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 * T n = T n T *. 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) = σ 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 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)) = ∅, 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.

Spectral properties of square hyponormal operators

“…(ii) Next we assume that λ 0. Then λ 2 is an isolated point of σ(T 2 ) by Lemma 2.1 of [5]. We will prove…”

“…In [4][5][6], the authors have studied spectral properties of n-normal operator, that is, an operator T such that T n is normal, in the cases that σ(T) (−σ(T)) = ∅ or σ(T) ∩ (−σ(T)) ⊂ {0}. Since an operator T such that T 2 is hyponormal is algebraically hyponormal, T is isoloid and Weyl's Theorem holds.…”

On the class of $k$-quasi-$(n,m)$-power normal operators

“…Very recently, several papers have appeared on n-normal operators. We refer to [4,5,18] for complete study. An operator T ∈ B(H) is called (n, m)-power normal if T n T * m − T * m T n = 0 and it is said to be (n, m)-power quasi-normal if T n T * m −T * m T n T = 0 where n, m be two nonnegative integers.…”

“…An operator T ∈ B(H) is called algebraic if there is a nonconstant polynomial Q ∈ C[z] for which Q(T ) = 0. An operator T ∈ B(H) is said to be isoloid [4] if every isolated point of σ(T ) belongs to the point spectrum of T . An operator T ∈ B(H) is called polaroid [7] if π(T ) = {µ ∈ iso σ(T )}, where iso σ(T ) is the set of isolated points of the spectrum of T and π(T ) is the set of poles of the resolvent of T. An operator…”

Remarks on n-normal operators

“…If an operator T ∈ L(H) satisfies (2), then T is invertible automatically. Recently, the authors in [4] have studied spectral properties of an n-normal operator T satisfying the following condition (3).…”