Mean ergodic convergence from near-diagonal subadditivity
- Primary subject
- Mathematics
- Subject subfields
- Functional Analysis · Dynamical Systems
- Paper type
- Research Article
- Manuscript stage
- Preprint
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. The corresponding lower-limit assertion remains open.