Projection operators are important in Analysis, Optimization and Algorithm. It is well known that these operators are firmly nonexpansive. In this paper, we provide an exact result that sharpens this well-known result. We develop the theory of averaged operators and provide a lower bound. We give a result on the avergedness of operator compositions. We also provide some nonlinear examples to illustrate our results.

翻译：投影算子在分析学、优化和算法中具有重要意义。众所周知，这些算子具有严格非扩张性。本文给出了一个精确结果，强化了这一已知结论。我们发展了平均算子理论，并给出了一个下界。我们还得到了关于算子复合平均性的结果，并通过一些非线性例子对结论进行了说明。