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 $n\in\mathbb{Z}$ .

References

Primary source

arXiv

Additional references

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

functional-analysis linear-dynamics

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims the conjecture is false for general Banach spaces but true for separable Hilbert spaces, although neither claim has been independently verified.

The conjecture asks whether shadowing forces generalized hyperbolicity for every invertible bounded operator on a Banach space. Its proposer and original date are not identified in the retrieved sources.

August 19, 2026 preprint claim

On August 19, 2026, Mihály Pituk’s preprint Resolving the Generalized Hyperbolicity Conjecture for Shadowing claims a counterexample on a complex Banach space: an invertible $T$ with shadowing but without generalized hyperbolicity. It also claims the implication holds for separable complex Hilbert spaces, while whether separability can be removed remains open. A separate preprint reports an independent Banach-space construction, including examples on $\ell^p(\mathbb{N})$ for $1<p<\infty$ and $p\ne2$ . These are preprint claims and remain unverified.

Current status (as of August 2026): The conjecture is claimed false for general Banach spaces and claimed true for separable Hilbert spaces, but both claims remain unverified; the nonseparable Hilbert-space case and other intermediate classes remain open.

Sources

No solutions have been posted yet.