Spectral properties of n-normal operators

Abstract

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.