On Certain Multivariable Subnormal Weighted Shifts and their Duals

Abstract

AbstractFor every subnormalm-variable weighted shift S (with bounded positive weights), there is a corresponding positive Reinhardt measure μ supported on a compact Reinhardt subset ofℂm. We show that, form≥ 2, the dimensions of the 1-st cohomology vector spaces associated with the Koszul complexes of S and its dual 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. We next prove that, form≥ 2, a Fredholm subnormalm-variable weighted shift S cannot be similar to its dual.