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.
Caixing GU. "Common reducing subspaces of several weighted shifts with operator weights." J. Math. Soc. Japan 70 (3) 1185 - 1225, July, 2018. https://doi.org/10.2969/jmsj/74677467