HILBERT THEOREM ON LEMNISCATE AND THE SPECTRUM OF THE PERTURBED SHIFT

V.L.Oleinik, B.S.Pavlov

Abstract

The spectrum of the perturbed shift operator $T: f(n)to al f(n+1)+a(n)f(n)$ in $l^2(Z)$ is considered for periodic $a(n)$ and fixed constant $al>0$. It is proven that the spectrum is continuous and fills a lemniscate. Some isospectral deformations of the sequence $a(n)$ are described. Similar facts for the perturbed shift in the spaces of sequences of some hypercomplex numbers is derived.

Keywords
Shift Operator, Spectrum, lemniscate

Math Review Classification

Last Updated

Length
11 pages

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