Nuh DURNA1, Mustafa YILDIRIM, Abbas KILIÇ

On the fine spectra of the Jacobi matrices on ????, ??, ???? (?? ≤ ?? ≤ ∞) and ?????? (?? ≤ ?? < ∞)

The spectrum and spectral divisions of band matrices are very new and popular topics of studies. In this paper, our aims are to investigate boundedness of Jacobi matrix which is a band matrix has important role in physics and give subdivisions of the spectra, which are approximate point spectrum, defect spectrum and compression spectrum, for a special type Jacobi matrix. Moreover, we will find the fine division of spectrum which is given by Goldberg with the help of it.

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[1] Amirov R. Kh., Durna N., Yildirim M., Subdivisions of the spectra for cesaro, rhaly and weighted mean operators on ??0, ?? and ℓ??, IJST, A3 (2011) 175-183.
[2] Appell J., Pascale E.D, Vignoli A., Nonlinear Spectral Theory, Berlin, New York: Walter de Gruyter, (2002).
[3] Stone M.H., Linear transformations in Hilbert space and their applications to analysis, New York (NY): American Mathematical Society; 1932.
[4] Goldberg S., Unbounded Linear Operators, New York:McGraw Hill, (1966).
[5] Brown A., Halmos P.R., Shields A.L., Cesàro operators, Acta Sci. Math. (Szeged) 26(1-2) (1965) 125-137.
[6] Cass, F.P.; Rhoades, B. E., Mercerian theorems via spectral theory, Pacific J. Math. 73(1) (1977) 63-71.
[7] Cartlidge J.P., Weighted Mean Matrices as Operator on ℓ??, Ph.D. Dissertation, Indiana University, 1978.
[8] Başar F., Durna N., Yildirim M., Subdivisions of the Spectra for Generalized Difference Operator over Certain Sequence Spaces, Thai J. Math. 9(2) (2011), 285-295.
[9] Durna N, Yıldırım M., Subdivision of the spectra for factorable matrices on ??0, GUJ Sci, 24(1) (2011) 45-49.
[10] Durna N, Yıldırım M., Subdivision of the spectra for factorable matrices on c and ℓ??, Math. Commun., 16 (2011) 519-530.
[11] Durna, N. , Yıldırım, 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.
[12] Akhmedov A.M., El-Shabrawy S.R., Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on ???? (1 ≤ ?? < ∞), Mathematica Slovaca 65(5) (2015) 1137-1152.
[13] Bilgiç H., Furkan H., On the fine spectrum of the generalized difference operator ??(??, ??) over the sequence spaces ℓ?? and ?????? (1 < ?? < ∞), Nonlinear Anal. 68(3) (2008) 499-506.
[14] Coskun C., The spectra and fine spectra for p- Cesàro operators, Turkish J. Math., 21 (1997) 207-212.
[15] Das R., On the spectrum and fine spectrum of the lower triangular matrix ??(±??, ±??) over the sequence space ??0, International Journal of Mathematical Archive 6(1) (2015) 229-240.
[16] Das R., On The fine spectra of the lower triangular matrix ??(??, ??) over the sequence space ℓ1, Internat. J. Functional Analysis, Operator Theory and Applications, 7(1) (2015) 1-18.
[17] Fathi J., On the ne spectrum of generalized upper triangular double-band matrices uv over the sequence spaces ??0 and c, Int. J. Nonlinear Anal. Appl, 7(1) (2015) 31-43.
[18] Karakaya V., Manafov M., Şimşek N., On the fine spectrum of the second order difference operator over the sequence spaces ℓ?? and ?????? (1 < ?? < ∞), Mathematical and Computer Modelling 55(3- 4) (2012) 426-431.
[19] Rhoades B.E, Yıldırım M., Spectra and fine spectra for factorable matrices, Integr.Equ. Oper. Theory, 53 (2005) 127-144.
[20] El-Shabrawy S.R., Spectra and fine spectra of certain lower triangular double-band matrices as operators on ??0, J. Inequal. Appl., 2014 (2014) 1- 9.
[21] Srivastava P.D., Kumar S., Fine spectrum of the generalized difference operator Δ???? on sequence space ℓ1, Appl. Math. Comput., 218(11) (2012) 6407-6414.
[22] Tripathy B.C., Das R., Spectrum and fine spectrum of the upper triangular matrix ??(??, ??) over the sequence space ????, Proyecciones Journal of Mathematics 34(2) (2015) 107-125.
[23] Tripathy B.C., Das R., Spectrum and fine spectrum of the lower triangular matrix ??(??, 0, ??) over the sequence space ????, Appl. Math. Inf. Sci. 9(4) (2015) 2139-2145.
[24] Yıldırım M., On the spectrum and fine spectrum of the compact Rhally operators, Indian J. Pure Appl. Math.,27(8) (1996) 779-784.
[25] Yıldırım M.E., The spectrum and fine spectrum of q-Cesaro matrices with 0 < ?? < 1 on ??0, Numerical functional analysis and optimization, 41(3) (2020) 361-377.
[26] Wilansky A., Summability through Functional Analysis, vol. 85 of North-Holland Mathematics Studies, North-Holland, The Netherlands, Amsterdam 1984.
[27] Altun M., Fine Spectra of Tridiagonal Symmetric Matrices, International Journal of Mathematics and Mathematical Sciences (2011) 161209.