When is hyponormality for 2-variable weighted shifts invariant under powers? R. CurtoJasang Yoon 47B2047B3747A1328A50generalized Hilbert matrixjointly hyponormal pairssubnormal 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4303 10.1512/iumj.2011.60.4303 en Indiana Univ. Math. J. 60 (2011) 997 - 1032 state-of-the-art mathematics http://iumj.org/access/