- The paper proves that the generalized hyperbolicity conjecture for shadowing does not hold on general Banach spaces but does hold on separable Hilbert spaces.
- Different behavior on Hilbert and Banach spaces is traced to spectral properties, specifically the inclusion $\sigma_s(T) \subseteq \sigma_r(T)$ on Banach spaces which becomes equality on Hilbert space
- The dichotomy is rooted in the complementability of kernels or the coincidence $\sigma_s(T) = \sigma_r(T)$
The generalized hyperbolicity conjecture for shadowing asked whether every invertible bounded linear operator on a complex Banach space that has the shadowing property must be generalized hyperbolic. This paper by Mihály Pituk settles the question with a two-part answer: the conjecture fails on general Banach spaces, but holds on separable Hilbert spaces (2608.19499). The dichotomy is traced precisely to the possible strictness of the inclusion σs(T)⊆σr(T) on Banach spaces, an inclusion that becomes equality on Hilbert spaces.
Background and the conjecture
For T∈GL(X) acting on a complex Banach space X, a δ-pseudotrajectory (xn)n∈Z satisfies ∥xn+1−Txn∥≤δ; the shadowing property requires that every δ-pseudotrajectory be ϵ-shadowed by an exact orbit Tnv uniformly over all n∈Z, for arbitrarily small T∈GL(X)0. Generalized hyperbolicity, introduced by Bernardes et al., weakens classical hyperbolicity: it requires a decomposition T∈GL(X)1 into closed invariant-under-the-appropriate-semigroup subspaces with T∈GL(X)2, T∈GL(X)3, and spectral radii T∈GL(X)4, T∈GL(X)5. Classical hyperbolicity additionally demands T∈GL(X)6 and T∈GL(X)7.
It was known since Bernardes et al. that generalized hyperbolicity implies shadowing, and the converse had been posed as Problem 5.0.3 of D'Aniello–Darji–Maiuriello and repeatedly stated as a conjecture in the literature (Antunes–Mantovani–Varão, Lee–Morales). Partial evidence existed: bilateral weighted shifts on T∈GL(X)8 and T∈GL(X)9 satisfy the equivalence, as do large classes of composition operators on X0.
Spectral characterization of shadowing
A central input is the recent characterization due to Dragičević and Pituk: for X1 on a Banach space, X2 has the shadowing property if and only if X3, where X4 is the unit circle. On Hilbert spaces this reduces to the right-spectrum condition X5, because every closed subspace of a Hilbert space is complemented, forcing X6. On general Banach spaces surjectivity does not imply right invertibility — X7 is right invertible exactly when it is surjective and X8 is complemented — so X9 can occur. This gap is the entire source of the counterexample.
A new characterization via right resolvent functions
The paper's structural contribution is a criterion for generalized hyperbolicity expressed through right resolvent functions. A right resolvent function on an open set δ0 is a continuous δ1 satisfying δ2 together with the full resolvent equation δ3. This is stronger than a holomorphic family of right inverses, whose existence on any connected open subset of δ4 follows from Allan's theorem but which need not obey the resolvent equation.
Main theorem C: δ5 is generalized hyperbolic if and only if there exists δ6 such that δ7 admits a right resolvent function on the annulus δ8.
The necessity direction is constructive: given the decomposition δ9, the explicit Laurent-type series
(xn)n∈Z0
(with (xn)n∈Z1 the projection onto (xn)n∈Z2 along (xn)n∈Z3) converges uniformly on compact subannuli by Gelfand's formula, satisfies the right-inverse identity, and its range equals the closed subspace (xn)n∈Z4 independently of (xn)n∈Z5; this common-range property forces the projection identity (xn)n∈Z6 to have constant range, from which the resolvent equation is derived algebraically. Conversely, expanding a right resolvent function into its Laurent series and matching coefficients yields a projection (xn)n∈Z7 whose ranges give the decomposition, with coefficient estimates (xn)n∈Z8 producing (xn)n∈Z9 and ∥xn+1−Txn∥≤δ0.
An immediate corollary is that generalized hyperbolic operators satisfy ∥xn+1−Txn∥≤δ1, recovering the known implication "generalized hyperbolic ∥xn+1−Txn∥≤δ2 shadowing" via the surjective-spectrum characterization.
Counterexample on a Banach space
Theorem A constructs ∥xn+1−Txn∥≤δ3 with ∥xn+1−Txn∥≤δ4 and an operator ∥xn+1−Txn∥≤δ5, where ∥xn+1−Txn∥≤δ6 involves the quotient map ∥xn+1−Txn∥≤δ7 and the backward shift ∥xn+1−Txn∥≤δ8 on ∥xn+1−Txn∥≤δ9. The design rests on two facts about δ0: it is surjective, indeed δ1 is surjective for all δ2 (the nonzero case handled via the forward shift δ3 and Neumann inversion of δ4); yet δ5 admits no bounded linear right inverse, because such a right inverse would compose to one for δ6, contradicting Phillips' classical theorem that δ7 is uncomplemented in δ8.
The doubling construction makes δ9 explicitly invertible (ϵ0). For ϵ1, surjectivity of ϵ2 reduces to surjectivity of ϵ3 with ϵ4, so ϵ5 and ϵ6 has shadowing. Conversely, a bounded right inverse of ϵ7 would yield one for ϵ8 via the embedding ϵ9, so Tnv0 and, by the corollary of Theorem C, Tnv1 is not generalized hyperbolic. The author notes that Theorem A was proved independently by Messaoudi et al. (Messaoudi et al., 17 Aug 2026).
Positive result on separable Hilbert spaces
Theorem B proves that on a separable complex Hilbert space, shadowing implies generalized hyperbolicity. Given shadowing, Corollary (right spectrum off Tnv2) provides Tnv3 for some Tnv4. The key additional step is showing that the nullity function Tnv5 is locally constant on Tnv6: near Tnv7, the family Tnv8 gives right inverses, the associated projections Tnv9 have constant range n∈Z0, and each kernel is isomorphic to the fixed quotient n∈Z1; Hilbert-space isomorphisms preserve dimension. Since n∈Z2 is connected, the nullity is constant there, so Proposition 9.17 of Apostol–Fialkow–Herrero–Voiculescu supplies a genuine right resolvent function, and Theorem C delivers the decomposition.
Two remarks qualify this argument. First, separability enters through the Apostol et al. proposition; the paper leaves open whether Theorem B extends to nonseparable Hilbert spaces. Second, Messaoudi et al. had earlier established equivalence between shadowing and pseudo-hyperbolicity on Hilbert spaces, but pseudo-hyperbolicity omits the closed direct-sum decomposition requirement and therefore does not resolve the conjecture.
Limitations and open questions
The paper concedes two points plainly. The positive theorem requires separability of the underlying Hilbert space, and removing this hypothesis remains open. Additionally, the counterexample necessarily exploits a non-complemented kernel structure tied to Phillips' theorem, so the failure of the conjecture cannot occur in settings where surjectivity and right invertibility coincide; whether analogous failures arise for other classes of Banach spaces where n∈Z3-type obstructions are absent is not addressed.
Conclusion
The paper resolves the generalized hyperbolicity conjecture for shadowing in full: negatively on general Banach spaces via an explicit invertible operator built from the quotient map n∈Z4, positively on separable Hilbert spaces. The technical core is the right-resolvent characterization of generalized hyperbolicity (Theorem C), which converts a functional-analytic condition on an annulus around n∈Z5 into a spectral-radius estimate for a pair of restrictions. The resulting picture identifies complementability of kernels — equivalently, the coincidence n∈Z6 — as exactly the mechanism separating the affirmative and negative cases.