Operator ergodic theorems with Möbius "weights"

Abstract: Motivated by Sarnak's conjecture in topological dynamics for the Möbius function $μ$, we study, for a power-bounded $T$ on a Banach space $E$, the weak convergence $$(<em>) \qquad \qquad \frac1N\sum_{n=1}^N μ(n)T^nv \to 0 \text{ weakly } \forall v\in E.$$ For that, we introduce a notion of dynamical entropy for operators, which we denote $h^</em><em>{top}(T)$, and show that if Sarnak's conjecture is true, then $h^*</em>{top}(T)=0$ implies the desired convergence (). We conclude an equivalent operator formulation of Sarnak's conjecture. For several classes of operators we prove that () holds, and that $h<sup>*_{top}(T)=0$.

• The paper establishes weak or norm convergence of Möbius-weighted operator averages for WAP, polynomially bounded, quasi-compact, rigid, and other structured operators, often with explicit logarithmic rates.
• The paper introduces dual and orbital topological entropy for contractions and proves that Sarnak’s conjecture is equivalent to Möbius convergence for separable Banach-space operators with zero dual entropy.
• The paper extends Möbius disjointness to pointwise and vector-valued ergodic theorems while identifying counterexamples and open problems involving non-reflexive spaces, rigidity, and entropy relationships.

Motivation and main problem

The paper studies, for a power-bounded operator $T$ on a Banach space $E$, the weak convergence

$$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$

where $\mu$ is the number-theoretic Möbius function. This is the natural operator-theoretic generalization of Sarnak's Möbius randomness law, which asserts that $\mu$ is orthogonal to every deterministic sequence, i.e., to every sequence of the form $f(\tau^n x)$ arising from a topological dynamical system $(X,\tau)$ of zero topological entropy. The authors follow Veech's program of formulating and attacking Sarnak-type questions for linear operators. Failure of Sarnak's conjecture would imply failure of Chowla's conjecture, which motivates the search for broad classes of systems (and operators) for which the orthogonality can be established unconditionally.

Modulated ergodic theorems for specific operator classes

The first body of results establishes strong convergence for several classes of power-bounded operators.

Weakly almost periodic operators. If $T$ is WAP on a Banach space $E$, then

$$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$

The proof uses the Jacobs–DeLeeuw–Glicksberg decomposition $E$0: on $E$1 the boundedness of $E$2 suffices, while on $E$3 Davenport's estimate,

$E$4

handles the unimodular eigenvectors. A complexification argument (Taylor norm) covers real spaces. As a corollary, a positive contraction $E$5 of $E$6 of a probability space with $E$7 satisfies $E$8; this follows since such $E$9 is WAP in $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$0 by Komorník's result. Notably, this corollary fails when $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$1: the shift on $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$2 admits a functional $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$3 built from $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$4 itself, giving $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$5. This example is important: it shows that even the weak convergence can fail without entropy control, yet it is not a counterexample to Sarnak's conjecture because the dual shift has positive topological entropy (the invariant set $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$6 carries positive-entropy Bernoulli measure).

Hilbert space contractions and polynomially bounded operators. For a contraction $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$7 on a Hilbert space, the unitary dilation theorem combined with Davenport's estimate yields the quantitative bound

$$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$8

The same rate holds for every polynomially bounded operator on any Banach space, via von Neumann's inequality and the maximum modulus principle applied to the polynomial $$\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv \to 0 \quad \text{weakly in } E,\ \forall v \in E,$$9. An appendix by Cuny improves the Hilbert space result to all power-bounded operators using Haase's transference theorem, at the cost of one logarithm: $\mu$0. The paper also records Algom–Wang's result that convergence in Sarnak's setting can be arbitrarily slow along prescribed rates, so no universal polynomial rate should be expected in general.

Rosenthal spaces and rigidity. If $\mu$1 does not contain an isomorphic copy of $\mu$2 and $\mu$3 is invertible with $\mu$4, then the weak convergence holds. The proof identifies the unit ball of $\mu$5 under $\mu$6 as a tame system (via Glasner–Megrelishvili), to which the Huang–Wang–Ye Möbius disjointness theorem for tame systems applies. Consequently, weakly rigid power-bounded operators on $\mu$7-free spaces satisfy the convergence. For rigid operators in general, only a weaker logarithmically weighted version is obtained:

$\mu$8

using Qiu–Wei–Xu's logarithmic Sarnak theorem; if rigidity holds along a sequence $\mu$9 with uniformly bounded prime-divisor sums $\mu$0, the full result follows from Kanigowski–Lemańczyk–Radziwiłł.

