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Zbl 1152.47024
Bilgiç, Hüseyin; Furkan, Hasan
On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces $\ell _{1}$ and $bv$.
(English)
[J] Math. Comput. Modelling 45, No. 7-8, 883-891 (2007). ISSN 0895-7177

Let $A=(a_{nk})_{n,k\in{\Bbb N}}$ be an infinite matrix. For a complex sequence $x=(x_k)_{k\in{\Bbb N}}$, let $Ax$ be, formally, the sequence with coefficients $(Ax)_n:=\sum_{k\in{\Bbb N}}a_{nk}x_k$. 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 $a_{n+2,n}=t$ ($n\in{\Bbb N}$) 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 $\Vert B(r,s,t)\Vert _{\ell_1\text{ or }b_v}=\vert r\vert +\vert s\vert +\vert t\vert$. 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 $$S:=\left\{\alpha\in{\Bbb C}: \left\vert \frac{2(r-\alpha)}{\sqrt{s^2}+\sqrt{s^2-4t(r-\alpha)}}\right\vert \le1\right\},$$ 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 {\it H.\,Furkan, H.\,Bilgiç} and {\it K.\,Kayaduman} [Hokkaido Math.\ J.\ 35, No.\,4, 893--904 (2006; Zbl 1119.47005)] and {\it H.\,Furkan} and {\it K.\,Kayaduman} [Int.\ Math.\ Forum 1, No.\,21--24, 1153--1160 (2006; Zbl 1119.47306)].
MSC 2000:
*47B37 Operators on sequence spaces, etc.
40C05 Matrix methods in summability
47A10 Spectrum and resolvent of linear operators

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

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