In this paper we lay the foundations for a new theory encompassing two natural extensions of the class of subnormal operators, namely the n–subnormal operators and the sub-n–normal operators, where \(n \in \mathbb {N}\) . We discuss inclusion relations among the above-mentioned classes and other related classes, e.g., n–quasinormal and quasi-n–normal operators. We show that sub-n–normality is stronger than n–subnormality, and produce a concrete example of a 3–subnormal operator which is not sub-2–normal. In (Curto et al., J. Funct. Anal. 278, 108342 2020), R.E. Curto, S.H. Lee and J. Yoon proved that if an operator T is subnormal, left-invertible, and such that \(T^2\) is quasinormal, then T is quasinormal. In subsequent work, (Putnam, C.R., Proc. Amer. Math. Soc. 8, 768–769 1957), P. Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility. In this paper we consider suitable analogs of this result for the case of operators in the above-mentioned classes. In particular, we prove that the weight sequence of an n–quasinormal unilateral weighted shift must be periodic with period at most n.
