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A note on the dichotomy spectrum. (English) Zbl 1180.39030

J. Difference Equ. Appl. 15, No. 10, 1021-1025 (2009); erratum ibid. 18, No. 7, 1257-1261 (2012).
Let \(X\) be a Banach space over the field of real or complex numbers and \(A=\{A_k\}_{k\in\mathbb Z}\), where \(\mathbb Z\) is the set of integers, be a sequence of bounded linear operators. Given \(\gamma >0\), consider the recurrence relations
\[ x_{k+1}=A_kx_k,\quad k\in\mathbb Z, \tag{1} \]
and
\[ x_{k+1}=\frac{1}{\gamma }A_kx_k,\quad k\in\mathbb Z. \tag{2} \]
The dichotomy resolvent and dichotomy spectrum of (1) are given as
\[ \rho (A)=\left\{ \gamma >0:\text{(2) admits an exponential dichotomy} \right\} \]
and
\[ \sum (A)=(0,\infty )\setminus \rho (A), \]
respectively. Some invariance and perturbation properties of \(\sum (A)\) are derived. These properties easily follow from the properties of weighted shift operators defined on sequence spaces.

MSC:

39A70 Difference operators
47A10 Spectrum, resolvent
47A25 Spectral sets of linear operators
47A55 Perturbation theory of linear operators
47B39 Linear difference operators
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