Journal of Operator Theory

Volume 58, Issue 1, Summer 2007 pp. 175-203.

Propagation phenomena for hyponormal 2-variable weighted shifts

Authors: Ra\'{u}l E. Curto (1) and Jasang Yoon (2)
Author institution: (1) Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA
(2) Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA

Summary: We study the class of hyponormal $2$-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for $1$-variable weighted shifts. In $1$-variable, it is well known that flat weighted shifts are necessarily subnormal (with finitely atomic Berger measures). By contrast, we exhibit a large collection of flat (i.e., horizontally and vertically flat) $2$-variable weighted shifts which are hyponormal but not subnormal. Moreover, we completely characterize the hyponormality and subnormality of symmetrically flat contractive $2$-variable weighted shifts.

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