Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$

Merve Esra Türkay

Research Article

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Year 2022, Volume: 10 Issue: 2 , 313 - 325 , 31.10.2022

Merve Esra Türkay

https://izlik.org/JA85YB98RZ

Abstract

References

[1] A.M. Akhmedov and S.R. El-Shabrawy, Spectra and fine spectra of lower triangular double-band matrices as operators on `p (1 p < ¥), Math. Slovaca, 65 (5) (2015), 1137–1152.
[2] A.M. Akhmedov and F. Bas¸ar, On the fine spectrum of the Ces`aro operator in c0, Math. J. Ibaraki Univ., 36 (2004), 25-32.
[3] A.M. Akhmedov and F. Bas¸ar, On the fine spectra of the difference operator D over the sequence space lp, (1 p < ¥), Demonstratio Math., 39 (3) (2006), 585-595.
[4] A.M. Akhmedov and F. Bas¸ar, The fine spectra of the Ces`aro operator C1 over the sequence space bvp, (1 p < ¥)), Math. J. Okayama Univ., 50 (2008), 135-147.
[5] A.M. Akhmedov and F. Bas¸ar, The fine spectra of the difference operator D over the sequence space bvp, (1 p < ¥)), Acta Math. Sin. Eng. Ser., 23 (10) (2007), 1757-1768.
[6] B. Altay and F. Bas¸ar, The fine spectrum and the matrix domain of the difference operator on the sequence space lp, (0 < p < 1), Commun. Math. Anal., 2 (2) (2007), 1-11.
[7] RKh.Amirov, N. Durna and M. Yildirim, Subdivisions of the spectra for Ces`aro, Rhaly and Weighted mean operator on `p, c and `p, IJST, A3 (2011), 175-183.
[8] J. Appell, E. De Pascale and A. Vignoli, Nonlinear Spectral Theory. Berlin, New York, Walter de Gruyter, 2004.
[9] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the Spectra for Generalized Difference Operator Dv on the Sequence Space `1, ICMS., 1309 (2010), 254-260.
[10] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the spectra for the triple band matrix over certain sequence spaces, Gen. Math. Notes, 4 (1) (2011), 35-48.
[11] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the spectra for genarilized difference operator over certain sequence spaces, Thai J. Math., 9 (1) (2011), 285-295.
[12] F. Bas¸ar, N. Durna and M. Yildirim, Subdivision of the spectra for difference operator over certain sequence space, Malays. J. Math. Sci., 6 (2012), 151-165.
[13] F. Bas¸ar, Summability Theory and its Applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton London New York, (in press) 2022.
[14] A. Brown, P.R. Halmos and A.L. Shields, Ces`aro operators. Acta Sci. Math., (Szeged). 26 (1965), 125–137.
[15] D.W. Boyd, The spectrum of the Ces`aro operator, Acta Sci. Math., (Szeged). 29 (1968), 31–34.
[16] G.P. Curbera and W.J. Ricker, Spectrum of the Ces`aro operator in `p, Arch. Math., 100 (2013), 267–271.
[17] N. Durna and M.cYildirim, Subdivision of the spectra for factorable matrices on c and `p, Math. Commun., 16 (2) (2011), 519-530.
[18] N. Durna and M. Yildirim, Subdivision of the Spectra for Factorable Matrices on c0, Gazi Univ. J. Sci., 24 (1) (2011), 45-49.
[19] N. Durna, M. Yildirim and C¸ . U¨ nal, On The Fine Spectrum of Generalized Lower Triangular Double Band Matrices Duv Over The Sequence Space c0, Cumhuriyet Sci. J., 37 (3) (2016), 281-291.
[20] N. Durna, Subdivision of the spectra for the generalized upper triangular double-band matrices Duv over the sequence spaces c0 and c, ADYUSCI, 6 (1) (2016), 31-43.
[21] N. Durna, Subdivision of the spectra for the generalized difference operator Da;b on the sequence space `p (1 < p < ¥), CBU J. Sci., 13 (2) (2017), 359–364.
[22] N. Durna, M. Yildirim and R. Kılıc¸, Partition of the spectra for the generalized difference operator B(r; s) on the sequence space cs, Cumhuriyet Sci. J., 39 (1) (2018), 7–15.
[23] N. Durna, Subdivision of spectra for some lower triangular doule-band matrices as operators on c0, Ukr. Mat. Zh., 70 (7) (2018), 913–922.
[24] N. Durna and M.E. T¨urkay, The spectrum of q-Ces`aro matrices on c and Its various spectral decomposition for 0 < q < 1, Oper. Matrices, 15 (3) (2020), 795-813.
[25] E. D¨undar and F. Bas¸ar, On the fine spectrum of the upper triangle double band matrix D+ on the sequence space c0, Math. Commun., 18 (2) (2013), 337–348.
[26] S.R. El-Shabrawy, On the fine spectrum of the generalized difference operator Da;b over the sequence space `p, (1 < p < ¥), Appl. Math. Inf. Sci., 6 (1S) (2012), 111–118.
