# Subdivision of Spectra for Some Lower Triangular Double-Band Matrices as Operators on *c*_{0}

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The generalized difference operator ∆where (

_{a,b}was defined by El-Shabrawy:$$ {\varDelta}_{a,b}x={\varDelta}_{a,b}\left({x}_n\right)={\left({a}_n{x}_n+{b}_{n-1}\right)}_{n=0}^{\infty}\;\mathrm{with}\;{x}_{-1}={b}_{-1}=0, $$

*a*_{k}) and (*b*_{k}) are convergent sequences of nonzero real numbers satisfying certain conditions. We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator ∆_{a,b}in a sequence space*c*_{0}*.*## Preview

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## References

- 1.A. M. Akhmedov and El-S. R. Shabrawy, “The spectrum of the generalized lower triangle double-band matrix ∆
_{a}over the sequence space*c*,”*Al-Azhar Univ. Eng. J. (Special Issue)*, 5, No. 9, 54–63 (2010).Google Scholar - 2.A. M. Akhmedov and El-S. R. Shabrawy, “On the fine spectrum of the operator ∆
_{v}over the sequence space*c*and*ℓ*_{p}(1*< p < ∞*)*,*”*Appl. Math. Inform. Sci.*, 5, No. 3, 635–654 (2011).MathSciNetGoogle Scholar - 3.B. Altay and F. Bașar, “On the fine spectrum of the difference operator on
*c*_{0}and*c*,”*Inform. Sci.*, 168, 217–224 (2004).MathSciNetzbMATHCrossRefGoogle Scholar - 4.J. Appell, E. D. Pascale, and A. Vignoli,
*Nonlinear Spectral Theory*, De Gruyter, Berlin; New York (2004).zbMATHCrossRefGoogle Scholar - 5.F. Bașar, N. Durna, and M. Yildirim, “Subdivisions of the spectra for generalized difference operator ∆
_{v}on the sequence space*ℓ*_{1},”*Int. Conf. Math. Sci.*, 254–260 (2010).Google Scholar - 6.F. Bașar, N. Durna, and M. Yildirim, “Subdivisions of the spectra for generalized difference operator over certain sequence spaces,”
*J. Thai J. Math.*, 9, No. 2, 285–295 (2011).MathSciNetzbMATHGoogle Scholar - 7.F. Bașar, N. Durna, and M. Yildirim, “Subdivision of the spectra for difference operator over certain sequence spaces,”
*Malays. J. Math. Sci.*, 6, 151–165 (2012).MathSciNetzbMATHGoogle Scholar - 8.N. Durna and M. Yildirim, “Subdivision of the spectra for factorable matrices on
*c*_{0},”*GUJ Sci.*, 24, No. 1, 45–49 (2011).zbMATHGoogle Scholar - 9.S. R. El-Shabrawy, “On the fine spectrum of the generalized difference operator ∆
_{a,b}over the sequence space*ℓ*_{p}(1*< p < ∞*)*,*”*Appl. Math. Inform. Sci.*, 6, No. 1, 111–118 (2012).MathSciNetzbMATHGoogle Scholar - 10.S. R. El-Shabrawy, “Spectra and fine spectra of certain lower triangular double band matrices as operators on
*c*_{0},”*J. Inequal. Appl.*, 241, No. 1, 1–9 (2014).MathSciNetzbMATHGoogle Scholar - 11.J. Fathi and R. Lashkaripour, “On the fine spectra of the generalized difference operator ∆
_{uv}over the sequence space*c*_{0},”*J. Mahani Math. Res. Cent.*, 1, No. 1, 1–12 (2012).zbMATHGoogle Scholar - 12.S. Goldberg,
*Unbounded Linear Operators*, McGraw Hill, New York (1966).zbMATHGoogle Scholar - 13.M. Gonzalez, “The fine spectrum of the Cesaro operator in
*ℓ*_{p}(1*< p < ∞*)*,*”*Arch. Math. (Basel)*, 44, 355–358 (1985).MathSciNetzbMATHCrossRefGoogle Scholar - 14.K. Kayaduman and H. Furkan, “On the fine spectrum of the difference operator ∆ over the sequence spaces
*ℓ*_{1}and*bv*,”*Int. Math. Forum*, 24, No. 1, 1153–1160 (2006).zbMATHCrossRefGoogle Scholar - 15.J. B. Reade, “On the spectrum of the Cesaro operator,”
*Bull. Lond. Math. Soc.*, 17, 263–267 (1985).MathSciNetzbMATHCrossRefGoogle Scholar - 16.B. E. Rhoades, “The fine spectra for weighted mean operators,”
*Pacific J. Math.*, 104, 263–267 (1983).MathSciNetzbMATHCrossRefGoogle Scholar - 17.M. Yildirim, “On the spectrum of the Rhaly operators on
*c*_{0}and*c*,”*Indian J. Pure Appl. Math.*, 29, 1301–1309 (1998).MathSciNetzbMATHGoogle Scholar - 18.M. Yildirim, “The fine spectra of the Rhaly operators on
*c*_{0},”*Turkish J. Math.*, 26, No. 3, 273–282 (2002).MathSciNetzbMATHGoogle Scholar - 19.R. B. Wenger, “The fine spectra of H¨older summability operators,”
*Indian J. Pure Appl. Math.*, 6, 695–712 (1975).MathSciNetzbMATHGoogle Scholar

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