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

Let $A={\left({a}_{nk}\right)}_{n,k\in ℕ}$ be an infinite matrix. For a complex sequence $x={\left({x}_{k}\right)}_{k\in ℕ}$, let $Ax$ be, formally, the sequence with coefficients ${\left(Ax\right)}_{n}:={\sum }_{k\in ℕ}{a}_{nk}{x}_{k}$. For any complex numbers $r$, $s$ and $t$ (with $s$ and $t$ not simultaneously null), let $A=B\left(r,s,t\right)$ be the infinite matrix with ${a}_{nn}=s$, ${a}_{n+1,n}=s$ and ${a}_{n+2,n}=t$ ($n\in ℕ$) and ${a}_{nk}=0$ otherwise. It is well-known that $B\left(r,s,t\right)$ defines a bounded linear operator over ${\ell }_{1}$ and ${b}_{v}$ with ${\parallel B\left(r,s,t\right)\parallel }_{{\ell }_{1}\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{b}_{v}}=|r|+|s|+|t|$.

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 ${\sigma }_{r}\left(B\left(r,s,t\right)\right)$ and the usual spectrum $\sigma \left(B\left(r,s,t\right)\right)$ of $B\left(r,s,t\right)$ over ${\ell }_{1}$ or ${b}_{v}$ coincide and are equal to

$S:=\left\{\alpha \in ℂ:\left|\frac{2\left(r-\alpha \right)}{\sqrt{{s}^{2}}+\sqrt{{s}^{2}-4t\left(r-\alpha \right)}}\right|\le 1\right\},$

so the point (discrete) spectrum ${\sigma }_{p}\left(B\left(r,s,t\right)\right)$ and the continuous spectrum ${\sigma }_{c}\left(B\left(r,s,t\right)\right)$ are empty (here, for a complex value $z$, $\sqrt{z}$ will denote the unique square root of $z$ with principal argument in $\left[0,\pi \right)$).

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 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 40C05 Matrix methods in summability 47A10 Spectrum and resolvent of linear operators