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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:
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