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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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