Residual-Weighted Decomposition of Positive Operators

TL;DR

This paper introduces a residual-weighted iterative scheme for decomposing positive operators into rank-one components, providing a constructive method to generate Parseval frames and analyze operator dynamics.

Contribution

It develops a novel residual-weighted decomposition approach for positive operators, including a canonical telescoping form and conditions for operator exhaustion and frame construction.

Findings

Residuals admit a canonical telescoping decomposition.

Constructive procedure for generating Parseval frames without spectral calculus.

Energy identities and conditions for operator exhaustion in trace-class cases.

Abstract

This paper investigates an iterative rank-one decomposition scheme for positive operators on a Hilbert space based on a residual-weighted congruence update. At each step the operator is compressed along a chosen unit vector while remaining inside the positive cone, and the resulting map defines a monotone dynamical system on the cone of positive operators. We prove that the associated residuals admit a canonical telescoping decomposition into rank-one terms and a limiting positive operator, and we identify this limit together with an exact energy identity expressing the defect between the initial and limiting operators as a convergent series of rank-one contributions. In the case where the iteration exhausts the operator, the residual directions form a Parseval frame for the natural range space, yielding a constructive procedure that produces Parseval frames without spectral calculus. We further solve the inverse problem by characterizing those decreasing chains with rank-one steps that arise from such dynamics via an intrinsic normalization condition involving the Moore-Penrose inverse. For trace-class operators we obtain a scalar energy identity and show that mild greedy or density conditions on the chosen directions guarantee exhaustion. An application to reproducing kernel Hilbert spaces illustrates the abstract results.