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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
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