On some new scenario of almost boundedness using matrices
Abstract
The authors M. F. Rahman and A. B. M. R. Karim have structured and studied the space space $r^w_g(t,s)$ and have computed its various properties like completeness, duals and many others as can be seen in \cite{[15]}. The basic structure of this paper is to further study it and investigate for the characterization with sequences of almost bounded $f_\infty$, almost convergent $f$ and almost sequences converging to zero $f_0$. Also, we will prove that $\widetilde{F}$ is not solid, where symbol $\widetilde{F}$ represents space having Riesz transform in $f$.
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References
\bibitem{Mas1} M. M. AlBaidani, Notion of new structure of uncertain sequences using $\Delta$-spaces, Journal of Mathematics, Volume 2022, Article ID 2615772, 6 pages.
\bibitem{Mas2i} M. M. AlBaidani, Statistical convergence of $\Delta$-spaces using fractional Order, Symmetry 2022, 14(8), 1685.
\bibitem{Mas2} M. M. AlBaidani and J. J. McDonald, On the block structure and frobenius normal form of powers of matrices, The electronic journal of linear algebra ELA, 35(1)(2019), 297-306.
\bibitem{Mas4} M. M. AlBaidani, H. M. Srivastava and A. H. Ganie, Notion of non-absolute family of spaces, Int. J. Nonlinear Anal. Appl. In Press, 1–11; http://dx.doi.org/10.22075/ijnaa.2021.24072.2664
\bibitem{[2]} S. Banach, The\"ories des operations lin\'earies, Warszawa, 1932.
\bibitem{[3]} J. Boos, Classical and modern methods in summability, Oxford University Press, Oxford, UK, 2001.
\bibitem{HD1} D. Fathima and A. H. Ganie, Almost convergence property of generalized Riesz spaces, Journal of Applied Mathematics and Computation, 4(4)(2020), 249-253.
\bibitem{D1i} D. Fathima and A. H. Ganie, On some new scenario of $\Delta$- spaces, J. Nonlinear Sci. Appl., 14 (2021), 163-167.
\bibitem{HD3} A. H. Ganie, Sigma bounded sequence and some matrix transformations, Algebra Letters, 3(2013), 1-7.
\bibitem{HD5} A. H. Ganie, and N. A. Sheikh, On some new sequence space of non-absolute type and matrix transformations, Jour. Egyptain Math. Soc., 21( 2013), 34-40.
\bibitem{HD4} A. H. Ganie and N. A. Sheikh, Infinite matrices and almost bounded sequences, Vietnam Journal of Mathematics, 42(2)(2014), 153-157.
\bibitem{[6]} A. H. Ganie and N. A. Sheikh, Infinite matrices and almost convergence, Filomat, 29(6)(2015), 1183-1188.
\bibitem{HD6} A. H. Ganie, B. C. Tripathy, N. A. Sheikh and M. Sen, Invariant means and matrix transformations, Functional Analysis: Theory, Methods \and Applications,2 (2016), 28-33.
\bibitem{HD7} A. H. Ganie, A. Mobin N. A. Sheikh and T. Jalal, New type of Riesz sequence space of non-absolute type, J. Appl. Comput. Math., 5(1)(2016), 1-4.
\bibitem{[9]} S. A. Gupkari, Some new sequence spaces and almost convergence, Filomat, 22(2)(2008), 59-64.
\bibitem{[10]} H. Kizmaz, On certain sequence spaces I. Can. Math. Bull. 25(2)(1981), 169-176.
\bibitem{[11]} G. G. Lorentz, A contribution to the theory of divergent series, Acta Math., (80)(1948), 167-190.
\bibitem{10ij} I. A. Malik and T. Jalal, Measures of noncompactness in $(\overline{N}^q_{\Delta^{-}})$ summable difference sequence space, J. Math. Ext.,
13(4)(2019), 155-171.
\bibitem{[12]} M. Mursaleen, Infinite matrices and almost convergent sequences, Southeast Asian Bull. Math., 19(1)(1995), 45-48.
\bibitem{[13]} S. Nanda, Matrix transformations and almost boundedness, Glas. Mat., 14(34) (1979) 99-107.
\bibitem{[14]} G. M. Petersen, Regular matrix transformations. McGraw-Hill Publishing Co. Ltd., London-New York-Toronto, 1966.
\bibitem{[15]} M. F. Rahman and A. B. M. R. Karim, Generalized Riesz sequence space of non-absolute type and some matrix mappings, Pure Appl. Math. J., 4(3)(2015), 90-95.
\bibitem{[16]} N. A. Sheikh and A. H. Ganie, A new paranormed sequence space and some matrix transformations, Acta Math. Acad. Paedago. Nygr., 28(1)(2012) 47-58.
\bibitem{[17]} N. A. Sheikh and A. H. Ganie, On the spaces of $\lambda$-convergent sequences and almost convergence, Thai J. Math., 11(2)(2013) 393-398.