Further classes. The paper proves norm convergence when $\mu$1 has finitely many unimodular eigenvalues and each $\mu$2 is mean ergodic (on $\mu$3-free spaces), hence for quasi-compact contractions. It also constructs counterexamples delineating the theory: an invertible isometry satisfying the "weak mixing" condition $\mu$4 that is nevertheless not WAP; a minimal non-uniquely ergodic system (Furstenberg's skew product, via Liu–Sarnak) whose Koopman operator satisfies the convergence but is not mean ergodic — showing that mean ergodicity is not necessary for the conclusion.

Operator entropy and equivalence with Sarnak's conjecture

The central conceptual contribution is a notion of dynamical topological entropy for contractions. Since every power-bounded operator is a contraction in an equivalent norm, define for a contraction $\mu$5 on $\mu$6

$\mu$7

where $\mu$8 is the unit ball of $\mu$9 with its weak* topology. This is well-defined because $f(\tau^n x)$0 maps $f(\tau^n x)$1 into itself continuously. The key structural facts are:

• Monotonicity: if $f(\tau^n x)$2 is $f(\tau^n x)$3-invariant, then $f(\tau^n x)$4, proved by exhibiting the quotient map as a factor.
• Sarnak transfer: if Sarnak's conjecture holds and $f(\tau^n x)$5, then the desired convergence holds for $f(\tau^n x)$6 (and all powers $f(\tau^n x)$7). In fact, invoking the known strengthening of Sarnak's conjecture to uniform norm convergence, one obtains norm convergence.
• Consistency with topology: for $f(\tau^n x)$8 on $f(\tau^n x)$9 with $(X,\tau)$0 compact metric, $(X,\tau)$1 if and only if $(X,\tau)$2. The non-invertible case requires a natural-extension argument (Appendix C): Rokhlin's extension preserves both Kolmogorov–Sinai entropy and topological entropy, and the induced map on probability measures is a factor.

This yields the paper's main structural theorem: Sarnak's conjecture for compact metric systems is equivalent to the statement that every contraction $(X,\tau)$3 on a separable Banach space with $(X,\tau)$4 satisfies the Möbius-weighted ergodic convergence. Thus the paper produces a genuine operator formulation of Sarnak's conjecture.

A second, orbital entropy is also introduced, inspired by Tao and Veech: for $(X,\tau)$5, let $(X,\tau)$6 be the weak* closure of the shifted sequences in $(X,\tau)$7, and set $(X,\tau)$8. One always has $(X,\tau)$9, so vanishing of the dual entropy implies vanishing of the orbital entropy, and either condition transfers Sarnak's conjecture to the operator. For minimal metric systems, all three entropies ($T$0, $T$1, $T$2) coincide; minimality is used only for the implication $T$3, and whether it can be dropped is left open.

The paper then verifies $T$4 for the classes handled earlier: WAP contractions (via a Stone–Weierstrass argument showing the induced system on $T$5 is WAP, hence tame, plus Glasner's theorem that minimal tame systems have zero entropy); contractions with some weakly compact power; contractions satisfying the weak-mixing condition on spaces with separable dual (where the dual system is uniquely ergodic with fixed point $T$6); weakly rigid contractions (rigidity forces every invariant measure to have zero entropy); and invertible isometries on $T$7-free spaces (via Kerr–Li's CA-entropy). The class $T$8 of zero-entropy contractions is shown to be convex and multiplicatively stable.

Reductions to shifts on $T$9

Following Veech, the paper reduces Sarnak's conjecture to statements about the shift. For $E$0, let $E$1 be the Gelfand space of the weak* closed algebra generated by the orbit of $E$2; call $E$3 shift deterministic (SD) if $E$4. Then Sarnak's conjecture for homeomorphisms is equivalent to Veech's conjecture that $E$5 for every SD sequence. Adapting the argument to $E$6 extends the equivalence to non-invertible maps, using the natural extension from Appendix C. A concrete consequence: for any $E$7 whose indicator generates a zero-entropy subshift (e.g., arithmetic progressions, where $E$8 is finite), Sarnak's conjecture implies $E$9.

