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 $c$

Yıl 2022, Cilt: 71 Sayı: 2, 554 - 565, 30.06.2022

Nuh Durna , Rabia Kılıç

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

Öz

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 $c$.

Anahtar Kelimeler

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

Kaynakça

• 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