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 $c$.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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 $x\equiv y~(mod3)$.
📊
Key Findings & Data: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 $s\neq 0$.
💡
Conclusion & Impact: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 $c$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Upper triangular band matrix · spectrum · fine spectrum
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 $c$.
DURNA, N., KILIÇ, 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, Cilt: 71 - Sayı: 2, 554–565. https://ajindex.com/articles/746659
Akhmedov, A. M., Başar, F., The fine spectra of the difference operator over the sequence space $bv_{p}, (1 ≤ p 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 subdiv