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Essential normality and the decomposability of homogeneous submodules

Author: Matthew Kennedy
Journal: Trans. Amer. Math. Soc. 367 (2015), 293-311
MSC (2010): Primary 47A13, 47A20, 47A99, 14Q99, 12Y05
Published electronically: July 17, 2014
MathSciNet review: 3271262
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Abstract: We establish the essential normality of a large new class of homogeneous submodules of the finite rank $ d$-shift Hilbert module. The main idea is a notion of essential decomposability that determines when a submodule can be decomposed into the algebraic sum of essentially normal submodules. We prove that every essentially decomposable submodule is essentially normal, and introduce methods for establishing that a submodule is essentially decomposable. It turns out that many submodules have this property. We prove that many of the submodules considered by other authors are essentially decomposable, and in addition establish the essential decomposability of a large new class of homogeneous submodules. Our results support Arveson's conjecture that every homogeneous submodule of the finite rank $ d$-shift Hilbert module is essentially normal.

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Additional Information

Matthew Kennedy
Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6

Received by editor(s): May 25, 2012
Received by editor(s) in revised form: December 17, 2012
Published electronically: July 17, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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