On Two Natural Extensions of n-normality.

A bounded linear operator T on a complex Hilbert space ℋ is called n-normal if T*Tn = TnT*. By Fuglede's theorem T is n-normal if and only if Tn is normal. Let k, n ∈ ℕ. Then a bounded linear operator T is said to be of type I k-quasi-n-normal if T*k{T*Tn − TnT*}Tk =0, and T is said to be of type II k-quasi-n-normal if T*k{T*nTn − TnT*n}Tk =0. 1-quasi-n-normal is called quasi-n-normal. We shall show that (1) type I quasi-2-normal and type II quasi-2-normal are different classes; (2) the intersection of the class of type I quasi-2-normal and the class of type II quasi-2-normal is equal to the class of 2-normal. We also give some examples of type I k-quasi-n-normal and type II k-quasi-n-normal. We also show that Weyl's theorem holds for this class of operators and every k-quasi-n-normal operator has a non trivial invariant subspace. [ABSTRACT FROM AUTHOR]