[Paper Review] The Scaled Relative Graph of a Linear Operator

Richard Pates|arXiv (Cornell University)|Jun 10, 2021

Matrix Theory and Algorithms8 references4 citations

TL;DR

This paper establishes a direct link between the scaled relative graph (SRG) of a linear operator and the numerical range of a related operator, enabling the use of numerical range theory to analyze SRG properties. The key contribution is a characterization showing that the SRG's Beltrami-Klein map is convex and satisfies an analogue of Hildebrant’s theorem, with applications to plotting SRG boundaries for matrices and differential operators in both real and complex Hilbert spaces.

ABSTRACT

The scaled relative graph (SRG) of an operator is a subset of the complex plane. It captures several salient features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related linear operator. Furthermore we demonstrate that the SRG of a linear operator has a range of spectral and convexity properties, and satisfies an analogue of Hildebrant's theorem.

Motivation & Objective

• To develop a geometric framework for analyzing linear operators using the scaled relative graph (SRG), particularly in the context of convergence analysis for fixed-point iterations.
• To resolve the challenge of determining the SRG for general linear operators, especially in finite-dimensional and infinite-dimensional Hilbert spaces.
• To extend existing SRG theory to real Hilbert spaces by leveraging complexification and known results from complex operator theory.
• To provide a computational method for plotting the boundary of the SRG using algorithms from numerical range computation.
• To establish convexity and spectral properties of the SRG, including an analogue of Hildebrant’s theorem.

Proposed method

• The SRG is related to the numerical range of a transformed operator via a Möbius transformation, specifically the Beltrami-Klein map.
• For complex Hilbert spaces, the SRG is characterized as the image of a joint numerical range of two self-adjoint operators derived from the original operator T.
• The method uses the complexification of real Hilbert spaces to apply complex operator theory, allowing the transfer of results from complex to real settings.
• Convexity of the SRG is established using known results on joint numerical ranges, particularly that of two Hermitian forms.
• Boundary plotting algorithms for the SRG are adapted from numerical range computation techniques, applied to the transformed operator.
• An analogue of Hildebrant’s theorem is derived by analyzing the spectral properties of the SRG under the Beltrami-Klein map.

Experimental results

Research questions

• RQ1 Can the SRG of a linear operator be fully characterized using the numerical range of a related operator?
• RQ2 What are the convexity properties of the SRG, and how do they differ from those of the numerical range?
• RQ3 How can the SRG be computed for operators on real Hilbert spaces, given that standard numerical range tools apply to complex spaces?
• RQ4 Does the SRG satisfy a theorem analogous to Hildebrant’s theorem on numerical ranges?
• RQ5 Can existing numerical range algorithms be adapted to compute the boundary of the SRG for matrices and differential operators?

Key findings

• The SRG of a linear operator on a complex Hilbert space is equal to the image of the joint numerical range of two self-adjoint operators derived from T under a Möbius transformation.
• The Beltrami-Klein map of the SRG is convex, with convexity holding in all dimensions except possibly in dimension 2, where it remains convex due to the elliptical nature of the joint range.
• An analogue of Hildebrant’s theorem holds for the SRG, linking its spectral properties to the approximate point spectrum rather than the spectrum.
• For real Hilbert spaces, the SRG can be computed via complexification, and its Beltrami-Klein image is convex unless the space is two-dimensional.
• The boundary of the SRG for matrix and linear differential operators can be plotted using adapted numerical range algorithms, leveraging the connection to the transformed operator’s joint numerical range.
• The SRG of a matrix is convex if the dimension is greater than 2; in dimension 2, it is an ellipse, circle, line, or point—each with a convex boundary.