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

TL;DR

This paper explores Hankel operators on Hardy spaces of quotient domains from bounded symmetric domains, revealing analogues of classical theorems and the non-projectivity of Hardy spaces in certain categories.

Contribution

It introduces the study of Hankel operators on quotient domains, demonstrating an analogue of Hartman's theorem and analyzing projectivity properties of Hardy spaces.

Findings

Hartman's theorem analogue holds for small Hankel operators.

Nehari's theorem fails for big Hankel operators on these domains.

Hardy space is not a projective object in relevant Hilbert module categories.

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.