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On the fine spectrum of the generalized difference operator $$B(r,s)$$ over the sequence space $$\ell_1$$ and $$bv$$. (English) Zbl 1119.47005
For two real numbers $$r$$ and $$s$$ such that $$s\not=0$$, denote by $$B(r,s)$$ the following band matrix $\begin{bmatrix} r & 0 & 0 & 0 & \dots \cr s & r & 0& 0 & \dots \cr 0& s&r&0& \dots \cr 0 & 0 & s & r & \dots \cr \vdots & \vdots & \vdots & \vdots& \ddots \end{bmatrix}.$ The authors describe the spectra of $$B(r,s)$$ when it is regarded as an operator acting on the space $$\ell_1$$ of all summable sequences and on the space $$bv$$ of all sequences of bounded variation.
Note that this operator is nothing but the sum of $$rI$$ and $$sU$$, where $$I$$ is the identity and $$U$$ is the unweighted unilateral shift.

##### MSC:
 47A10 Spectrum, resolvent 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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