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We aim to find conditions on two Hilbert space operators $A$ and $B$ underwhich the expression $AX-XB$ having low rank forces the operator $X$ itself toadmit 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 relaxthis normality condition, using the idea of operator dilations. The basicproblem then becomes the lifting of Sylvester equations, which is reminiscentof the classical commutant lifting theorem and its variations. Our approachalso allows us to show that the (factored) alternating direction implicitmethod for solving Sylvester equaftions $AX-XB=C$ does not require too manyiterations, even without requiring $A$ to be normal.

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