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Fine spectra of lacunary matrices. (English) Zbl 1176.47008
Let \((k_n)_{n\geq0}\) be a lacunary sequence which is an increasing sequence of nonnegative integers for which \(k_0=0\) and \(h_r:=k_r-k_{r-1}\to\infty\) as \(r\to\infty\). The corresponding lacunary operator \(L\) is defined by \(L((x_1,x_2,x_3,\dots))=(y_1,y_2,y_3,\dots)\), where
\[ y_r=\frac{1}{h_r}\sum_{i\in(k_{r-1},k_r]}x_i,\quad r\geq 1. \]
The authors determine the spectra and fine spectra of \(L\) when regarded as an operator on \(c_0\), \(c\) and \(l_\infty\), the spaces of all null, convergent and bounded sequences, respectively.

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