Abstract
We establish sufficient conditions for the existence of invariant subspaces for operators on real Banach spaces, and we investigate the behaviour of operators without such subspaces. All proofs are elementary.
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References
Ansari, S.I.: Hypercyclic and cyclic vectors. J. Funct. Anal. 128, 374–383 (1995)
Atzmon, A., Godefroy, G.: An application of the smooth variational principle to the existence of nontrivial invariant subspaces. Note aux C.R.A.S. Paris t. 332(1), 151–156 (2001)
Atzmon, A., Godefroy, G., Kalton, N.J.: Invariant subspaces and the exponential map. Positivity 8, 101–107 (2004)
Chalendar, I., Partington, J.: Modern Approaches to the Invariant-Subspace Problem. Cambridge Tracts in Mathematics, vol. 188. Cambridge University Press, Cambridge (2011)
Esterle, J.: Personal communication, October (2018)
Grivaux, S., Roginskaya, M.: A general approach to Read’s type constructions of operators without non-trivial invariant closed subspaces. Proc. Lond. Math. Soc. (3) 109(3), 596–652 (2014)
Lomonosov, V.: A counterexample to the Bishop–Phelps theorem in complex spaces. Isr. J. Math. 115, 25–48 (2000)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes, vol. 1364. Springer, Berlin (1989)
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Godefroy, G. Convex analysis and non-trivial invariant subspaces. Positivity 24, 369–372 (2020). https://doi.org/10.1007/s11117-019-00682-4
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DOI: https://doi.org/10.1007/s11117-019-00682-4