Common reducing subspaces of several weighted shifts with operator weights

We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

我们刻画了具有算子权的几个加权移位的公共约化子空间。作为应用，我们研究了多变量全纯函数的希尔伯特空间上坐标函数幂的乘法算子的公共约化子空间。约化子空间的确定也导致了由这些乘法算子生成的冯·诺依曼代数的换位子的结构定理。这种一般方法统一适用于加权哈代空间、加权伯格曼空间、德鲁里 - 阿弗森空间以及单位球或多圆盘的狄氏空间。