Abstract
We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the unit circle. Finally, we prove that if a linear operator is expansive and has the shadowing property, then the origin is the only nonwandering point.
Highlights
The dynamics of linear operators on Banach spaces has been studied in the literature [1]
We prove that a linear operator has a shadowable point if and only if it has the shadowing property
We prove that an equicontinuous linear operator has no shadowing property and its spectrum is contained in the unit circle
Summary
The dynamics of linear operators on Banach spaces has been studied in the literature [1].For instance, Eisenberg and Hedlund [2,3] characterized uniformly expansive operators in terms of spectral theory, and Ombach [4] as well as Bernardes et al [5] tried to obtain a similar result for operators with the shadowing property. The dynamics of linear operators on Banach spaces has been studied in the literature [1]. We prove that a linear operator has a shadowable point (in the sense of Reference [6]) if and only if it has the shadowing property. We prove that an equicontinuous linear operator has no shadowing property and its spectrum is contained in the unit circle.
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