Quasinormality of powers of commuting pairs of bounded operators

Abstract: We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal $2$-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator $T$ whose square $T^{2}$ is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal $n$-tuples. Fourth, we show that if a $2$-variable weighted shift $W_{\left(\alpha ,\beta \right) }$ and its powers $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ are all spherically quasinormal, then $W_{\left( \alpha ,\beta \right)}$ may not necessarily be jointly quasinormal. Moreover, it is possible for both $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ to be spherically quasinormal without $W_{\left(\alpha ,\beta \right)}$ being spherically quasinormal. Finally, we prove that, for $2$-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal.