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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.)