2024/08/07 by Das, Jyotirmay, Hazra, Somnath
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2408.03711
Let Möb be the biholomorphic automorphism group of the unit disc of the complex plane, H be a complex separable Hilbert space and U(H) be the group of all unitary operators. Suppose H is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc \mathbb Dn and contains all the polynomials. If π: Möb → U(H) is a multiplier representation, then we prove that there exist λ1, λ2, …, λn > 0 such that π is unitarily equivalent to (⊗i=1n Dλi+)|Möb, where each Dλi+ is a holomorphic discrete series representation of Möb. As an application, we prove that if (T1, T2) is a Möb - homogeneous pair in the Cowen - Douglas class of rank 1 over the bi-disc, then each Ti posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each Ti, the diagonal operators are identified. We also prove that if H consists of symmetric (resp. anti-symmetric) holomorphic functions over \mathbb D2 and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists λ> 0 such that π≅ ⊕m = 0^∞ D+λ+ 4m (resp. π≅ ⊕m=0^∞ D+λ+ 4m + 2).