Hankel operators and Projective Hilbert modules on quotients of bounded symmetric domains

Abstract

Consider a bounded symmetric domain $\Omega$ 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 $\theta$ which by virtue of being proper maps map the \v Silov boundary of $\Omega$ to the \v Silov boundary of $\theta(\Omega)$. Thus, the natural measure on the \v Silov boundary of $\Omega$ 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 $\Omega$ is the polydisc $\mathbb D^d$, the {\em Hardy space} is not a projective object in the category of all Hilbert modules over the algebra $\mathcal A (\theta(\mathbb D^d))$ of functions which are holomorphic in the quotient domain and continuous on the closure $\overline {\theta(\mathbb D^d)}$. 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 $\mathcal A (\theta(\mathbb D^d))$, is projective.