On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$

Nuh Durna; Rabia Kılıç

doi:10.31801/cfsuasmas.977593

Research Article

On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$

Year 2022, Volume: 71 Issue: 2, 554 - 565, 30.06.2022

Nuh Durna Rabia Kılıç

https://doi.org/10.31801/cfsuasmas.977593

Abstract

The upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ is defined on a Banach sequence space by

$U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})(x_{n})=(a_{n}x_{n}+b_{n}x_{n+1})_{n=0}^{\infty}$

where $a_{x}=a_{y},~b_{x}=b_{y}$ for $x\equiv y~(mod3)$. The class of the operator

$U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$

includes, in particular, the operator $U(r,s)$ when $a_{k}=r$ and $b_{k}=s$ for all $k\in\mathbb{N}$, with $r,s\in\mathbb{R}$ and $s\neq 0$. Also, it includes the upper difference operator; $a_{k}=1$ and $b_{k}=-1$ for all $k\in\mathbb{N}$. In this paper, we completely determine the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$.

Keywords

Upper triangular band matrix, spectrum, fine spectrum, approximate point spectrum

References

Akhmedov, A. M., Başar, F., The fine spectra of the difference operator over the sequence space $bv_{p}, (1 ≤ p < ∞)$, Acta. Math. Sin. (Engl Ser), 23(10) (2007), 1757–1768. doi.org/10.1007/s10114-005-0777-0
Appell, J., De Pascale, E., Vignoli, A., Nonlinear Spectral Theory, Walter de Gruyter, Berlin, New York, 2004.
Wilansky, A., Summabilitiy Through Functional Analysis, Amsterdam, North Holland, 1984.
Başar, F., Durna, N., Yildirim, M., Subdivisions of the spectra for generalized difference operator over certain sequence spaces, Thai J. Math., 9(1) (2011), 285-295.
Das, R., On the spectrum and fine spectrum of the upper triangular matrix $U(r_{1},r_{2};s_{1},s_{2})$ over the sequence space $c_{0}$, Afr. Math., 28 (2017), 841-849. doi.org/10.1007/s13370-017-0486-8
Durna, N., Yildirim, M., Subdivision of the spectra for factorable matrices on $c_{0}$, GU J. Sci., 24(1) (2011), 45-49.
Durna, N., Subdivision of the spectra for the generalized upper triangular double-band matrices $\Delta^{uv}$ over the sequence spaces $c_{0}$ and $c$, ADYU Sci., 6(1) (2016), 31-43.
Durna, N., Yildirim, M., Kılıç, R., 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.
Durna, N., Kılıç, R., Spectra and fine spectra for the upper triangular band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c_{0}$, Proyecciones J. Math., 38(1) (2019), 145-162.
Durna, N., Subdivision of spectra for some lower triangular double-band matrices as operators on $c_{0}$, Ukr. Mat. Zh., 70(7) (2018), 913-922.
Dündar, E., Başar, F., On the fine spectrum of the upper triangular double band matrix $\Delta^{+}$ on sequence space $c_{0}$, Math. Commun., 18 (2013), 337–348.
El-Shabrawy, S. R., Abu-Janah, S. H., Spectra of the generalized difference operator on the sequence spaces and $bv_{0}$ and $h$, Linear and Multilinear Algebra, 66(1) (2017), 1691-1708.doi.org/10.1080/03081087.2017.1369492
Goldberg, S., Unbounded Linear Operators, McGraw Hill, New York, 1966.
Karakaya, V., Altun, M., Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math., 234 (2010), 1387–1394. doi.org/10.1016/j.cam.2010.02.014
Tripathy, B. C., Das, R., Fine spectrum of the upper triangular matrix $U(M, 0, 0, s)$ over the sequence spaces $c_{0}$ and $c$, Proyecciones J. Math., 37(1) (2018), 85-101.
Yeşilkayagil, M., Kirisci, M., On the fine spectrum of the forward difference operator on the Hahn space, Gen. Math. Notes, 33(2) (2016), 1-16.
Yildirim, M., Durna, N., The spectrum and some subdivisions of the spectrum of discrete generalized Ces`aro operators on $ℓ_{p}$, $(1<p<\infty)$, J. Inequal. Appl., 2017(193) (2017), 1-13.DOI 10.1186/s13660-017-1464-2
Yildirim, M., Mursaleen, M., Doğan, Ç, The spectrum and fine spectrum of generalized Rhaly-Cesaro matrices on $c_{0}$ and $c$, Operators and Matrices, 12(4) (2018), 955-975. doi:10.7153/oam-2018-12-58
Yildirim, M., The spectrum and fine spectrum of $q$-Cesaro matrices with $0<q<1$ on $c_{0}$, Numer. Func. Anal. Optim., 41(3) (2020), 361-377. doi.org/10.1080/01630563.2019.1633666

