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On the duals of Szegö and Cauchy tuples

Authors: Ameer Athavale and Pramod Patil
Journal: Proc. Amer. Math. Soc. 139 (2011), 491-498
MSC (2010): Primary 47B20; Secondary 33C55
Published electronically: July 12, 2010
MathSciNet review: 2736332
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Abstract: The tuple of multiplications by coordinate functions on the Hardy space of the open unit ball $ \mathbb{B}^{2m}$ in $ \mathbb{C}^m$ (resp. open unit polydisk $ \mathbb{D}^m$ in $ \mathbb{C}^m$) is referred to as the Szegö tuple (resp. Cauchy tuple) and is a well-known example of a subnormal operator tuple. Naturally associated with the Szegö tuple (resp. Cauchy tuple) is its dual whose coordinates act on the orthocomplement of the Hardy space of the ball (resp. polydisk) in an appropriate $ L^2$ space. We examine the Koszul complexes associated with the duals of the Szegö and Cauchy tuples and determine their Betti numbers. We explicitly verify that, for $ m \geq 2$, the $ m$'th cohomology vector space associated with the Koszul complex of either the dual of the Szegö tuple or the dual of the Cauchy tuple is zero-dimensional. It follows in particular that, for $ m \geq 2$, neither the Szegö $ m$-tuple nor the Cauchy $ m$-tuple is quasisimilar to its dual; this is in contrast with the case $ m=1$ where both the Szegö tuple and the Cauchy tuple reduce to the Unilateral Shift, which is known to be unitarily equivalent to its dual.

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Additional Information

Ameer Athavale
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

Pramod Patil
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

Keywords: Subnormal, quasisimilar, Koszul complex, spherical harmonics
Received by editor(s): July 15, 2009
Received by editor(s) in revised form: March 3, 2010
Published electronically: July 12, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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