Subnormal n-th roots of matricially and spherically quasinormal pairs

Abstract

In a recent paper, Curto et al.\ [J. Funct. Anal.  278(3) (2020) 108342] asked the following question: \textit{``Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that T is quasinormal?"}. Pietrzycki and Stochel have answered this question in the affirmative \cite{Pietr} and proved an even stronger result. Namely, the authors have showed that the subnormal $n$-th roots of a quasinormal operator must be quasinormal. In the present paper, using an elementary technique, we present a much simpler proof of this result and generalize some other results from \cite{Curto}. We also show that we can relax a condition in the definition of matricially quasinormal $n$-tuples and we give a correction for one of the results from \cite{Curto}. Finally, we give sufficient conditions for the equivalence of matricial and spherical quasinormality of $\mathbf{T}^{(n,n)}:=(T_1^n,T_2^n)$ and matricial and spherical quasinormality of $\mathbf{T}=(T_1,T_2)$, respectively.