[27] S.R. El-Shabrawy, Spectra and fine spectra of certain lower triangular double-band matrices as operators on c0, J. Inequal. Appl., 2014 (1) (2014), 1-9.
[28] S.R. El-Shabrawy and S.H. Abu-Janah, Spectra of the generalized difference operator on the sequence spaces bv0 and h, Linear and Multilinear Algebra 66 (8) (2018), 1691–1708.
[29] S.R. El-Shabrawy, On q-Ces`aro Operators:Boundness, Compactness and Spectra, Numer.Funct.Anal.Optim., 41 (2) (2021), 1019-1037.
[30] J. Fathi and L. Rahmatollah, On the fine spectra of the generalized difference operator Duv over the sequence space `p, JMMRC, 1 (1) (2012), 1-12.
[31] H. Furkan, H. Bilgic¸ and F. Bas¸ar, On the fine spectrum of the operator B(r; s; t) over the sequence spaces `p and bvp, (1 < p < ¥), Comput. Math. Appl., 60 (7) (2010), 2141–2152.
[32] S. Goldberg, Unbounded Linear Operators, McGraw Hill, New York, 1966.
[33] M. Gonzalez, The fine spectrum of Ces`aro operator in `p (1 < p < ¥), Arch. Math., 44 (1985), 355-358.
[34] V. Karakaya and M. Altun, Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math., 234 (2010), 1387–1394.
[35] V. Karakaya, M.D. Manafov and N. S¸ims¸ek, On the fine spectrum of the second order difference operator over the sequence spaces `p and bvp, (1 < p < ¥). Math. Comput. Modelling, 55 (3-4) (2012), 426–431.
[36] V. Karakaya, M.D. Manafov and N. S¸ims¸ek, On fine spectra and subspectrum (approximate point, defect and compression) of operator with periodic coefficients, J. Nonlinear Convex Anal., 18 (4) (2017), 709–717.
[37] E. Kreyszing, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York Chichester Brisbane Toronto, 1978.
[38] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1970.
[39] M. Mursaleen, M. Yildirim and N. Durna, On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices. Commun. Fac. Sci. Univ. Ank. Series A1, 68 (1) (2019),712–723.
[40] M. Mursaleen and F. Bas¸ar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton London New York, 2020.
[41] J.I. Okutoyi, On the spectrum of Ces`aro operator, P.h.D. Thesis Birmingham University, 1986.
[42] J.I. Okutoyi, On the spectrum of C1 as an operator on bv0, J. Aust. Math. Soc., A 48 (1) (1990), 79–86.
[43] J.T. Okutoyi, On the spectrum of C1 as an operator on bv, Commun. Fac. Sci. Univ. Ank. Series A1, 41 (1992), 197–207.
[44] J.B. Reade, On the spectrum of the Ces`aro operator, Bull. Lond. Math. Soc., 17 (3) (1985), 263–267.
[45] B.E. Rhoades and M. Yildirim, Spectra and fine spectra for factorable matrices, Integral Equations Operator Theory, 53 (1) (2005), 127–144.
[46] B.E. Rhoades, The fine spectra for weighted mean operators in B(`p), Integral Equations Operator Theory, 12 (1) (1989), 82–98.
[47] B.C. Tripathy and R. Das, Fine spectrum of the upper triangular matrix U(r;0;0; s) over the sequence spaces c0 and c, Proyecciones, 37 (1) (2018), 85–101.
[48] T. Yaying, B. Hazarika and M. Mursaleen, On sequence space derived by the domain of q Ces`aro matrix in `p space and the associated operator ideal, J. Math. Anal. Appl., 493 (1) (2021), 124453.
[49] M. Yes¸ilkayagil and F. Bas¸ar, On the ne spectrum of the operator defined by a lambda matrix over the sequence spaces of null and convergent sequences, Abstr. Appl. Anal., 2013, Article ID 687393 (2013), 13 pages.
[50] M. Yes¸ilkayagil and F. Bas¸ar, A survey for the spectrum of triangles over sequence spaces, Numer. Funct. Anal. Optim., 40 (16) (2019), 1898-1917.
[51] M.E. Yildirim, The spectrum and fine spectrum of q􀀀Ces`aro matrices with 0 < q < 1 on c0, Numer. Func.Anal. Optim., 41 (3) (2020), 361–377.
[52] M. Yildirim, The spectrum and fine spectrum of the compact Rhaly operator, Indian J. Pure Appl. Math., 27 (8) (1996), 779-784.
[53] M. Yildirim, The spectrum of Rhaly operator on `p, Indian J. Pure Appl. Math., 32 (2) (2001), 191-198.
[54] M. Yildirim and N. Durna, The spectrum and some subdivisions of the spectrum of discrete generalized Ces`aro operators on `p, (1 < p < ¥), J. Inequal. Appl. 193 (2017), 1–13.
[55] M. Yildirim, M. Mursaleen and C¸ . Do˘gan, The Spectrum and fine spectrum of generalized Rhaly-Ces`aro matrices on c0 and c, Oper. Matrices, 12 (4) (2018), 955–975.
[56] R.B. Wenger, The fine spectra of the H¨older summability operator, Indian J. Pure Appl. Math., 6 (6) (1975), 695–712.
[57] A. Wilansky, Summability Through Functional Analysis, North Holland, 1984.

Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$

Year 2022, Volume: 10 Issue: 2 , 313 - 325 , 31.10.2022

Merve Esra Türkay

https://izlik.org/JA85YB98RZ

Abstract

The main purpose of the this paper is to investigate the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the genaralized $\alpha q$-Ces\`{a}ro matrix $% C_{q}^{\alpha }$ with $\alpha ,q\in \left( 0,1\right) $ on the sequence space $c_{0}$.

Keywords

q-Hausdorff matrices , Lower bound problem , q-Cesaro matrices , spectrum , fine spectrum.

References

[1] A.M. Akhmedov and S.R. El-Shabrawy, Spectra and fine spectra of lower triangular double-band matrices as operators on `p (1 p < ¥), Math. Slovaca, 65 (5) (2015), 1137–1152.
[2] A.M. Akhmedov and F. Bas¸ar, On the fine spectrum of the Ces`aro operator in c0, Math. J. Ibaraki Univ., 36 (2004), 25-32.
[3] A.M. Akhmedov and F. Bas¸ar, On the fine spectra of the difference operator D over the sequence space lp, (1 p < ¥), Demonstratio Math., 39 (3) (2006), 585-595.
[4] A.M. Akhmedov and F. Bas¸ar, The fine spectra of the Ces`aro operator C1 over the sequence space bvp, (1 p < ¥)), Math. J. Okayama Univ., 50 (2008), 135-147.
[5] A.M. Akhmedov and F. Bas¸ar, The fine spectra of the difference operator D over the sequence space bvp, (1 p < ¥)), Acta Math. Sin. Eng. Ser., 23 (10) (2007), 1757-1768.
[6] B. Altay and F. Bas¸ar, The fine spectrum and the matrix domain of the difference operator on the sequence space lp, (0 < p < 1), Commun. Math. Anal., 2 (2) (2007), 1-11.
[7] RKh.Amirov, N. Durna and M. Yildirim, Subdivisions of the spectra for Ces`aro, Rhaly and Weighted mean operator on `p, c and `p, IJST, A3 (2011), 175-183.
[8] J. Appell, E. De Pascale and A. Vignoli, Nonlinear Spectral Theory. Berlin, New York, Walter de Gruyter, 2004.
[9] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the Spectra for Generalized Difference Operator Dv on the Sequence Space `1, ICMS., 1309 (2010), 254-260.
[10] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the spectra for the triple band matrix over certain sequence spaces, Gen. Math. Notes, 4 (1) (2011), 35-48.
[11] F. Bas¸ar, N. Durna and M. Yildirim, Subdivisions of the spectra for genarilized difference operator over certain sequence spaces, Thai J. Math., 9 (1) (2011), 285-295.
[12] F. Bas¸ar, N. Durna and M. Yildirim, Subdivision of the spectra for difference operator over certain sequence space, Malays. J. Math. Sci., 6 (2012), 151-165.
[13] F. Bas¸ar, Summability Theory and its Applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton London New York, (in press) 2022.
[14] A. Brown, P.R. Halmos and A.L. Shields, Ces`aro operators. Acta Sci. Math., (Szeged). 26 (1965), 125–137.
[15] D.W. Boyd, The spectrum of the Ces`aro operator, Acta Sci. Math., (Szeged). 29 (1968), 31–34.
[16] G.P. Curbera and W.J. Ricker, Spectrum of the Ces`aro operator in `p, Arch. Math., 100 (2013), 267–271.
