Lifting Sylvester equations: singular value decay for non-normal coefficients

TL;DR

This paper investigates conditions under which low-rank Sylvester equations imply low-rank solutions, extending known results from normal operators to non-normal cases using operator dilations, and analyzing iterative solution methods.

Contribution

It introduces a dilation-based approach to relax normality assumptions in Sylvester equations, enabling low-rank approximations for broader classes of operators.

Findings

Low-rank solutions are achievable for non-normal operators under certain conditions.

Operator dilations can be used to extend Sylvester equation results beyond normal operators.

The ADOI method converges efficiently without requiring normality of A.

Abstract

We aim to find conditions on two Hilbert space operators $A$ and $B$ under which the expression $AX-XB$ having low rank forces the operator $X$ itself to admit a good low rank approximation. It is known that this can be achieved when $A$ and $B$ are normal and have well-separated spectra. In this paper, we relax this normality condition, using the idea of operator dilations. The basic problem then becomes the lifting of Sylvester equations, which is reminiscent of the classical commutant lifting theorem and its variations. Our approach also allows us to show that the (factored) alternating direction implicit method for solving Sylvester equaftions $AX-XB=C$ does not require too many iterations, even without requiring $A$ to be normal.