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arXiv · 2608.19499

Resolving the generalized hyperbolicity conjecture for shadowing

Mihály Pituk
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Abstract

It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.

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Mihály Pituk. 2026-08-19. Resolving the generalized hyperbolicity conjecture for shadowing. https://arxiv.org/abs/2608.19499

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