[17] N. Durna and M.cYildirim, Subdivision of the spectra for factorable matrices on c and `p, Math. Commun., 16 (2) (2011), 519-530.
[18] N. Durna and M. Yildirim, Subdivision of the Spectra for Factorable Matrices on c0, Gazi Univ. J. Sci., 24 (1) (2011), 45-49.
[19] N. Durna, M. Yildirim and C¸ . U¨ nal, On The Fine Spectrum of Generalized Lower Triangular Double Band Matrices Duv Over The Sequence Space c0, Cumhuriyet Sci. J., 37 (3) (2016), 281-291.
[20] N. Durna, Subdivision of the spectra for the generalized upper triangular double-band matrices Duv over the sequence spaces c0 and c, ADYUSCI, 6 (1) (2016), 31-43.
[21] N. Durna, Subdivision of the spectra for the generalized difference operator Da;b on the sequence space `p (1 < p < ¥), CBU J. Sci., 13 (2) (2017), 359–364.
[22] N. Durna, M. Yildirim and R. Kılıc¸, Partition of the spectra for the generalized difference operator B(r; s) on the sequence space cs, Cumhuriyet Sci. J., 39 (1) (2018), 7–15.
[23] N. Durna, Subdivision of spectra for some lower triangular doule-band matrices as operators on c0, Ukr. Mat. Zh., 70 (7) (2018), 913–922.
[24] N. Durna and M.E. T¨urkay, The spectrum of q-Ces`aro matrices on c and Its various spectral decomposition for 0 < q < 1, Oper. Matrices, 15 (3) (2020), 795-813.
[25] E. D¨undar and F. Bas¸ar, On the fine spectrum of the upper triangle double band matrix D+ on the sequence space c0, Math. Commun., 18 (2) (2013), 337–348.
[26] S.R. El-Shabrawy, On the fine spectrum of the generalized difference operator Da;b over the sequence space `p, (1 < p < ¥), Appl. Math. Inf. Sci., 6 (1S) (2012), 111–118.
[27] S.R. El-Shabrawy, Spectra and fine spectra of certain lower triangular double-band matrices as operators on c0, J. Inequal. Appl., 2014 (1) (2014), 1-9.
[28] S.R. El-Shabrawy and S.H. Abu-Janah, Spectra of the generalized difference operator on the sequence spaces bv0 and h, Linear and Multilinear Algebra 66 (8) (2018), 1691–1708.
[29] S.R. El-Shabrawy, On q-Ces`aro Operators:Boundness, Compactness and Spectra, Numer.Funct.Anal.Optim., 41 (2) (2021), 1019-1037.
[30] J. Fathi and L. Rahmatollah, On the fine spectra of the generalized difference operator Duv over the sequence space `p, JMMRC, 1 (1) (2012), 1-12.
[31] H. Furkan, H. Bilgic¸ and F. Bas¸ar, On the fine spectrum of the operator B(r; s; t) over the sequence spaces `p and bvp, (1 < p < ¥), Comput. Math. Appl., 60 (7) (2010), 2141–2152.
[32] S. Goldberg, Unbounded Linear Operators, McGraw Hill, New York, 1966.
[33] M. Gonzalez, The fine spectrum of Ces`aro operator in `p (1 < p < ¥), Arch. Math., 44 (1985), 355-358.
[34] V. Karakaya and M. Altun, Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math., 234 (2010), 1387–1394.
[35] V. Karakaya, M.D. Manafov and N. S¸ims¸ek, On the fine spectrum of the second order difference operator over the sequence spaces `p and bvp, (1 < p < ¥). Math. Comput. Modelling, 55 (3-4) (2012), 426–431.
