Generalized hyperbolicity conjecture for shadowing

For every Banach space $X$ and every invertible bounded linear operator $T\in\mathcal{B}(X)$ , if $T$ has the shadowing property, then $T$ is generalized hyperbolic. In particular, the shadowing property means that for every $\varepsilon>0$ there exists $\delta>0$ such that every bi-infinite sequence $(x_n)_{n\in\mathbb{Z}}\subset X$ satisfying $\lVert Tx_n-x_{n+1}\rVert<\delta$ for all $n\in\mathbb{Z}$ is $\varepsilon$ -shadowed by an orbit: there exists $x\in X$ such that $\lVert T^n x-x_n\rVert<\varepsilon$ for all $n\in\mathbb{Z}$ .

Progress summary

Solved

A new preprint claims the conjecture is false in general Banach spaces but true for separable Hilbert spaces; the claim has not yet been independently verified.

The conjecture asks whether every operator with the shadowing property must be generalized hyperbolic. The retrieved material does not identify its proposer or original date.

August 2026 preprint claim

Mihály Pituk’s preprint claims a negative answer in general Banach spaces and a positive answer for separable Hilbert spaces. A related construction gives, for every $1<p<\infty$ with $p\ne2$ , an invertible operator on $\ell^p(\mathbb{N})$ with shadowing but without pseudo-hyperbolicity, hence without generalized hyperbolicity. The preprint also claims equivalence of the relevant stability properties when unimodular eigenspaces have closed complements, a condition automatic in Hilbert spaces. No independent verification or referee report was found.

Current status (as of August 2026): The conjecture is claimed false for general Banach spaces, with a claimed positive result for separable Hilbert spaces, but the preprint claim is unverified and intermediate classes remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Resolving the generalized hyperbolicity conjecture for shadowing — arXiv — Pituk, Mihály

functional-analysis linear-dynamics

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