Speaker
Description
The talk is based on Dirichlet-type spaces on the unit bidisc. We shall begin with a notion of Dirichlet-type spaces $\mathcal{D}(\mu_1, \mu_2)$ on the unit bidisc $\mathbb{D}^2$ with harmonic weights corresponding to finite positive Borel measures $\mu_1$ and $\mu_2$ supported on the unit circle. Then we show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal{D}(\mu_1, \mu_2)$ and that the complex polynomials are dense in $\mathcal{D}(\mu_1, \mu_2)$. Furthermore, we solve Gleason's problem for $\mathcal{D}(\mu_1, \mu_2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr{M}_z$ of the multiplication operators $\mathscr{M}_{z_1}$ and $\mathscr{M}_{z_2}$ on $\mathcal{D}(\mu_1, \mu_2)$ defines a cyclic toral $2$-isometry. Moreover, show that a cyclic analytic toral $2$-isometric pair $T$ with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu_1, \mu_2)$ for some $\mu_1,\mu_2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$
Furthermore, we shall discuss a more general class of Dirichlet-type space ${\bf D}(\mu_1, \mu_2)$ on $\mathbb D^2$ for finite positive Borel measures $\mu_1, \mu_2$ on $\overline{\mathbb{D}}.$ Similar to the previous case, the multiplication operator $\mathscr{M}_z = (\mathscr{M}_{z_1}, \mathscr{M}_{z_2})$ is bounded on ${\bf D}(\mu_1, \mu_2),$ and the set of polynomials is dense in ${\bf D}(\mu_1, \mu_2).$ We show that the commuting pair $\mathscr{M}_z$ is a cyclic analytic completely hyperexpansive $2$-tuple on ${\bf D}(\mu_1, \mu_2).$
In particular, we establish a representation theorem for cyclic analytic completely hyperexpansive operator $2$-tuple $T = (T_1, T_2)$ satisfying
$ I - T^*_1 T_1 - T^*_2 T_2 + T^*_1 T^*_2 T_1 T_2 = 0.$