Positive Operator Cocycles in Shrinking Corridors

Abstract

We characterize trace-norm convergence of positive operator sequences whose cocycle subadditivity is required only near the diagonal. For a positive trace-preserving map on trace class, every such sequence has convergent normalized slopes exactly when the map is strongly mean ergodic and the scalar corridor's square-width integral diverges. The scalar width threshold is inherited; the operator transfer combines recurrent-support absorption, compact positive orbits and a regularized root-fidelity energy with a linear trace-slack bound. The criterion covers faithful stationary states and absorbing channels with nonfaithful stationary states. A summable dyadic trace-error condition gives robustness for positive errors with varying directions. We also prove a finite-dimensional selfadjoint criterion, recover reducing-corner convergence, and obtain the unitary classification by pure-point spectrum and square width. Explicit examples describe infinite-dimensional escape and the role of trace preservation. The inequalities are imposed in both orders on a measurable nonincreasing relative-width corridor with eventual adjacent integer access. Strong mean ergodicity means pointwise trace-norm convergence of the Cesaro averages on every trace-class input. The proof identifies a recurrent support carrying a faithful stationary density and controls transient mass and coherences before synchronizing the normalized slopes.