article

Title: When is hyponormality for 2-variable weighted shifts invariant under powers?

Authors: Raul E. Curto and Jasang Yoon

Issue: Volume 60 (2011), Issue 3, 997-1032

Abstract:

$For $2$-variable weighted shifts $W_{(\alpha,\beta)} \equiv (T_1,T_2)$ we study the invariance of (joint) $k$-hyponormality under the action $(h,\ell) \mapsto W_{(\alpha,\beta)}^{(h,\ell)} := (T_1^h,T_2^{\ell})$ ($h,\ell \geq 1$). We show that for every $k \geq 1$ there exists $W_{(\alpha,\beta)}$ such that $W_{(\alpha,\beta)}^{(h,\ell)}$ is $k$-hyponormal (all $h \geq 2$, $\ell \geq 1$) but $W_{(\alpha,\beta)}$ is not $k$-hyponormal. On the positive side, for a class of $2$-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invariant under \emph{all} powers; moreover, for this class $2$-hyponormality automatically implies subnormality. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness.$