Wandering subspace property for homogeneous invariant subspaces

Abstract

For shift-like commuting tuples $T\in B(H{)}^n$ 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 $(H_k{)}_{k\ge 0}$ of $H$, the algebraic direct sum $\tilde{M}={\oplus }_{k\ge 0}M\cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb{C}\left[z\right]$ in $n$ complex variables. We show that the wandering subspace $W_T\left(M\right)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T\left(M\right)$ forms a minimal set of generators for $\tilde{M}$. We describe an algorithm that transforms each set of homogeneous generators of $\tilde{M}$ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals $I\subset \mathbb{C}\left[z\right]$. We prove that each finitely generated $\gamma$-graded commuting row contraction $T\in B(H{)}^n$ admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.