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.

In press

References in corpus (1)

• Aluthge transforms of 2-variable weighted shifts

Cited by in corpus (8)

• Subnormal $n$th roots of quasinormal operators are quasinormal
• The Spectral Picture and Joint Spectral Radius of the Generalized Spherical Aluthge Transform
• On $n$th roots of bounded and unbounded quasinormal operators
• Two-moment characterization of spectral measures on the real line
• Polynomial embeddings of unilateral weighted shifts into $2$-variable weighted shifts
• Classes of operators related to subnormal operators
• Unbounded operators having self-adjoint or normal powers and some related results
• Semi-hyponormality of commuting pairs of Hilbert space operators