On the duals of Szegö and Cauchy tuples
Authors:
Ameer Athavale and Pramod Patil
Journal:
Proc. Amer. Math. Soc. 139 (2011), 491498
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 wellknown 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 zerodimensional. 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/S000299392010104823
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.
