2023/08/22 by Raphaël Clouâtre, Clouâtre, Raphaël, Brock Klippenstein +3 · 2 citations
Engineering · Computer Science · Mathematics · #Sparse and Compressive Sensing Techniques #Matrix Theory and Algorithms #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2308.11533
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.