Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators

Abstract

We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\mathcal D(μ_1,μ_2)$ on the unit bidisc $\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $μ_1,μ_2$ on $\overline{\mathbb D}.$ The multiplication operators $\mathscr M_{z_1}$ and $\mathscr M_{z_2}$ by the coordinate functions $z_1$ and $z_2,$ respectively, are bounded on $\mathcal D(μ_1,μ_2)$ and the set of polynomials is dense in $\mathcal D(μ_1,μ_2).$ We show that the commuting pair $\mathscr M_{z}=(\mathscr M_{z_1},\mathscr M_{z_2})$ is a cyclic analytic toral completely hyperexpansive $2$-tuple on $\mathcal D(μ_1,μ_2).$ Unlike the one variable case, not all cyclic analytic toral completely hyperexpansive pairs arise as multiplication $2$-tuple $\mathscr M_z$ on these spaces. In particular, we establish that a cyclic analytic toral completely hyperexpansive operator $2$-tuple $T=(T_1,T_2)$ satisfying $I-T^*_1 T_1-T^*_2T_2+T^*_1T^*_2T_1T_2=0$ and having a cyclic vector $f_0$ is unitarily equivalent to $\mathscr{M}_z$ on $\mathcal{D}(μ_1, μ_2)$ for some finite positive Borel measures $μ_1$ and $μ_2$ on $\overline{\mathbb{D}}$ if and only if $\ker T^*$, spanned by $f_0$, is a wandering subspace for $T$.