A Note on ∗-Paranormal Operators and Related Classes of Operators

Abstract

Abstract. We shall show that the Riesz idempotent E λ of every * -paranormal operator T on a complex Hilbert space H with respect to each isolated point λ of its spectrum σ(T ) is self-adjoint and satisfies E λ H = ker(T − λ) = ker(T − λ) * . Moreover, Weyl's theorem holds for * -paranormal operators and more general for operators T satisfying the norm condition T x n ≤ T n x x n−1 for all x ∈ H. Finally, for this more general class of operators we find a sufficient condition such that E λ H = ker(T − λ) = ker(T − λ) * holds.

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

“…Since T 1 is * -paranormal we conclude that T 1 = 0. Also, since T k 3 = 0, a computation shows that Proof Since T is invertible * -paranormal, the assertion holds by [15]. …”

“…Proof Suppose that T is a k-quasi- * -paranormal operator. (i) If the range R(T k ) is dense, then T is * -paranormal operator and so the result follows by [15,Theorem 3]. Therefore, we may assume that R(T k ) is not dense.…”

“…Proof If R(T ) is dense, then T is * -paranormal and μ is a simple pole of the resolvent of T by [15]. So we may assume that R(T ) is not dense.…”

Finite Operators and Weyl Type Theorems for Quasi-*-n-Paranormal Operators

“…In [14] the authors obtained that Weyl's theorem holds for * -paranormal operators. In [11] the authors obtained that Weyl's theorem holds for quasi- * -paranormal operators.…”

ON n-*-PARANORMAL OPERATORS

“…For n = 2, n-paranormal and n- * -paranormal operators are simply called paranormal and * -paranormal operators, respectively. The inclusion relations between the above mentioned classes of operators are shown below (see [7,13,14,29]). hyponormal⊂ * -class A ⊂ * -paranormal⊂ n- * -paranormal⊂ n + 1-paranormal and hyponormal ⊂ paranormal ⊂ n + 1-paranormal.…”