[36] V. Karakaya, M.D. Manafov and N. S¸ims¸ek, On fine spectra and subspectrum (approximate point, defect and compression) of operator with periodic coefficients, J. Nonlinear Convex Anal., 18 (4) (2017), 709–717.
[37] E. Kreyszing, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York Chichester Brisbane Toronto, 1978.
[38] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1970.
[39] M. Mursaleen, M. Yildirim and N. Durna, On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices. Commun. Fac. Sci. Univ. Ank. Series A1, 68 (1) (2019),712–723.
[40] M. Mursaleen and F. Bas¸ar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton London New York, 2020.
[41] J.I. Okutoyi, On the spectrum of Ces`aro operator, P.h.D. Thesis Birmingham University, 1986.
[42] J.I. Okutoyi, On the spectrum of C1 as an operator on bv0, J. Aust. Math. Soc., A 48 (1) (1990), 79–86.
[43] J.T. Okutoyi, On the spectrum of C1 as an operator on bv, Commun. Fac. Sci. Univ. Ank. Series A1, 41 (1992), 197–207.
[44] J.B. Reade, On the spectrum of the Ces`aro operator, Bull. Lond. Math. Soc., 17 (3) (1985), 263–267.
[45] B.E. Rhoades and M. Yildirim, Spectra and fine spectra for factorable matrices, Integral Equations Operator Theory, 53 (1) (2005), 127–144.
[46] B.E. Rhoades, The fine spectra for weighted mean operators in B(`p), Integral Equations Operator Theory, 12 (1) (1989), 82–98.
[47] B.C. Tripathy and R. Das, Fine spectrum of the upper triangular matrix U(r;0;0; s) over the sequence spaces c0 and c, Proyecciones, 37 (1) (2018), 85–101.
[48] T. Yaying, B. Hazarika and M. Mursaleen, On sequence space derived by the domain of q Ces`aro matrix in `p space and the associated operator ideal, J. Math. Anal. Appl., 493 (1) (2021), 124453.
[49] M. Yes¸ilkayagil and F. Bas¸ar, On the ne spectrum of the operator defined by a lambda matrix over the sequence spaces of null and convergent sequences, Abstr. Appl. Anal., 2013, Article ID 687393 (2013), 13 pages.
[50] M. Yes¸ilkayagil and F. Bas¸ar, A survey for the spectrum of triangles over sequence spaces, Numer. Funct. Anal. Optim., 40 (16) (2019), 1898-1917.
[51] M.E. Yildirim, The spectrum and fine spectrum of q􀀀Ces`aro matrices with 0 < q < 1 on c0, Numer. Func.Anal. Optim., 41 (3) (2020), 361–377.
[52] M. Yildirim, The spectrum and fine spectrum of the compact Rhaly operator, Indian J. Pure Appl. Math., 27 (8) (1996), 779-784.
[53] M. Yildirim, The spectrum of Rhaly operator on `p, Indian J. Pure Appl. Math., 32 (2) (2001), 191-198.
[54] M. Yildirim and N. Durna, The spectrum and some subdivisions of the spectrum of discrete generalized Ces`aro operators on `p, (1 < p < ¥), J. Inequal. Appl. 193 (2017), 1–13.
[55] M. Yildirim, M. Mursaleen and C¸ . Do˘gan, The Spectrum and fine spectrum of generalized Rhaly-Ces`aro matrices on c0 and c, Oper. Matrices, 12 (4) (2018), 955–975.
[56] R.B. Wenger, The fine spectra of the H¨older summability operator, Indian J. Pure Appl. Math., 6 (6) (1975), 695–712.
[57] A. Wilansky, Summability Through Functional Analysis, North Holland, 1984.

