Remarks on n-normal operators

Abstract

Let T be a bounded linear operator on a complex Hilbert space and n, m ∈ N. Then T is said to be n-normal if T * T n = T n T * and (n, m)-normal if T * m T n = T n T * 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 σ(T) (−σ(T)) ⊂ {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 σ(T) which is not constant on each of the components of its domain.

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

“…Proof. If the range T k (H) is dense, then T is an (n, m)-power normal operator and the theorem holds by Theorem 3.6 in [5]. Assume that T k (H) ̸ = H. Let…”

“…Following [5,Corollary 4.3], it was observed that if T ∈ B(H) is (n, m)-power normal, then T is isoloid and polaroid. Corollary 2.30.…”

“…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 (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. We refer the reader to [1], [2] and [5] for complete details on these families of operators.…”

Spectral properties of square hyponormal operators

“…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.…”