Abstract
To a linear differential operator (respectively, equation) with unbounded periodic coefficients in a Banach space of vector functions defined on the entire real line, we assign a difference operator (respectively, a difference equation) with a constant operator coefficient defined in the corresponding Banach space of two-sided vector sequences. For the differential and difference operators, we prove the coincidence of the dimensions of their kernels and coranges, the simultaneous complementability of kernels and ranges, the simultaneous invertibility, and a relationship between the spectra.
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