In this paper we solve the Sylvester equation \(AX-XB=C\), where A and B are closed densely defined self-adjoint operators, and C is a linear operator. We obtain sufficient conditions for the existence of infinitely many solutions and we manage to classify them. These results generalize the previously known results regarding singular Sylvester equations. Finally, we illustrate our results on an operator equation that appears in quantum mechanics, called the position-momentum operator equation.

中文翻译：

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具有无界自伴A和B的奇异Sylvester方程

在本文中，我们解决了Sylvester方程\（AX-XB = C \），其中A和B是闭合的密集定义的自伴算子，而C是线性算子。我们为存在众多解决方案获得了充分的条件，并设法对其进行分类。这些结果概括了关于奇异Sylvester方程的先前已知结果。最后，我们说明了在量子力学中出现的一个算子方程（称为位置动量算子方程）的结果。