Pötzsche, Christian 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. Reviewer: Sui Sun Cheng (Hsinchu) Cited in 1 ReviewCited in 13 Documents MSC: 39A70 Difference operators 47A10 Spectrum, resolvent 47A25 Spectral sets of linear operators 47A55 Perturbation theory of linear operators 47B39 Linear difference operators Keywords:dichotomy spectrum; exponential dichotomy; nonautonomous difference equation; invariance; perturbation; Banach space; bounded linear operators; dichotomy resolvent; sequence spaces PDFBibTeX XMLCite \textit{C. Pötzsche}, J. Difference Equ. Appl. 15, No. 10, 1021--1025 (2009; Zbl 1180.39030) Full Text: DOI References: [1] DOI: 10.1080/10236190108808310 · Zbl 1001.39003 · doi:10.1080/10236190108808310 [2] DOI: 10.1080/10236199608808060 · Zbl 0880.39009 · doi:10.1080/10236199608808060 [3] DOI: 10.1006/jmaa.1994.1248 · Zbl 0806.39005 · doi:10.1006/jmaa.1994.1248 [4] DOI: 10.1007/BF01200554 · Zbl 0751.47014 · doi:10.1007/BF01200554 [5] DOI: 10.1512/iumj.1993.42.42031 · Zbl 0809.47013 · doi:10.1512/iumj.1993.42.42031 [6] Chicone C., Mathematical Surveys and Monographs 70 (1999) [7] DOI: 10.1016/j.jmaa.2007.04.052 · Zbl 1138.39007 · doi:10.1016/j.jmaa.2007.04.052 [8] Henry D., Lect. Notes Math 840, in: Geometric Theory of Semilinear Parabolic Equations (1981) · doi:10.1007/BFb0089647 [9] Kato T., Perturbation Theory for Linear Operators, 2. ed. (1980) · Zbl 0435.47001 [10] DOI: 10.1016/0022-0396(79)90072-X · Zbl 0438.34008 · doi:10.1016/0022-0396(79)90072-X [11] DOI: 10.1016/0022-0396(78)90057-8 · Zbl 0372.34027 · doi:10.1016/0022-0396(78)90057-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.