Topological shadowing for linear operators

Abstract: We prove a linear operator of a finite-dimensional Banach space has the topological shadowing property, as introduced in \cite{lny}, if and only if its spectrum lies either within the open unit complex disk or outside the closure of that disk. Moreover, the spectrum of every uniformly expansive linear operator with the topological shadowing property lies either within the open unit complex disk or outside the closure of that disk. Lastly, we prove that a normal operator with the topological shadowing property on a Hilbert space does not have any nonzero nonwandering points.

• The paper proves that an invertible linear operator on a finite-dimensional Banach space has topological shadowing exactly when its spectrum lies entirely inside the unit disk or entirely outside its closure.
• The paper shows in arbitrary dimensions that uniformly expansive operators with topological shadowing satisfy the same spectral dichotomy, while normal Hilbert-space operators have only the zero point in their nonwandering set.
• The results establish strong rigidity for variable-radius shadowing and leave open the full infinite-dimensional conjecture, its relationship with classical shadowing, and whether bounded shadowing follows from it.

Overview

This paper, "Topological shadowing for linear operators" (2608.12862), studies the topological shadowing property — a variable-radius refinement of the classical shadowing property in which the error tolerances $\epsilon$ and $\delta$ are positive continuous functions $C^+(X)$ rather than constants — for invertible bounded linear operators on Banach spaces. The authors (Yang and Morales) resolve a conjecture in the finite-dimensional case and establish two further spectral and dynamical rigidity results. The central message is that topological shadowing is extremely restrictive for linear dynamics: it forces the spectrum to lie entirely inside the open unit disk $\mathbb{D}$ or entirely outside its closure.

Background and definitions

For a bijective map $f$ of a metric space $X$, a sequence $(x_n)_{n\in\mathbb{Z}}$ is a $\delta$-pseudo orbit if $d(f(x_n), x_{n+1}) < \delta(f(x_n))$ for all $n$, and it is $\delta$0-shadowed by some $\delta$1 if $\delta$2 for all $\delta$3. The map has the topological shadowing property if for every $\delta$4 there is $\delta$5 shadowing every $\delta$6-pseudo orbit. This notion was introduced by Lee–Nguyen–Yang under the name "shadowing property" and is distinct from the classical Bowen shadowing property with constant tolerances; the terminology follows Das–Lee–Richeson–Wiseman.

The setting is $\delta$7, the group of invertible bounded linear operators with bounded inverse on a nontrivial complex Banach space. The guiding conjecture states that $\delta$8 has the topological shadowing property if and only if $\delta$9 or $C^+(X)$0.

Main results

Theorem A proves the conjecture in finite dimensions: a linear operator of a finite-dimensional Banach space has the topological shadowing property if and only if its spectrum lies in $C^+(X)$1 or in $C^+(X)$2. This is a complete spectral characterization: in finite dimensions, topological shadowing holds precisely for uniform contractions and uniform expansions (up to equivalent norms).

Theorem B gives a one-sided version valid in arbitrary dimension: every uniformly expansive operator (in the sense of Hedlund) with the topological shadowing property has spectrum in $C^+(X)$3 or $C^+(X)$4.

Theorem C concerns normal operators on Hilbert spaces: if a normal operator has the topological shadowing property, then its nonwandering set is trivial, $C^+(X)$5. This parallels Mazur's theorem that normal operators have the classical shadowing property exactly when hyperbolic, but extends the conclusion to the weaker topological shadowing hypothesis.

Key technical ingredients

The proofs rest on several lemmas of independent interest:

• Hyperbolicity plus topological shadowing implies spectral dichotomy (Lemma sim): if a hyperbolic operator on any Banach space has topological shadowing, then $C^+(X)$6 lies wholly in $C^+(X)$7 or wholly outside $C^+(X)$8. The proof constructs a pseudo orbit that switches from the unstable component to the stable component at time zero and derives a contradiction using the exponentially decaying tolerance $C^+(X)$9.
• Expansivity from shadowing: if $\mathbb{D}$0 has topological shadowing and the set $\mathbb{D}$1 of points with bounded full orbits is closed, then $\mathbb{D}$2 is expansive. The argument renorms $\mathbb{D}$3 via $\mathbb{D}$4 so that the restriction becomes a linear isometry, contradicting the fact that no linear isometry of a nontrivial Banach space has the topological bounded shadowing property — a strong rigidity statement proved here by a chaining argument showing every point would lie within distance 2 of the origin.
• Uniform expansivity plus topological shadowing implies hyperbolicity (Lemma putin): the stable and unstable subspaces $\mathbb{D}$5 and $\mathbb{D}$6 are closed with spectral radii below 1, and a pseudo-orbit/shadowing argument shows $\mathbb{D}$7.

The converse of the last lemma fails: hyperbolic operators need not have topological shadowing unless they are uniform contractions or expansions, as Lemma sim already shows.

Proof strategy for Theorem A

Necessity follows because finite-dimensional subspaces are closed, so $\mathbb{D}$8 is closed; Lemmas baja and sim then force the spectral dichotomy. Sufficiency is proved directly for $\mathbb{D}$9: after passing to an adapted norm with $f$0, the authors construct a shadowing point via the convergent series $f$1 and verify the estimate $f$2 using a carefully chosen $f$3 satisfying monotonicity and decay conditions outside a compact ball. Compactness of closed balls in finite dimensions supplies the constant $f$4. The case $f$5 reduces to the inverse, and local compactness upgrades positive shadowing to full two-sided shadowing.

Limitations and open questions

Several gaps remain explicitly open. The full conjecture in infinite dimensions is unresolved: the paper proves only the uniformly expansive case (Theorem B). It is unknown whether the topological shadowing property implies the topological bounded shadowing property, and whether the latter passes to restrictions on $f$6 without the closedness assumption. Whether every operator with topological shadowing also has the classical shadowing property remains open — a consequence of the conjecture if true. Finally, Theorem C addresses only normal operators; the behavior of general operators on Hilbert spaces with topological shadowing is not settled.

Conclusion

The paper establishes that topological shadowing for linear operators is essentially equivalent to uniform contraction or expansion: exactly characterized in finite dimensions, and shown to impose the same spectral dichotomy under uniform expansivity in general, while forcing trivial nonwandering sets for normal Hilbert-space operators. The remaining infinite-dimensional conjecture, and the relationship between topological and classical shadowing for operators, constitute the natural next targets.