\bibitem{[18]} J. Tanweer and A. H. Ganie, Almost Convergence and some matrix transformation, Shekhar (New Series)- Int. J. Math., 1(1)(2009) 133-138.
\bibitem{[19]} J. Tanweer, S. A. Gupkari and A. H. Ganie,, Infinite matrices and $\sigma$-convergent sequences, Southeast Asian Bull. Math. 36(6)
(2012), 825-830.
\bibitem{Mas2i} M. M. AlBaidani, Statistical convergence of $\Delta$-spaces using fractional Order, Symmetry 2022, 14(8), 1685.
\bibitem{Mas2} M. M. AlBaidani and J. J. McDonald, On the block structure and frobenius normal form of powers of matrices, The electronic journal of linear algebra ELA, 35(1)(2019), 297-306.
\bibitem{Mas4} M. M. AlBaidani, H. M. Srivastava and A. H. Ganie, Notion of non-absolute family of spaces, Int. J. Nonlinear Anal. Appl. In Press, 1–11; http://dx.doi.org/10.22075/ijnaa.2021.24072.2664
\bibitem{[2]} S. Banach, The\"ories des operations lin\'earies, Warszawa, 1932.
\bibitem{[3]} J. Boos, Classical and modern methods in summability, Oxford University Press, Oxford, UK, 2001.
\bibitem{HD1} D. Fathima and A. H. Ganie, Almost convergence property of generalized Riesz spaces, Journal of Applied Mathematics and Computation, 4(4)(2020), 249-253.
\bibitem{D1i} D. Fathima and A. H. Ganie, On some new scenario of $\Delta$- spaces, J. Nonlinear Sci. Appl., 14 (2021), 163-167.
\bibitem{HD3} A. H. Ganie, Sigma bounded sequence and some matrix transformations, Algebra Letters, 3(2013), 1-7.
\bibitem{HD5} A. H. Ganie, and N. A. Sheikh, On some new sequence space of non-absolute type and matrix transformations, Jour. Egyptain Math. Soc., 21( 2013), 34-40.
\bibitem{HD4} A. H. Ganie and N. A. Sheikh, Infinite matrices and almost bounded sequences, Vietnam Journal of Mathematics, 42(2)(2014), 153-157.
\bibitem{[6]} A. H. Ganie and N. A. Sheikh, Infinite matrices and almost convergence, Filomat, 29(6)(2015), 1183-1188.
\bibitem{HD6} A. H. Ganie, B. C. Tripathy, N. A. Sheikh and M. Sen, Invariant means and matrix transformations, Functional Analysis: Theory, Methods \and Applications,2 (2016), 28-33.
\bibitem{HD7} A. H. Ganie, A. Mobin N. A. Sheikh and T. Jalal, New type of Riesz sequence space of non-absolute type, J. Appl. Comput. Math., 5(1)(2016), 1-4.
\bibitem{[9]} S. A. Gupkari, Some new sequence spaces and almost convergence, Filomat, 22(2)(2008), 59-64.
\bibitem{[10]} H. Kizmaz, On certain sequence spaces I. Can. Math. Bull. 25(2)(1981), 169-176.
\bibitem{[11]} G. G. Lorentz, A contribution to the theory of divergent series, Acta Math., (80)(1948), 167-190.
\bibitem{10ij} I. A. Malik and T. Jalal, Measures of noncompactness in $(\overline{N}^q_{\Delta^{-}})$ summable difference sequence space, J. Math. Ext.,
13(4)(2019), 155-171.
\bibitem{[12]} M. Mursaleen, Infinite matrices and almost convergent sequences, Southeast Asian Bull. Math., 19(1)(1995), 45-48.
\bibitem{[13]} S. Nanda, Matrix transformations and almost boundedness, Glas. Mat., 14(34) (1979) 99-107.
\bibitem{[14]} G. M. Petersen, Regular matrix transformations. McGraw-Hill Publishing Co. Ltd., London-New York-Toronto, 1966.
\bibitem{[15]} M. F. Rahman and A. B. M. R. Karim, Generalized Riesz sequence space of non-absolute type and some matrix mappings, Pure Appl. Math. J., 4(3)(2015), 90-95.
\bibitem{[16]} N. A. Sheikh and A. H. Ganie, A new paranormed sequence space and some matrix transformations, Acta Math. Acad. Paedago. Nygr., 28(1)(2012) 47-58.
\bibitem{[17]} N. A. Sheikh and A. H. Ganie, On the spaces of $\lambda$-convergent sequences and almost convergence, Thai J. Math., 11(2)(2013) 393-398.
\bibitem{[18]} J. Tanweer and A. H. Ganie, Almost Convergence and some matrix transformation, Shekhar (New Series)- Int. J. Math., 1(1)(2009) 133-138.
\bibitem{[19]} J. Tanweer, S. A. Gupkari and A. H. Ganie,, Infinite matrices and $\sigma$-convergent sequences, Southeast Asian Bull. Math. 36(6)
(2012), 825-830.
Published
2025-12-04
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