Remarks on n-normal Operators

Let T be a bounded linear operator on a complex Hilbert space and n, m is an element of N. Then T is said to be n-normal if T*T-n= (TT)-T-n* and (n, m)-normal if T*T-m(n) = (TT)-T-n *(m). In this paper, we study several properties of n-normal, (n, m)-normal operators. In particular, we prove that if T is 2-normal with sigma(T) boolean AND (-sigma(T)) subset of {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n, m)-normal, then T is decomposable and Weyl's theorem holds for f(T), where f is an analytic function on sigma(T) which is not constant on each of the components of its domain.