On k-quasi- $$*$$ ∗ -paranormal operators

Abstract

For a positive integer k, an operator T ∈ B(H ) is called k-quasi- * -paranormal if T * T k x 2 ≤ T k x T k+2 x for all x ∈ H , which is a common generalization of * -paranormal and quasi- * -paranormal. In this paper, firstly we prove some inequalities of this class of operators; secondly we give a necessary and sufficient condition for T to be kquasi- * -paranormal. Using these results, we prove that: (1) if T * T n = T n T for some positive integer n ≥ k, then a k-quasi- * -paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point μ 0 of the spectrum of a k-quasi- * -paranormal operator T , then (i) if μ 0 = 0, then R(E) = ker(T − μ 0 ) (ii) if μ 0 = 0, then R(E) = ker(T k ).

Spectrum of k-quasi-class A_n operators

“…In [16], Stampfli showed that if T satisfies the growth condition G 1 , then E is self-adjoint and R(E) = ker(T − µ). Recently, Jeon and Kim [11], Uchiyama [18] and Rashid [15] obtained Stampfli's result for quasi-class A operators, paranormal operators and k-quasi- * -paranormal operators. In general even if T is a paranormal operator, the Riesz idempotent E of T with respect to µ is not necessarily self-adjoint.…”

Some results on totally n-star-paranormal operators

“…This class of operators introduced and studied by K. Tanahashi and A. Uchiyama. The references [12,13,14] are among the various extensions of these classes of operators.…”