LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Hilbert cesaro averages converge to the fixed subspace

Statement

Assume AC. Let $U$ be a linear isometry on a closed complex $L^{2}$ subspace $H$ , and put $F = \ker (I - U)$ . For each $f \in H$ , $A_{N} f = \frac{1}{N} \sum_{n = 0}^{N - 1} U^{n} f ⟶ P_{F} f in norm as N \to \infty .$ Also $R = U (H)$ is closed, and $V = U^{- 1} P_{R}$ is a linear contraction satisfying $⟨ U f, g ⟩ = ⟨ f, V g ⟩$ and $V U = I$ . Here $U^{- 1}$ means the inverse from $R$ to $H$ , not a surjectivity assumption on $U$ .

Facts & Assumptions

[F1]

Closed subspaces have unique orthogonal projections and orthogonal decompositions under AC Closed l two subspaces have orthogonal projections.

[F2]

Isometries preserve the pairing and the norm; adjoint and invariant-subspace conventions are fixed locally L two operator conventions for weak mixing.

[F3]

The complex pairing is sesquilinear and satisfies Cauchy–Schwarz The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz.

[F4]

Assume AC The Axiom of Choice, as required for the projections and completeness used in their proof.

[F5]

Under countable choice, complex $L^{2}$ is complete Complex Lp completeness and almost-everywhere subsequences.

Proof

Given: $H$ , $U$ and AC as stated; $N$ is a positive integer.

1.1F1F2F4F5

F5 and AC make the ambient complex $L^{2}$ complete. Hence its closed subspace $H$ is complete: an $H$ -valued Cauchy sequence converges in the ambient space by F5, and closedness puts its limit in $H$ . If $U f_{n}$ converges in $H$ , then $∥ f_{n} - f_{m} ∥ = ∥ U f_{n} - U f_{m} ∥$ makes $(f_{n})$ Cauchy. Its limit $f \in H$ satisfies $U f_{n} \to U f$ by isometry, so $R$ is closed. Isometry makes $U$ injective; its inverse on $R$ is linear and isometric. The projection $P_{R}$ therefore defines the linear contraction $V = U^{- 1} P_{R}$ .

1.2F2F3

Let $M = \overline{(I - U) H}$ . This is a closed subspace: sums and scalar multiples of limits remain limits by the norm inequalities. If $h ⊥ M$ , then $⟨ h, h - U h ⟩ = 0$ , whence $⟨ h, U h ⟩ = ∥ h ∥^{2}$ . Expansion and isometry give $∥ h - U h ∥^{2} = 2 ∥ h ∥^{2} - 2 Re ⟨ h, U h ⟩ = 0$ , so $U h = h$ . Conversely, if $U h = h$ , then for each $g \in H$ , $⟨ h, U g ⟩ = ⟨ U h, U g ⟩ = ⟨ h, g ⟩$ . Thus $h$ is orthogonal to $(I - U) H$ , and Cauchy–Schwarz extends orthogonality to its closure. Consequently $M^{⊥} = F$ .

2.1F1F2step 1.1

Write $P_{R} g = U h$ . Orthogonality gives $⟨ U f, g ⟩ = ⟨ U f, U h ⟩ = ⟨ f, h ⟩ = ⟨ f, V g ⟩$ . Since $P_{R} U f = U f$ , we have $V U f = f$ . This proves the adjoint identity without a representation theorem or an inverse of $U$ on all of $H$ .

2.2F1step 1.2

By orthogonal decomposition, $H = M \oplus F$ . The subspace $F$ is closed, either as $M^{⊥}$ or directly by continuity of $I - U$ . In the decomposition $f = m + h$ , $m \in M$ , $h \in F$ , the vector $m$ is orthogonal to $F$ , so uniqueness of projection gives $h = P_{F} f$ .

2.3F2step 1.2

Isometry and the triangle inequality give $∥ A_{N} ∥ \leq 1$ . For $g \in H$ , cancellation of the finite sum gives $A_{N} (I - U) g = (g - U^{N} g) / N$ , of norm at most $2 ∥ g ∥ / N$ . For $m \in M$ and $ε > 0$ , choose one $g$ with $∥ m - (I - U) g ∥ < ε$ . Hence ${lim sup}_{N} ∥ A_{N} m ∥ \leq ε$ . As $ε$ is arbitrary, $A_{N} m \to 0$ . No sequence of such approximants is needed.

3.1F4step 1.1step 2.2step 2.3∎

For $h \in F$ , every $U^{n} h = h$ , so $A_{N} h = h$ . Applying this and the previous limit to $f = m + h$ gives $A_{N} f \to h = P_{F} f$ . For $N = 1$ the average is the identity; zero vectors and $H = {0}$ obey every formula without division by a vector norm. AC is inherited from the projection/completeness argument in step 1.1 and the projections in step 2.2.

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sarig Theorem 2.1 pp.35–36 (standard reference, not scraped)
Axler 8.37–8.40 projection route (standard reference, not scraped)