2024/04/15 by Santu Bera, Sameer Chavan, Soumitra Ghara · 4 citations
Mathematics · #Holomorphic and Operator Theory #Advanced Topics in Algebra #Advanced Algebra and Geometry
paper · pdf · doi:10.4153/s0008414x24000300
Abstract We introduce and study Dirichlet-type spaces \mathcal D(μ 1, μ 2) of the unit bidisc \mathbb D2, where μ 1, μ 2 are finite positive Borel measures on the unit circle. We show that the coordinate functions z1 and z2 are multipliers for \mathcal D(μ 1, μ 2) and the complex polynomials are dense in \mathcal D(μ 1, μ 2). Further, we obtain the division property and solve Gleason’s problem for \mathcal D(μ 1, μ 2) 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(μ 1, μ 2) defines a cyclic toral 2 -isometry and \mathscr M^*z belongs to the Cowen–Douglas class \mathbf 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(μ 1, μ 2) for some μ 1,μ 2 if and only if ker T^*, spanned by f0, is a wandering subspace for T.