NON-SVEP, Right-Inversion Point Spectrum and Chaos

We discuss some relations between the local existence of analytic selections of eigenvectors (LSP = “NON-SVEP”) for an operator in Banach space and some chaoticity properties of linear dynamical system (with discrete or continuous time) generated by this operator. Our main goal is to prove the existence of a strong connection of some results known for many years in the local spectral theory to some important problems (which seem to be not solved so far) in the linear chaos theory. We also find a simple particular solution of the problem formulated in “Eigenvectors Selection Conjecture” (Banasiak and Moszyński in Discrete Contin Dyn Syst A 20(3):577–587, 2008, Conjecture 4.3, p. 585) and we formulate a new convenient spectral criterion for linear chaos. To make the assumptions more clear we introduce some special parts of the point spectrum of a closed operator, including the right-inversion point spectrum. Using this new criterion we prove chaoticity of a large class of super-upper-triangular operators in lp and c0 spaces and also of some strongly continuous semi-groups generated by such operators.