2022/04/28 by Ghosh, Gargi
#30H10 #47B35 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2205.00883
Let Ω be either the unit polydisc \mathbb Dd or the unit ball \mathbb Bd in \mathbb Cd and G be a finite pseudoreflection group which acts on Ω. Associated to each one-dimensional representation \varrho of G, we provide a notion of the (weighted) Hardy space H2_\varrho(Ω/G) on Ω/G. Subsequently, we show that each H2_\varrho(Ω/G) is isometrically isomorphic to the relative invariant subspace of H2(Ω) associated to the representation \varrho. For Ω=\mathbb Dd, G=\mathfrakSd, the permutation group on d symbols and \varrho = the sign representation of \mathfrakSd, the Hardy space H2_\varrho(Ω/G) coincides to well-known notion of the Hardy space on the symmetrized polydisc. We largely use invariant theory of the group G to establish identities involving Toeplitz operators on H2(Ω) and H2_\varrho(Ω/G) which enable us to study algebraic properties (such as generalized zero product problem, characterization of commuting Toeplitz operators, compactness etc.) of Toeplitz operators on H2_\varrho(Ω/G).