论文标题
在poly-disc上，Cowen-Douglas班级的均质Hermitian Holomorphic Vector束和操作员
Homogeneous Hermitian Holomorphic Vector Bundles And Operators In The Cowen-Douglas Class Over The Poly-disc
论文作者
论文摘要
在本文中，我们获得了两组结果。第一组完整的结果专门用于双盘案例，而第二组结果部分描述了，其中哪一组将其延续到多盘的一般情况下： *对$ \ mbox {möb} \ times \ times \ mbox {möb} $，对不可约的Hermitian Helomorphic vector捆绑包的分类，相对于$ \ mbox {Möb} \ times \ times \ times \ times \ times \ times \ mbox {möb} $。在这些引起$ \ mathbb {d}^2 $等级$ 1,2 $或$ 3 $的Cowen-Douglas类中的操作员的那些。 * Any hermitian holomorphic vector bundle of rank $2$ over $\mathbb{D}^n$, homogeneous with respect to the $n$-fold product of the group $\mbox{Möb}$ is shown to be a tensor product of $n-1$ hermitian holomorphic line bundles, each of which is homogeneous with respect to $\mbox{Möb}$ and Hermitian Holomorphic Vector Bundle $ 2 $，相对于$ \ Mbox {Möb} $均匀。 *将不可约的均质Hermitian Holomororphic Vector Buldles分类超过$ \ Mathbb {d}^2 $等级$ 3 $（以及相应的Cowen-Cowen-Douglas Offerators of Operators of Operators）的分类范围扩展到$ \ \ \ \ \ Mathbb {d}^n $，$ n $ n> 2 $。 *显示出Cowen -Douglas类$ \ Mathrm b_2（\ Mathbb {d}^n）$中的运算符中没有不可约合的$ N $ - 与$ \ mbox {autbb {autbb {autbb {autbb {\ mathbb {d}^n）$，$ n> 1 $ 1 $均均均均具有同质性。另外，在$ \ mathrm b_3（\ mathbb {d}^2）中的一对操作员相对于$ \ mbox {aut}（\ mathbb {d}^2）$均质，而不是$ \ no $ n $ - $ \ mathrm b_3（d hom bbb^bbb^n of $ n $ tuple） $ \ mbox {aut}（\ mathbb {d}^n）$，$ n> 2 $。
In this article, we obtain two sets of results. The first set of complete results are exclusively for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc: * A classification of irreducible hermitian holomorphic vector bundles over $\mathbb{D}^2$, homogeneous with respect to $\mbox{Möb}\times \mbox{Möb}$, is obtained assuming that the associated representations are \textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\mathbb{D}^2$ of rank $1,2$ or $3$ is determined. * Any hermitian holomorphic vector bundle of rank $2$ over $\mathbb{D}^n$, homogeneous with respect to the $n$-fold product of the group $\mbox{Möb}$ is shown to be a tensor product of $n-1$ hermitian holomorphic line bundles, each of which is homogeneous with respect to $\mbox{Möb}$ and a hermitian holomorphic vector bundle of rank $2$, homogeneous with respect to $\mbox{Möb}$. * The classification of irreducible homogeneous hermitian holomorphic vector buldles over $\mathbb{D}^2$ of rank $3$ (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of $\mathbb{D}^n$, $n>2$. * It is shown that there is no irreducible $n$ - tuple of operators in the Cowen-Douglas class $\mathrm B_2(\mathbb{D}^n)$ that is homogeneous with respect $\mbox{Aut}(\mathbb{D}^n)$, $n >1$. Also, pairs of operators in $\mathrm B_3(\mathbb{D}^2)$ homogeneous with respect to $\mbox{Aut}(\mathbb{D}^2)$ are produced, while it is shown that no $n$ - tuple of operators in $\mathrm B_3(\mathbb{D}^n)$ is homogeneous with respect to $\mbox{Aut}(\mathbb{D}^n)$, $n > 2$.
