On Two Natural Extensions of n-normality

Abstract

A bounded linear operator T on a complex Hilbert space ${\cal H}$ 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*}T^k =0, and T is said to be of type II k-quasi-n-normal if T*^k{T*ⁿTⁿ − TⁿT*ⁿ}T^k =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.