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On Certain Multivariable Subnormal Weighted Shifts and their Duals

Published online by Cambridge University Press:  20 November 2018

Ameer Athavale  and
Pramod Patil
Ameer Athavale
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: athavale@math.iitb.ac.in; pramodp@math.iitb.ac.in
Pramod Patil
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India e-mail: athavale@math.iitb.ac.in; pramodp@math.iitb.ac.in
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Abstract

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For every subnormal $m$-variable weighted shift $S$ (with bounded positive weights), there is a corresponding positive Reinhardt measure $\mu $ supported on a compact Reinhardt subset of ${{\mathbb{C}}^{m}}$. We show that, for $m\,\ge \,2$, the dimensions of the 1-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual $\widetilde{S}$ are different if a certain radial function happens to be integrable with respect to μ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to $\widetilde{S}$. We next prove that, for $m\,\ge \,2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual.

Keywords

47B20subnormalReinhardtBetti numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 3 , 01 September 2013 , pp. 459 - 465
Copyright
Copyright © Canadian Mathematical Society 2013

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