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On the fine spectrum of the operator $$B(r,s,t)$$ over the sequence spaces $$\ell _{1}$$ and $$bv$$. (English) Zbl 1152.47024
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)].

MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 40C05 Matrix methods for summability 47A10 Spectrum, resolvent
Citations:
Zbl 1119.47005; Zbl 1119.47306
Full Text:
References:
  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  A.M. Akhmedov, F. Başar, The fine spectra of the difference operator $$\Delta$$ over the sequence space $$b v_p$$, $$(1 \leq p < \infty)$$, Acta Math. Sin. Engl. Ser. (2006) (in press)  H. Bilgiç, H. Furkan, On the fine spectrum of the generalized difference operator $$B(r, s)$$ over the sequence spaces $$\ell_p$$ and $$b v_p$$ (under communication)  J.P. Cartlidge, Weighted Mean Matrices as Operators on $$\ell^p$$, Ph.D. Dissertation, Indiana University, 1978  H. Furkan, H. Bilgiç, K. Kayaduman, On the fine spectrum of the generalized difference operator $$B(r, s)$$ over the sequence spaces $$\ell_1$$ and $$b v$$, Hokkaido Math. J. (in press)  Goldberg, S., Unbounded lineer operators, (1985), Dover Publications New York  Gonzàlez, M., The fine spectrum of the Cesàro operator in $$\ell_p(1 < p < \infty)$$, Arch. math., 44, 355-358, (1985) · Zbl 0568.47021  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  Kreyszig, E., Introductory functional analysis with applications, (1978), John Wiley & Sons New York · Zbl 0368.46014  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  Okutoyi, J.T., On the spectrum of $$C_1$$ as an operator on $$b v$$, Commun. fac. sci. univ. ank. ser. $$\operatorname{A}_1$$, 41, 197-207, (1992) · Zbl 0831.47020
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