Pointwise convergence in $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$0 and vector-valued settings

Independently of entropy considerations, the paper establishes almost everywhere convergence. Building on Bourgain's observation (elaborated by Cuny–Weber and el Abdalaoui et al.) that $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$1 a.e. for every $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$2 and every measure-preserving $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$3, the paper proves:

• For every Dunford–Schwartz contraction $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$4 on $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$5 of a probability space, $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$6 a.e. for all $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$7, via Baxter–Olsen's transference principle; Markov operators with invariant probability are covered as a corollary.
• For positive contractions of $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$8, $$\Big\|\frac{1}{N}\sum_{n=1}^{N}\mu(n)T^nv\Big\| \to 0 \quad \forall v \in E.$$9, the same a.e. convergence holds.
• Vector-valued extensions: for reflexive $E$00, power-bounded $E$01 on $E$02 gives norm convergence; for $E$03 on $E$04, pointwise convergence holds for arbitrary Banach $E$05; and for Hilbert-space-valued Dunford–Schwartz-type contractions, a.e. convergence is proved for $E$06 functions using the quantitative Hilbert-space bound and a dyadic subsequence argument. For real Hilbert lattices, positivity and Chacon's domination technique extend the result to all of $E$07.

Whether the Hilbert-lattice result extends to arbitrary reflexive $E$08, or to all of $E$09 under Chacon's hypotheses, is posed as an open problem.

Other modulations

Appendix D observes that most proofs use the Möbius function only through Davenport-type exponential sum estimates, so they extend verbatim to other weight sequences: the Liouville function (Bateman–Chowla), the Rudin–Shapiro sequence (with the stronger bound $E$10, yielding $E$11 on Hilbert space), and Rademacher random signs for almost every $E$12, including a pointwise ergodic theorem with random signs for non-positive $E$13 contractions and for Dunford–Schwartz operators. The appendix notes an instructive contrast: the topological entropy of the Möbius sequence itself is positive, while that of Rudin–Shapiro is zero — so these limit theorems do not depend on the entropy of the modulating sequence, only on its exponential-sum behavior.

Limitations and open problems

Several limitations are stated plainly. The unconditional results require structural hypotheses (WAP, quasi-compactness, $E$14-freeness, rigidity, finite unimodular spectrum); the general zero-entropy statement depends on Sarnak's conjecture, which remains open. The failure example on $E$15 shows the $E$16-free hypothesis in the Rosenthal-space theorem cannot simply be dropped. Specific open problems include: whether every power-bounded operator on a Banach space satisfying the convergence must act on a reflexive space (Problem 1); whether rigid contractions on separable spaces always satisfy the convergence (Problem 2); whether finite unimodular spectrum can be relaxed to countable (Problem 3); whether $E$17 for Koopman operators of non-invertible maps beyond the zero-entropy case (Problem 4); whether quasi-compact contractions have $E$18 (Problem 5); whether $E$19 implies $E$20 (Problem 6); and whether the orbital entropy characterization of zero topological entropy extends beyond minimal systems (Problem 7). The appendix also exhibits weakly rigid contractions that are not rigid (via Körner's minimal recurrent-but-not-uniformly-recurrent system) and rigid contractions that are not mean ergodic, showing these pathologies persist within the class relevant to the main theorems.

Conclusion

The paper develops a systematic operator-theoretic framework for the Möbius randomness law: it introduces a dual (and an orbital) topological entropy for power-bounded operators, proves that Sarnak's conjecture is equivalent to its operator formulation restricted to zero-entropy contractions, establishes unconditional norm or weak convergence for WAP, polynomially bounded, quasi-compact, rigid-on-$E$21-free, and related operator classes with explicit logarithmic rates, and provides pointwise (including vector-valued) analogues requiring no entropy assumptions. The reduction of Sarnak's conjecture to Veech's shift-deterministic formulation, together with the identified open problems on the relationship between the various entropies and on the necessity of reflexivity, delineates precisely what remains unresolved in the operator-theoretic approach.