On Some Normality-Like Properties and Bishop's Property (β) for a Class of Operators on Hilbert Spaces

We prove some further properties of the operator T∈[nQN] (n-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator T∈[nQN] satisfying the translation invariant property is normal and that the operator T∈[nQN] is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator T∈[2QN] is subscalar of order m; that is, it is similar to the restriction of a scalar operator of order m to an invariant subspace.