Residual-Weighted Decomposition of Positive Operators
James Tian
TL;DR
The paper addresses decomposing positive operators without exiting the positive cone or performing spectral decompositions. It introduces a multiplicative residual-weighted update that drives a monotone dynamics, yielding a canonical rank-one telescoping decomposition and a limiting operator. It characterizes the inverse problem via an intrinsic Moore-Penrose normalization, establishes exhaustion criteria leading to Parseval frames, and demonstrates applications to RKHS with iterative kernel decompositions that avoid spectral calculus. Collectively, it provides a constructive, diagonal-free framework to obtain Parseval frames and operator-valued energy identities for positive operators.
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.