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Note on spectra of non-selfadjoint operators over dynamical systems. (English) Zbl 07091610
Summary: We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

MSC:
47A10 Spectrum, resolvent
47A35 Ergodic theory of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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