Year 2022, Volume: 71 Issue: 2, 554 - 565, 30.06.2022

Nuh Durna Rabia Kılıç

https://doi.org/10.31801/cfsuasmas.977593

Abstract

References

Akhmedov, A. M., Başar, F., The fine spectra of the difference operator over the sequence space $bv_{p}, (1 ≤ p < ∞)$, Acta. Math. Sin. (Engl Ser), 23(10) (2007), 1757–1768. doi.org/10.1007/s10114-005-0777-0
Appell, J., De Pascale, E., Vignoli, A., Nonlinear Spectral Theory, Walter de Gruyter, Berlin, New York, 2004.
Wilansky, A., Summabilitiy Through Functional Analysis, Amsterdam, North Holland, 1984.
Başar, F., Durna, N., Yildirim, M., Subdivisions of the spectra for generalized difference operator over certain sequence spaces, Thai J. Math., 9(1) (2011), 285-295.
Das, R., On the spectrum and fine spectrum of the upper triangular matrix $U(r_{1},r_{2};s_{1},s_{2})$ over the sequence space $c_{0}$, Afr. Math., 28 (2017), 841-849. doi.org/10.1007/s13370-017-0486-8
Durna, N., Yildirim, M., Subdivision of the spectra for factorable matrices on $c_{0}$, GU J. Sci., 24(1) (2011), 45-49.
Durna, N., Subdivision of the spectra for the generalized upper triangular double-band matrices $\Delta^{uv}$ over the sequence spaces $c_{0}$ and $c$, ADYU Sci., 6(1) (2016), 31-43.
Durna, N., Yildirim, M., Kılıç, R., 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.
Durna, N., Kılıç, R., Spectra and fine spectra for the upper triangular band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c_{0}$, Proyecciones J. Math., 38(1) (2019), 145-162.
Durna, N., Subdivision of spectra for some lower triangular double-band matrices as operators on $c_{0}$, Ukr. Mat. Zh., 70(7) (2018), 913-922.
Dündar, E., Başar, F., On the fine spectrum of the upper triangular double band matrix $\Delta^{+}$ on sequence space $c_{0}$, Math. Commun., 18 (2013), 337–348.
El-Shabrawy, S. R., Abu-Janah, S. H., Spectra of the generalized difference operator on the sequence spaces and $bv_{0}$ and $h$, Linear and Multilinear Algebra, 66(1) (2017), 1691-1708.doi.org/10.1080/03081087.2017.1369492
Goldberg, S., Unbounded Linear Operators, McGraw Hill, New York, 1966.
Karakaya, V., Altun, M., Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math., 234 (2010), 1387–1394. doi.org/10.1016/j.cam.2010.02.014
Tripathy, B. C., Das, R., Fine spectrum of the upper triangular matrix $U(M, 0, 0, s)$ over the sequence spaces $c_{0}$ and $c$, Proyecciones J. Math., 37(1) (2018), 85-101.
Yeşilkayagil, M., Kirisci, M., On the fine spectrum of the forward difference operator on the Hahn space, Gen. Math. Notes, 33(2) (2016), 1-16.
Yildirim, M., Durna, N., The spectrum and some subdivisions of the spectrum of discrete generalized Ces`aro operators on $ℓ_{p}$, $(1<p<\infty)$, J. Inequal. Appl., 2017(193) (2017), 1-13.DOI 10.1186/s13660-017-1464-2
Yildirim, M., Mursaleen, M., Doğan, Ç, The spectrum and fine spectrum of generalized Rhaly-Cesaro matrices on $c_{0}$ and $c$, Operators and Matrices, 12(4) (2018), 955-975. doi:10.7153/oam-2018-12-58
Yildirim, M., The spectrum and fine spectrum of $q$-Cesaro matrices with $0<q<1$ on $c_{0}$, Numer. Func. Anal. Optim., 41(3) (2020), 361-377. doi.org/10.1080/01630563.2019.1633666

There are 19 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Articles
Authors	Nuh Durna 0000-0001-5469-7745 Rabia Kılıç This is me 0000-0002-3415-1945
Publication Date	June 30, 2022
Submission Date	August 2, 2021
Acceptance Date	January 20, 2022
Published in Issue	Year 2022 Volume: 71 Issue: 2

Cite

APA	Durna, N., & Kılıç, R. (2022). On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 554-565. https://doi.org/10.31801/cfsuasmas.977593
AMA	Durna N, Kılıç R. On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2022;71(2):554-565. doi:10.31801/cfsuasmas.977593
Chicago	Durna, Nuh, and Rabia Kılıç. “On the Spectrum of the Upper Triangular Double Band Matrix $U(a_{0},a_{1},a_{2};B_{0},b_{1},b_{2})$ over the Sequence Space $c$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 554-65. https://doi.org/10.31801/cfsuasmas.977593.
EndNote	Durna N, Kılıç R (June 1, 2022) On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 554–565.
IEEE	N. Durna and R. Kılıç, “On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 554–565, 2022, doi: 10.31801/cfsuasmas.977593.
ISNAD	Durna, Nuh - Kılıç, Rabia. “On the Spectrum of the Upper Triangular Double Band Matrix $U(a_{0},a_{1},a_{2};B_{0},b_{1},b_{2})$ over the Sequence Space $c$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 554-565. https://doi.org/10.31801/cfsuasmas.977593.
JAMA	Durna N, Kılıç R. On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:554–565.
MLA	Durna, Nuh and Rabia Kılıç. “On the Spectrum of the Upper Triangular Double Band Matrix $U(a_{0},a_{1},a_{2};B_{0},b_{1},b_{2})$ over the Sequence Space $c$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 554-65, doi:10.31801/cfsuasmas.977593.
Vancouver	Durna N, Kılıç R. On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(2):554-65.