2024/09/06 by Tirthankar Bhattacharyya, Bhattacharyya, Tirthankar, Mainak Bhowmik +3
Mathematics · #46M20. Secondary: 46M10 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary: 47B35 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2409.04582
openalex publication_date 2024/09/06 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28
Consider a bounded symmetric domain Ω with a finite pseudo-reflection group acting on it as a subgroup of the group of automorphisms. This gives rise to quotient domains by means of basic polynomials θ which by virtue of being proper maps map the \v Silov boundary of Ω to the \v Silov boundary of θ(Ω). Thus, the natural measure on the \v Silov boundary of Ω can be pushed forward. This gives rise to Hardy spaces on the quotient domain. The study of Hankel operators on the Hardy spaces of the quotient domains is introduced. The use of the weak product space shows that an analogue of Hartman's theorem holds for the small Hankel operator. Nehari's theorem fails for the big Hankel operator and this has the consequence that when the domain Ω is the polydisc \mathbb Dd, the \em Hardy space is not a projective object in the category of all Hilbert modules over the algebra \mathcal A (θ(\mathbb Dd)) of functions which are holomorphic in the quotient domain and continuous on the closure θ(\mathbb Dd). It is not a projective object in the category of cramped Hilbert modules either. Indeed, no projective object is known in these two categories. On the other hand, every normal Hilbert module over the algebra of continuous functions on the \v Silov boundary, treated as a Hilbert module over the algebra \mathcal A (θ(\mathbb Dd)), is projective.