Decomposability of nonnegative r-potent operators on L2(X)

TL;DR

This paper studies the structure of nonnegative r-potent operators on L2(X), providing an algorithm to construct nonnegative orthogonal basis functions and establishing conditions for their decomposability.

Contribution

It introduces a constructive algorithm for basis functions and characterizes decomposability of nonnegative r-potent operators based on the dimension of their range spaces.

Findings

Basis functions can be chosen to be nonnegative and orthogonal.

Operators with range space dimension > r-1 are decomposable.

Provides a criterion for decomposability based on range space dimension.

Abstract

We investigate the decomposability of nonnegative compact r-potent operators on a separable Hilbert space L2(X). We provide a constructive algorithm to prove that basis functions of range spaces of nonnegative r-potent operators can be chosen to be all nonnegative and mutually orthogonal. We use this orthogonality to establish that nonnegative compact r-potent operators with range spaces of dimension strictly greater than r-1 are decomposable.