicz-Lorentz sequence spaces, Int. J. Math. Anal. (Ruse) 7 (2013), no. 29–32, 1461–1469, DOI: http://dx.doi.org/10.12988/ijma.2013.3360.
• [13] B. Sagggir, O. Ogggur, and C. Duyar, Multiplication operators on Orlicz-Lorentz sequence spaces, J. Math. Comput. Sci. 5 (2015), no. 3, 265–272, DOI: http://scik.org/index.php/jmcs/article/download/2112/1122.
• [14] B. S. Komal, S. Pandoh, and K. Raj, Multiplication operators on Cesàro sequence spaces, Demonstr. Math. 49 (2016), no. 4, 430–436, DOI: https://doi.org/10.1515/dema-2016-0037.
• [15] K. Raj, C. Sharma, and S. Pandoh, Multiplication operators on Cesàro-Orlicz sequence spaces, Fasc. Math. 57 (2016), 137–145, DOI: https://doi.org/10.1515/fascmath-2016-0021.
• [16] R. E. Castillo, J. C. Ramos-Fernández, and M. Salas-Brown, The essential norm of multiplication operators on Lorentz sequence spaces, Real Anal. Exchange. 41 (2016), no. 1, 245–251, https://projecteuclid.org/journalArticle/Download?urlId=rae%2F1490752826.
• [17] J. C. Ramos-Fernández, M. A. Rivera-Sarmiento, and M. Salas-Brown, On the essential norm of multiplications operators acting on Cesàro sequence spaces, J. Funct. Spaces 2019 (2019), no. 5069610, 1–5, DOI: https://doi.org/10.1155/2019/5069610.
• [18] J. C. Ramos-Fernández and M. Salas-Brown, The essential norm of multiplication operators acting on Orlicz sequence spaces, Proyecciones 39 (2020), no. 6, 1407–1414, DOI: https://doi.org/10.22199/issn.0717-6279-2020-06-0086.
• [19] G. Köthe and O. Toeplitz, Lineare Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math. 171 (1934), 193–226, DOI: https://doi.org/10.1515/crll.1934.171.193.
• [20] G. Köthe, Die Stufenräume, eine einfache Klasse lineare vollkommener Räume, Math. Z. 51 (1948), 317–345, DOI: https://doi.org/10.1007/BF01181598.
• [21] G. Köthe, Neubegrü ndung der Theorie der vollkommenen Räume, Math. Nachr. 4 (1950), 70–80, DOI: https://doi.org/10.1002/mana.3210040109.
• [22] J. Dieudonné, Sur les espaces de Köthe, J. Analyse Math. 1 (1951), 81–115, DOI: https://doi.org/10.1007/BF02790084.
• [23] M.J. Cooper, Coordinated linear spaces, Proc. London Math. Soc. 3 (1953), no. 3, 305–327, DOI: https://doi.org/10.1112/plms/s3-3.1.305.
• [24] G. G. Lorentz and D. G. Wertheim, Representation of linear functionals on Köthe spaces, Canad. J. Math. 5 (1953), 568–575, DOI: https://doi.org/10.4153/CJM-1953-064-4.
• [25] L. V. Kantorovich and G. P. Akilov. Functional Analysis, Nauka, Moscow, 1977.
• [26] J. B. Conway, A Course in Functional Analysis, 2nd edition, Springer International Publishing, Switzerland, 1990.
• [27] S. Axler, N. Jewell, and A. Shields, The essential norm of an operator and its adjoint, Trans. Amer. Math. Soc. 261 (1980), no. 1, 159–167, DOI: https://doi.org/10.1090/S0002-9947-1980-0576869-9.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).