A Robust Approach About Compact Operators on Some Generalized Fibonacci Difference Sequence Spaces

Year 2024, Volume: 5 Issue: 1, 48 - 59, 31.01.2024

Murat Candan

https://doi.org/10.54974/fcmathsci.1303769

Abstract

In this new study, which
deals with the different properties of $\ell_{p}(\widehat{F}(r,s))$ $(1\leq
p<\infty)$ and $\ell_{\infty}(\widehat{F}(r,s))$ spaces defined by Candan and
Kara in 2015 by using Fibonacci numbers according to a certain rule, we have
tried to review all the qualities and features that the authors of the
previous editions have found most useful. This document provides everything
needed to characterize the matrix class $(\ell_{1},\ell_{p}(\widehat{F}%
(r,s)))$ $(1\leq p<\infty)$. Using the Hausdorff measure of noncompactness, we
simultaneously provide estimates for the norms of the bounded linear operators
$L_{A}$ defined by these matrix transformations and identify requirements to
derive the corresponding subclasses of compact matrix operators. The results of the current research can be regarded as to be more inclusive and broader
when compared to the similar results available in the literature.

Keywords

Sequence spaces, Fibonacci numbers, Compact operators, Hausdorff measure of noncompactness

References

• [1] Başar F., Summability Theory and Its Applications, 2nd Ed., CRC Press, Taylor and Francis Group, 2022.
• [2] Başar F., Dutta H., Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties, CRC Press, Taylor and Francis Group, Monographs and Research Notes in Mathematics, 2020.
• [3] Mursaleen M., Başar F., Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor and Francis Group, Series: Mathematics and Its Applications, 2020.
• [4] Mursaleen M., Applied Summability Methods, Springer Briefs, 2014.
• [5] De Malafosse B., Malkowsky E., Rakocevic V., Operators Between Sequence Spaces and Applications, Springer, 2022.
• [6] Başar F., Altay B., On the space of sequences of p−bounded variation and related matrix mappings, Ukrains; Matematychnyi Zhurnal, 55(1), 136-147, 2003.
• [7] Altay B., Başar F., Mursaleen M., On the Euler sequence spaces which include the spaces ℓp and ℓ∞, Information Sciences, 176(10), 1450-1462, 2006.
• [8] Altay B., Başar F., Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, Journal of Mathematical Analysis and Applications, 336(1), 632-645, 2007.
• [9] Başar F., Malkowsky E., Altay B., Matrix trasformations on the matrix domains of triangles in the spaces of strongly C1−summable and bounded sequences, Publicationes Mathematicae Debrecen, 73(1-2), 193-213, 2008.
• [10] Başarır M., Başar F., Kara E.E., On the spaces of Fibonacci difference absolutely p−summable, null and convergent sequences, Sarajevo Journals of Mathematics, 12(25), 167-182, 2016.
• [11] Banaś J., Mursaleen M., Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, 2014.
• [12] Wilansky A., Summability Through Functional Analysis, North-Holland Mathematics Studies 85, Elsevier Science Publishers, 1984.
• [13] Goldenštein L.S., Gohberg I.T., Markus A.S., Investigations of some properties of bounded linear operators in connection with their q -norms, Učen Zap Kishinevsk Universty, 29, 29-36, 1957.
• [14] Goldenštein L.S., Markus A.S., On a measure of noncompactness of bounded sets and linear operators, Studies in Algebra and Mathematical Analysis, 45-54, 1965.
• [15] Kuratowski K., Sur les espaces complets, Fundamenta Mathematicae, 15, 301-309, 1930.
• [16] Darbo G., Punti uniti in transformazioni a condominio non compatto, Rendiconti del Seminario Matematico della Università di Padova, 24, 84-92, 1955.
• [17] Akhmerov R.R., Kamenskij M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N., Measures of Noncompactness and Condensing Operators, Operator Theory Advances and Applications, 1992.
• [18] Ayerbe Toledano J.M., Domínguez Benavides T., López Azedo G., Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory Advances and Applications, 1997.
• [19] Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, 1980.
• [20] Malkowsky E., Rakočević V., An introduction into the theory of sequence spaces and measures of noncompactness, Zbornik Radova, Matematički Institut SANU, 9(17), 143-234, 2000.
• [21] Alotaibi A., Malkowsky E., Mursaleen M., Measure of noncompactness for compact matrix operators on some BK spaces, Filomat, 28, 1081-1086, 2014.
• [22] Başarır M., Kara E.E., On compact operators on the Riesz B(m) -difference sequence spaces, Iranian Journal of Science and Technology, 35(A4), 279–285, 2011.
• [23] Başarır M., Kara E.E., On some difference sequence spaces of weighted means and compact operators, Annals of Functional Analysis, 2, 114-129, 2011.
• [24] Başarır M., Kara E.E., On the B-difference sequence space derived by generalized weighted mean and compact operators, Journal of Mathematical Analysis and Applications, 391, 67-81, 2012.
• [25] Kara E.E., Başarır M., On compact operators and some Euler B(m)-difference sequence spaces, Journal of Mathematical Analysis and Applications, 379, 499-511, 2011.
• [26] De Malafosse B., Malkowsky E., Rakočević V., Measure of noncompactness of operators and matrices on the spaces c and c0 , International Journal of Mathematics and Mathematical Sciences, 2006, 1-5, 2006.
• [27] De Malafosse B., Rakočević V., Applications of measure of noncompactness in operators on the spaces sα, Journal of Mathematical Analysis and Applications, 323, 131-145, 2006.
• [28] Mursaleen M., Karakaya V., Polat H., Simsek N., Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Computers and Mathematics with Applications, 62, 814-820, 2011.
• [29] Mursaleen M., Mohiuddine S.A., Applications of measures of noncompactness to the infinite system of differential equations in ℓp spaces, Nonlinear Analysis: Theory, Methods and Applications, 75, 2111-2115, 2012.
• [30] Mursaleen M., Noman A.K., Compactness by the Hausdorff measure of noncompactness, Nonlinear Analysis: Theory, Methods and Applications, 73, 2541-2557, 2010.
• [31] Mursaleen M., Noman A.K., Compactness of matrix operators on some new difference sequence spaces, Linear Algebra and Its Applications, 436, 41-52, 2012.
• [32] Koshy T., Fibonacci and Lucas Numbers with Applications, Wiley, 2001.
• [33] Kara E.E., Some topological and geometrical properties of new Banach sequence spaces, Journal of Inequalities and Applications, 38, 15 pages, 2013.
• [34] Candan M., Kara E.E., A study on topological and geometrical characteristics of new Banach sequence spaces, Gulf Journal of Mathematics 3(4), 67-84, 2015.
• [35] Alotaibi A., Mursaleen M., Alamri B.A.S., Mohiuddine S.A., Compact operators on some Fibonacci difference sequence spaces, Journal of Inequalities and Applications, 203, 8 pages, 2015.
• [36] Kara E.E., Başarır M., Mursaleen M., Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers, Kragujevac Journal of Mathematics, 39(2), 217-230, 2015.
• [37] Malkowsky E., Rakočević V., On matrix domains of triangles, Applied Mathematics and Computation, 187, 1146-1163, 2007.