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On Certain Multivariable Subnormal Weighted Shifts and their Duals

Published:2011-11-15
Printed: Sep 2013
• Ameer Athavale,
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
• Pramod Patil,
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
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Abstract

To every subnormal $m$-variable weighted shift $S$ (with bounded positive weights) corresponds a positive Reinhardt measure $\mu$ supported on a compact Reinhardt subset of $\mathbb C^m$. We show that, for $m \geq 2$, the dimensions of the $1$-st cohomology vector spaces associated with the Koszul complexes of $S$ and its dual ${\tilde S}$ are different if a certain radial function happens to be integrable with respect to $\mu$ (which is indeed the case with many classical examples). In particular, $S$ cannot in that case be similar to ${\tilde S}$. We next prove that, for $m \geq 2$, a Fredholm subnormal $m$-variable weighted shift $S$ cannot be similar to its dual.
 Keywords: subnormal, Reinhardt, Betti numbers
 MSC Classifications: 47B20 - Subnormal operators, hyponormal operators, etc.