A local theory for operator tuples in the Cowen-Douglas class

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• 作者：Li Chen^a ; ^{lichen1019@gmail.com} ; Ronald G. Douglas^b ; ^{rdouglas@math.tamu.edu}
• 关键词：46E22 ; 47A65 ; 32Axx ; 55Rxx
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：754-779
• 全文大小：500 K
• 卷排序：307

文摘

We present a local theory for a commuting m  -tuple S=(S1,S2,⋯,Sm)S=(S1,S2,⋯,Sm) of Hilbert space operators lying in the Cowen–Douglas class. By representing S on a Hilbert module MM consisting of vector-valued holomorphic functions over CmCm, we identify and study the localization of S on an analytic hyper-surface in CmCm. We completely determine unitary equivalence of the localization and relate it to geometric invariants of the Hermitian holomorphic vector bundle associated to S. It turns out that the localization coincides with an important class of quotient Hilbert modules, and our result concludes its classification problem in full generality.