In this article, we consider the singular value asymptotics of compositions of compact linear operators mapping in the real Hilbert space of quadratically integrable functions over the unit interval. Specifically, the composition is given by the compact simple integration operator followed by the non-compact Ces`aro operator possessing a non-closed range. We show that the degree of ill-posedness of that composition is two, which means that the Ces`aro operator increases the degree of illposedness by the amount of one compared to the simple integration operator.

翻译：本文考虑在单位区间上平方可积函数的实希尔伯特空间中，紧线性算子复合的奇异值渐近性质。具体而言，该复合由紧简单积分算子后接具有非闭值域的非紧Cesàro算子构成。我们证明该复合问题的不适定程度为二，这意味着与简单积分算子相比，Cesàro算子将不适定程度增加了一。