Singular value decay for solutions of Sylvester equations

Abstract

A Sylvester equation is an operator equation of the form $AX-XB=C$. A fact that has been proven multiple times before, see \cite{penzl2000eigenvalue,sabino2006solution}, is that if $C$ has low rank, then $A$ and $B$ satisfying certain conditions imply $X$ has a low rank approximation. Another set of conditions was given by Beckermann and Townsend in 2019, \cite{beckermann2019bounds}, where they impose the conditions that $A$ and $B$ are both normal, and have disjoint and well-separated spectra. In this thesis, we explore cases where the normality condition can be relaxed. Our main tool is unitary operator dilations, whereby one can realize a given operator as the corner of a unitary operator acting on a bigger space. The basic problem becomes the lifting of the original Sylvester equation to a new one involving the unitary dilation. This is reminiscent of the so-called intertwining dilation theorem, but it requires a completely new analysis as we require additional conditions if we wish to be able to guarantee the solution has a low rank approximation.

Our main result states that if we trade in normality of $A$ for a norm condition on $A$ and $B$, then we can unitarily dilate $A$. This in turn allows us to conclude that $X$ has a low rank approximation provided our condition is satisfied, $B$ is normal, and $C$ has low rank. Due to the similarity in the conditions for the theorem by Beckermann and Townsend and the conditions required to solve a Sylvester equation quickly using algorithms such as the alternating direction implicit (ADI) method, our dilation method also allows us to show the ADI method does not require too many iterations without requiring $A$ to be normal.