A $Z^k$-invariant subspace without the wandering property

TL;DR

This paper demonstrates that in certain Dirichlet-type spaces, there exist invariant subspaces under multiplication by z^k that do not have the wandering property, revealing new insights into the structure of these spaces.

Contribution

It establishes the existence of specific k and alpha where z^k-invariant subspaces lack the wandering property in Dirichlet-type spaces, a novel finding in operator theory.

Findings

Existence of k and alpha with non-wandering invariant subspaces

Any Dirichlet-type space can be normed to lose the wandering property

The wandering property fails for multiplication by z^k for k ≥ 6

Abstract

We study operators of multiplication by $z^k$ in Dirichlet-type spaces $D_\alpha$. We establish the existence of $k$ and $\alpha$ for which some $z^k$-invariant subspaces of $D_\alpha$ do not satisfy the wandering property. As a consequence of the proof, any Dirichlet-type space accepts an equivalent norm under which the wandering property fails for some space for the operator of multiplication by $z^k$, for any $k \geq 6$.