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.

翻译：我们旨在寻找希尔伯特空间算子$A$和$B$满足的条件，使得表达式$AX-XB$具有低秩时，算子$X$本身也能获得良好的低秩逼近。已知当$A$和$B$为正规算子且其谱充分分离时，这一结果成立。在本文中，我们利用算子扩张的思想放宽了这一正规性条件。基本问题因此转化为Sylvester方程的提升，这使人联想到经典的交换子提升定理及其变体。我们的方法还表明，即使不要求$A$正规，求解Sylvester方程$AX-XB=C$的（因式分解）交替方向隐式方法也无需过多迭代。