Entropy for compact operators and results on entropy and specification

We investigate the topological entropy of operators. More precisely, in the Banach space setting, we show that compact operators have finite entropy, which depends solely on their point spectrum. Moreover, for operators on (F)-spaces, we explore the relationship between the specification property and entropy. In particular, we show that the specification property implies infinite topological entropy, while the operator specification property implies positive entropy. We also show that the invariance principle fails for the class of compact operators.

💡 Research Summary

The paper investigates the topological entropy of linear operators, focusing on two distinct settings: compact operators on Banach spaces and operators on general F‑spaces. Using Bowen’s definition of entropy (based on (n,ε)‑separated sets), the authors first establish that any compact operator T acting on a Banach space has finite topological entropy, and that this entropy is completely determined by the part of the point spectrum lying outside the unit disc. Specifically, they prove

 h_top(T) = Σ_{λ∈σ(T), |λ|>1} log|λ|,

where σ(T) denotes the spectrum of T. The proof hinges on classical spectral facts about compact operators: σ(T) consists of 0 together with a sequence of eigenvalues λ_n→0, each with finite‑dimensional eigenspace. By applying the Riesz decomposition theorem, the space X splits into two T‑invariant closed subspaces M₁ and M₂ corresponding respectively to eigenvalues with |λ|≤1 and |λ|>1. On M₁ the spectral radius is ≤1, so a sufficiently high power of T becomes a contraction, yielding zero entropy. On M₂ the restriction of T is a finite‑dimensional linear map, and Bowen’s theorem for finite‑dimensional endomorphisms gives the explicit logarithmic sum. Lemma 3.1 shows that entropy of a direct sum does not exceed the sum of the entropies of the summands, which together with the previous observations yields the formula above.

A striking corollary is that the classical variational principle (which equates topological entropy with the supremum of metric entropies over all invariant probability measures) fails for compact operators. Because the entropy can be arbitrarily large while the only invariant measures are supported on finite‑dimensional eigenspaces, the supremum of metric entropies is zero, contradicting the topological value.

The second part of the work moves to F‑spaces (complete metrizable topological vector spaces) and studies two specification‑type properties. The ordinary specification property (SP) requires that any finite collection of orbit pieces can be ε‑shadowed by a single periodic point after a uniformly bounded transition time. The operator specification property (OSP) adapts this notion to linear operators by demanding a sequence of invariant sets whose restrictions satisfy SP. The authors prove two complementary results (Theorem B):

1. If an operator T on an F‑space has OSP, then T possesses at least one eigenvalue with modulus greater than one, and consequently h_top(T) > 0.
2. If T satisfies the full specification property, then its topological entropy is infinite.

The intuition behind (2) is that SP allows the construction of arbitrarily many disjoint orbit segments, leading to (n,ε)‑separated sets whose cardinality grows faster than any exponential rate, forcing the entropy to diverge. For (1), OSP forces the existence of a non‑trivial expanding direction, which by the earlier compact‑operator analysis yields positive entropy.

Concrete examples are provided via weighted shift operators on sequence spaces (ℓ^p, 1≤p<∞). A backward weighted shift B_w defined by B_w e_n = w_n e_{n‑1} has spectrum {0}∪{w_n : n∈ℕ}. If the weights satisfy |w_n|>1 infinitely often, B_w exhibits infinite entropy and satisfies SP; if the weights are uniformly less than one, B_w is a contraction with zero entropy. The authors also construct an F‑space and a weighted shift that fulfills OSP but not SP, illustrating the strict hierarchy between the two properties.

Overall, the paper makes three major contributions:

• It provides an exact entropy formula for compact operators, showing that entropy is a purely spectral invariant in this class.
• It demonstrates the breakdown of the variational principle for compact operators, highlighting a fundamental difference between linear and nonlinear dynamics.
• It clarifies the relationship between specification‑type properties and entropy in infinite‑dimensional linear dynamics, establishing that SP forces infinite entropy while OSP guarantees only positivity.

These results deepen the connection between operator theory, spectral analysis, and dynamical complexity, and they open avenues for further research on entropy in non‑compact, non‑linear, or non‑metrizable settings.