A calibrated Jordan-block crossover for the Aluthge transform

We present a candidate crossover law for the ordinary half Aluthge iteration near a change of nilpotent Jordan type. Two Jordan blocks of fixed sizes p,q, with unit internal links, are joined by a positive link epsilon. At iteration number m of order epsilon^(-4/(p+q)), a specified moving triangular representative has a limiting sqrt(m)-scaled profile, uniformly on compact positive crossover windows. Its first-link energies follow a positive nearest-neighbor system, with entrance coefficient [2^(p+q-2)(p-1)!(q-1)!]^(-1) in the original link convention. The Frobenius-energy consequence applies to the actual Aluthge iterates. The argument combines an all-later-time intrinsic discrete-to-flow comparison, the full real triangular boundary spectrum, dominant-mode passage and exact entrance calibration. The result retains candidate status.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-27
DOI
https://doi.org/10.5281/zenodo.23691758
Primary Topic
Spectral Theory in Mathematical Physics
Type
preprint
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preprint

A calibrated Jordan-block crossover for the Aluthge transform

Robert Barritz
5 links
Zenodo (CERN European Organization for Nuclear Research)
Spectral Theory in Mathematical Physics
preprint

A calibrated Jordan-block crossover for the Aluthge transform

Robert Barritz
preprint en

Abstract

We present a candidate crossover law for the ordinary half Aluthge iteration near a change of nilpotent Jordan type. Two Jordan blocks of fixed sizes p,q, with unit internal links, are joined by a positive link epsilon. At iteration number m of order epsilon^(-4/(p+q)), a specified moving triangular representative has a limiting sqrt(m)-scaled profile, uniformly on compact positive crossover windows. Its first-link energies follow a positive nearest-neighbor system, with entrance coefficient [2^(p+q-2)(p-1)!(q-1)!]^(-1) in the original link convention. The Frobenius-energy consequence applies to the actual Aluthge iterates. The argument combines an all-later-time intrinsic discrete-to-flow comparison, the full real triangular boundary spectrum, dominant-mode passage and exact entrance calibration. The result retains candidate status.

Zenodo (CERN European Organization for Nuclear Research)
Spectral Theory in Mathematical Physics
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