There are 57 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Article
Authors	Merve Esra Türkay 0000-0003-4429-2685
Submission Date	August 1, 2022
Acceptance Date	August 16, 2022
Publication Date	October 31, 2022
IZ	https://izlik.org/JA85YB98RZ
Published in Issue	Year 2022 Volume: 10 Issue: 2

Cite

APA	Türkay, M. E. (2022). Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$. Konuralp Journal of Mathematics, 10(2), 313-325. https://izlik.org/JA85YB98RZ
AMA	1.Türkay ME. Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$. Konuralp J. Math. 2022;10(2):313-325. https://izlik.org/JA85YB98RZ
Chicago	Türkay, Merve Esra. 2022. “Some Spectrum Estimates of the $ \alpha Q$-Cesaro Matrices With $0<\alpha ,q<1$ on $c_{0}$”. Konuralp Journal of Mathematics 10 (2): 313-25. https://izlik.org/JA85YB98RZ.
EndNote	Türkay ME (October 1, 2022) Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$. Konuralp Journal of Mathematics 10 2 313–325.
IEEE	[1]M. E. Türkay, “Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$”, Konuralp J. Math., vol. 10, no. 2, pp. 313–325, Oct. 2022, [Online]. Available: https://izlik.org/JA85YB98RZ
ISNAD	Türkay, Merve Esra. “Some Spectrum Estimates of the $ \alpha Q$-Cesaro Matrices With $0<\alpha ,q<1$ on $c_{0}$”. Konuralp Journal of Mathematics 10/2 (October 1, 2022): 313-325. https://izlik.org/JA85YB98RZ.
JAMA	1.Türkay ME. Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$. Konuralp J. Math. 2022;10:313–325.
MLA	Türkay, Merve Esra. “Some Spectrum Estimates of the $ \alpha Q$-Cesaro Matrices With $0<\alpha ,q<1$ on $c_{0}$”. Konuralp Journal of Mathematics, vol. 10, no. 2, Oct. 2022, pp. 313-25, https://izlik.org/JA85YB98RZ.
Vancouver	1.Merve Esra Türkay. Some Spectrum Estimates of the $ \alpha q$-Cesaro Matrices with $0<\alpha ,q<1$ on $c_{0}$. Konuralp J. Math. [Internet]. 2022 Oct. 1;10(2):313-25. Available from: https://izlik.org/JA85YB98RZ

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