Wandering subspace property for homogeneous invariant subspaces

Abstract

For shift-like commuting tuples T∈B(H)n on graded Hilbert spaces H, we show that each homogeneous invariant subspace M of T has finite index and is generated by its wandering subspace. Under suitable conditions on the grading (Hk)k≥0 of H, the algebraic direct sum M˜=⊕k≥0M∩Hk becomes a finitely generated module over the polynomial ring C[z] in n complex variables. We show that the wandering subspace WT(M) of M is contained in M˜ and that each linear basis of WT(M) forms a minimal set of generators for M˜. We describe an algorithm that transforms each set of homogeneous generators of M˜ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals I⊂C[z]. We prove that each finitely generated γ-graded commuting row contraction T∈B(H)n admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.