doiSerbia

Spectral properties of n-normal operators

Chō Muneo
Načevska Biljana

For a bounded linear operator T on a complex Hilbert space and n ϵ 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 σ(T) ∩ (-σ(T))  {0}, then T is isoloid and σ(T) = σa(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 ϵ C is an isolated point of σ(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 σ(T) ∩(-σ(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.

Keywords: Hilbert space, linear operator, normal operator, spectrum, Weyl’s Theorem