Subnormal n-th roots of matricially and spherically quasinormal pairs

In a recent paper, Curto et al. [4] asked the following question: "Let T be a subnormal operator, and assume that T2 is quasinormal. Does it follow that T is quasinormal?". Pietrzycki and Stochel have answered this question in the affirmative [18] 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 [4]. 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 [4]. Finally, we give sufficient conditions for the equivalence of matricial and spherical quasinormality of T(n,n) := (T1n, T2n) and matricial and spherical quasinormality of T = (T1, T2), respectively.