For shift-like commuting tuples on graded Hilbert spaces , we show that each homogeneous invariant subspace of has finite index and is generated by its wandering subspace. Under suitable conditions on the grading of , the algebraic direct sum becomes a finitely generated module over the polynomial ring in complex variables. We show that the wandering subspace of is contained in and that each linear basis of forms a minimal set of generators for . We describe an algorithm that transforms each set of homogeneous generators of into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals . We prove that each finitely generated -graded commuting row contraction admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.
"Wandering subspace property for homogeneous invariant subspaces." Banach J. Math. Anal. 13 (2) 486 - 505, April 2019. https://doi.org/10.1215/17358787-2018-0052