Mean ergodic convergence from near-diagonal subadditivity
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Abstract
We prove a mean ergodic theorem under subadditivity required only for nearly equal block lengths. The operator is positive, unital and integral-preserving, without ergodicity or invertibility. Convergence of invariant projections of uniformly integrable normalized inputs, with both ordered adjacent-two inequalities, forces L1 convergence. An existing scalar shrinking-corridor theorem supplies invariant Lp convergence for every finite p when the nonincreasing relative width has divergent square integral and eventually admits length differences two. Nonnegative inputs require individual Lp membership; for signed inputs, uniform integrability of normalized negative-part pth powers is necessary and sufficient. Under the same scalar-admissible width and norm criterion, signed integrable inputs have almost-sure limsup equal to the invariant limit, uniformly over macroscopic operator powers and lengths.
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Research details
- Subjects
- Mathematics · Functional Analysis · Dynamical Systems
- Keywords
- Functional Analysis, Dynamical Systems, mathematical research
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
General nonergodic positive unital integral-preserving Markov norm theorem and exact negative-part UI criterion retained for every scalar-admissible width. Signed individual-L1 inputs with both eventual ordered adjacent-two inequalities and explicit finite almost-sure invariant projected slopes satisfy the conditional pointwise upper and uniform macroscopic operator-power upper laws; norm convergence gives almost-sure limsup equality. The existing scalar-width theorem supplies these projected slopes and norm convergence for every nonincreasing width with divergent square integral and eventual n*w(log n)>=2, including nonnegative individual-Lp inputs.
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