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 in (resp. open unit polydisk in ) 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 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 , the '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 , neither the Szegö -tuple nor the Cauchy -tuple is quasisimilar to its dual; this is in contrast with the case 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

Email:
athavale@math.iitb.ac.in

**Pramod Patil**

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

Email:
pramodp@math.iitb.ac.in

DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10482-3

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.