有界对称域商空间上的 Hankel 算子与投射 Hilbert 模
Hankel operators and Projective Hilbert modules on quotients of bounded symmetric domains
AI总结：
本文研究有限伪反射群作用下有界对称域商空间的 Hardy 空间，证明小 Hankel 算子满足 Hartman 型定理，而大 Hankel 算子不满足 Nehari 定理，并据此说明多圆盘商域上的 Hardy 空间非投射 Hilbert 模。
摘要：
考虑一个有界对称域 $Ω$，其上有一个有限伪反射群作为自同构群的子群作用。这通过基本多项式 $θ$ 产生商域；由于这些基本多项式是真映射，它们将 $Ω$ 的 Šilov 边界映到 $θ(Ω)$ 的 Šilov 边界。因此，$Ω$ 的 Šilov 边界上的自然测度可以被前推。这就在商域上产生了 Hardy 空间。文中引入了对商域 Hardy 空间上 Hankel 算子的研究。弱乘积空间的使用表明，Hartman 定理的一个类似结论对小 Hankel 算子成立。Nehari 定理对大 Hankel 算子不成立，其后果是：当域 $Ω$ 为多圆盘 $\mathbb D^d$ 时，在代数 $\mathcal A(θ(\mathbb D^d))$ 上的所有 Hilbert 模构成的范畴中，Hardy 空间不是投射对象；该代数由在商域上全纯且在闭包 $\overline{θ(\mathbb D^d)}$ 上连续的函数组成。它在狭窄 Hilbert 模范畴中也不是投射对象。事实上，在这两个范畴中尚不存在已知的投射对象。另一方面，Šilov 边界上连续函数代数上的每个正规 Hilbert 模，在被视为代数 $\mathcal A(θ(\mathbb D^d))$ 上的 Hilbert 模时，都是投射的。
英文摘要：
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 $\mathbb D^d$, the {\em Hardy space} is not a projective object in the category of all Hilbert modules over the algebra $\mathcal A (θ(\mathbb D^d))$ of functions which are holomorphic in the quotient domain and continuous on the closure $\overline {θ(\mathbb D^d)}$. 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 $\mathcal A (θ(\mathbb D^d))$, is projective.
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