Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert \(C^*\)-module. An example is constructed to show that in the Hilbert \(C^*\)-module case, \(\mathfrak {X}\) and its subset \(\mathfrak {X}_\bot \) can be different, where \(\mathfrak {X}\) denotes the set of all products of two projections on a Hilbert \(C^*\)-module, and \(\mathfrak {X}_\bot \) consists of those elements in \(\mathfrak {X}\) that have the polar decomposition. Some new phenomena are revealed when an operator is taken in \(\mathfrak {X}\) instead of \(\mathfrak {X}_\bot \). Some refinements are made on norms of differences of two projections.

将希尔伯特空间上两个投影的乘积的某些特征推广到希尔伯特\（C ^ * \）-模上有限个投影的乘积的情况。构造了一个示例，以说明在希尔伯特\（C ^ * \）模块的情况下，\（\ mathfrak {X} \）及其子集\（\ mathfrak {X} _ \ bot \）可以不同，其中\（\ mathfrak {X} \）表示集合的上一个希尔伯特两个突起的所有产品\（C ^ * \） -模，和\（\ mathfrak {X} _ \机器人\）由在那些元件的\ （\ mathfrak {X} \）具有极性分解。当运算符采用\（\ mathfrak {X} \）时，会发现一些新现象。而不是\（\ mathfrak {X} _ \ bot \）。对两个预测的差异范数进行了一些改进。