Universal Kernel Models for Iterated Completely Positive Maps

TL;DR

This paper introduces a unified framework for analyzing iterated completely positive maps on operator-valued kernels, providing explicit formulas, limit behaviors, and structural decompositions within a single Hilbert space setting.

Contribution

It develops a comprehensive model that captures the dynamics of iterated completely positive maps on kernels, including explicit limit kernels and Radon-Nikodym representations, enhancing understanding of their asymptotic properties.

Findings

Explicit limit kernel for unital maps

Stein-type decomposition and Radon-Nikodym representation

Almost-sure growth law for random compositions

Abstract

We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This model yields a direct formula for every iterated kernel and allows pointwise limits, contractive behavior, and kernel domination to be read as standard operator facts. The main results include an explicit limit kernel for unital maps, a Stein-type decomposition, a Radon-Nikodym representation under subunitality, and an almost-sure growth law for random compositions. The construction keeps all iterates in one space, making their comparison and asymptotic analysis transparent.