Bilgiç, Hüseyin; Furkan, Hasan On the fine spectrum of the operator \(B(r,s,t)\) over the sequence spaces \(\ell _{1}\) and \(bv\). (English) Zbl 1152.47024 Math. Comput. Modelling 45, No. 7-8, 883-891 (2007). Let \(A=(a_{nk})_{n,k\in{\mathbb N}}\) be an infinite matrix. For a complex sequence \(x=(x_k)_{k\in{\mathbb N}}\), let \(Ax\) be, formally, the sequence with coefficients \((Ax)_n:=\sum_{k\in{\mathbb N}}a_{nk}x_k\). For any complex numbers \(r\), \(s\) and \(t\) (with \(s\) and \(t\) not simultaneously null), let \(A=B(r,s,t)\) be the infinite matrix with \(a_{nn}=s\), \(a_{n+1,n}=s\) and \(a_{n+2,n}=t\) (\(n\in{\mathbb N}\)) and \(a_{nk}=0\) otherwise. It is well-known that \(B(r,s,t)\) defines a bounded linear operator over \(\ell_1\) and \(b_v\) with \(\| B(r,s,t)\| _{\ell_1\text{ or }b_v}=| r| +| s| +| t| \).The paper under review deals with spectral properties of this operator over \(\ell_1\) and \(b_v\). In particular, the authors show that the residual spectrum \(\sigma_r(B(r,s,t))\) and the usual spectrum \(\sigma(B(r,s,t))\) of \(B(r,s,t)\) over \(\ell_1\) or \(b_v\) coincide and are equal to \[ S:=\left\{\alpha\in{\mathbb C}: \left| \frac{2(r-\alpha)}{\sqrt{s^2}+\sqrt{s^2-4t(r-\alpha)}}\right| \leq1\right\}, \]so the point (discrete) spectrum \(\sigma_p(B(r,s,t))\) and the continuous spectrum \(\sigma_c(B(r,s,t))\) are empty (here, for a complex value \(z\), \(\sqrt{z}\) will denote the unique square root of \(z\) with principal argument in \([0,\pi)\)).Some results of this paper extend other ones by H.Furkan, H.Bilgiç and K.Kayaduman [Hokkaido Math.J.35, No.4, 893–904 (2006; Zbl 1119.47005)] and H.Furkan and K.Kayaduman [Int.Math.Forum 1, No.21–24, 1153–1160 (2006; Zbl 1119.47306)]. Reviewer: José A. Prado-Bassas (Sevilla) Cited in 28 Documents MSC: 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 40C05 Matrix methods for summability 47A10 Spectrum, resolvent Keywords:spectrum of an operator; difference operator; generalized difference operator; spectral mapping theorem; sequence spaces \(\ell_1\) and \(b_v\) Citations:Zbl 1119.47005; Zbl 1119.47306 PDFBibTeX XMLCite \textit{H. Bilgiç} and \textit{H. Furkan}, Math. Comput. Modelling 45, No. 7--8, 883--891 (2007; Zbl 1152.47024) Full Text: DOI References: [1] Akhmedov, A. M.; Başar, F., On the fine spectra of the difference operator \(\Delta\) over the sequence space \(\ell_p,(1 \leq p < \infty)\), Demonstratio Math., XXXIX, 3, 585-595 (2006) · Zbl 1118.47303 [2] A.M. Akhmedov, F. Başar, The fine spectra of the difference operator \(\Deltab v_p ( 1 \leq p < \infty )\); A.M. Akhmedov, F. Başar, The fine spectra of the difference operator \(\Deltab v_p ( 1 \leq p < \infty )\) [3] H. Bilgiç, H. Furkan, On the fine spectrum of the generalized difference operator \(B ( r , s ) \ell_pb v_p\); H. Bilgiç, H. Furkan, On the fine spectrum of the generalized difference operator \(B ( r , s ) \ell_pb v_p\) [4] J.P. Cartlidge, Weighted Mean Matrices as Operators on \(\ell^p\); J.P. Cartlidge, Weighted Mean Matrices as Operators on \(\ell^p\) [5] H. Furkan, H. Bilgiç, K. Kayaduman, On the fine spectrum of the generalized difference operator \(B ( r , s ) \ell_1b v\); H. Furkan, H. Bilgiç, K. Kayaduman, On the fine spectrum of the generalized difference operator \(B ( r , s ) \ell_1b v\) [6] Goldberg, S., Unbounded Lineer Operators (1985), Dover Publications: Dover Publications New York [7] Gonzàlez, M., The fine spectrum of the Cesàro operator in \(\ell_p(1 < p < \infty)\), Arch. Math., 44, 355-358 (1985) · Zbl 0568.47021 [8] Kayaduman, K.; Furkan, H., The fine spectra of the difference operator \(\Delta\) over the sequence spaces \(\ell_1\) and \(b v\), Int. Math. Forum, 1, 21-24, 1153-1160 (2006) · Zbl 1119.47306 [9] Kreyszig, E., Introductory Functional Analysis with Applications (1978), John Wiley & Sons: John Wiley & Sons New York · Zbl 0368.46014 [10] Okutoyi, J. I., On the spectrum of \(C_1\) as an operator on \(b v_0\), J. Austral. Math. Soc. Ser. A, 48, 79-86 (1990) · Zbl 0691.40004 [11] Okutoyi, J. T., On the spectrum of \(C_1\) as an operator on \(b v\), Commun. Fac. Sci. Univ. Ank. Ser. \(A_1, 41, 197-207 (1992)\) · Zbl 0831.47020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.