Wandering subspace property for homogeneous invariant subspaces

TL;DR

This paper investigates the structure of homogeneous invariant subspaces in graded Hilbert spaces with shift-like operators, showing they are finitely generated by their wandering subspaces and providing algorithms for minimal generator sets.

Contribution

It establishes the finite generation of homogeneous invariant subspaces and develops an algorithm to find minimal generators, extending the understanding of operator theory in graded Hilbert spaces.

Findings

Homogeneous invariant subspaces have finite index and are generated by their wandering subspaces.

The algebraic sum of intersections with grading components forms a finitely generated module.

An algorithm is provided to compute minimal generator sets for these modules.

Abstract

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