2023/06/12 by Bera, Santu, Chavan, Sameer, Ghara, Soumitra
#31C25 #32A36 #46E20 (Secondary) #47A13 #47B38 (Primary) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2306.07022
We introduce and study Dirichlet-type spaces \mathcal D(μ1, μ2) of the unit bidisc \mathbb D2, where μ1, μ2 are finite positive Borel measures on the unit circle. We show that the coordinate functions z1 and z2 are multipliers for \mathcal D(μ1, μ2) and the complex polynomials are dense in \mathcal D(μ1, μ2). Further, we obtain the division property and solve Gleason's problem for \mathcal D(μ1, μ2) over a bidisc centered at the origin. In particular, we show that the commuting pair \mathscr Mz of the multiplication operators \mathscr Mz1, \mathscr Mz2 on \mathcal D(μ1, μ2) defines a cyclic toral 2-isometry and \mathscr M^*z belongs to the Cowen-Douglas class \bf B1(\mathbb D2r) for some r >0. Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter's representation theorem for cyclic analytic 2-isometries. In particular, we show that a cyclic analytic toral 2-isometric pair T with cyclic vector f0 is unitarily equivalent to \mathscr Mz on \mathcal D(μ1, μ2) if and only if ker T^*, spanned by f0, is a wandering subspace for T.