For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-$*$-paranormal operators satisfying 

$$\begin{aligned} ||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \ge ||T^*(T^{k}x)|| \text { for all } x \in H. \end{aligned}$$

This class includes the class of $n$-$*$-paranormal operators and the class of $(1,k)$-quasi-$*$-paranormal operators contains the class of $k$-quasi-$*$-class $A$ operators. We study the basic properties of $(n,k)$-quasi-$*$-paranormal operators, like relations of this new class of operators with other classes known in the literature, their matrix representation, and properties of their spectra.

On (n,k)-quasi-*-paranormal operators

A bounded linear operator T on a complex Hilbert space H is said to be k∗- paranormal ifT ∗ xkT k xfor every unit vector x ∈ H where k is a natural number with 2k. This class of operators is an extension of hyponormal operators and have many interesting properties. We show that k∗-paranormal operators have Bishop's property (β ), i.e., if fn(λ) is an analytic function on some open set D ⊂ C such that (T −z)fn(z) → 0 uniformly on every compact subset K ⊂ D ,t henfn(z) → 0 uniformly on K. In case of k = 2, this means that ∗-paranormal operators have Bishop's property (β ).

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SVEP AND BISHOP'S PROPERTY FOR k∗-PARANORMAL OPERATORS

For nonnegative integers $n$ and $k$, we introduce in this paper a new class of $(n,k)$-quasi-*-paranormal operators satisfying 
$$||T^{1+n}(T^{k}x)||^{1/(1+n)}||T^{k}x||^{n/(1+n)} \geq 
||T^*(T^{k}x)|| \makebox{\ for all} x \in H.$$ 
This class includes the class of $n$-*-paranormal operators and the class of $(1,k)$-quasi-*-paranormal operators contains the class of $k$-quasi-*-class $A$ operators. We study basic properties of $(n,k)$-quasi-*-paranormal operators: (1) inclusion relations and examples; (2) a matrix representation; (3) joint (approximate) point spectrum and single valued extension property.

On $(n,k)$-quasi-*-paranormal operators

Riesz idempotent of (n, k)-quasi-*-paranormal operators

In this paper, we mainly obtain the following assertions: (1) If T is an al- gebraically quasi-∗-n-paranormal operator, then T is polaroid. (2) If T is an algebraically quasi-∗-n-paranormal operator, then Weyl's theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the space of analytic functions on an open neighborhood of σ(T). (3) If T ∗ is an algebraically quasi-∗-n-paranormal operator, then generalized α-Weyl's theorem holds for f(T) for every f ∈ H(σ(T)). Let H be an infinite-dimensional separable Hilbert space, denote by B(H) the algebra of all bounded linear operators on H ,w riteN (T )a ndR(T ) for the null space and range space of T ∈ B(H), and also, write σ(T )a ndσa(T ) for the spectrum of T and the approximate point spectrum of T , respectively. An operator T is called Fredholm if R(T )i s closed,α(T )=d imN (T ) < ∞ and β(T )= dim H/R(T ) < ∞ .M oreover, ifi(T )= α(T ) − β(T )=0 , thenT is called Weyl. The essential spectrum σe(T ) and the Weyl spectrum σW(T ) are defined by

Cheon-Seoung Ryoo

Papers

Some Remarks on The Structure of $k^*$-Paranormal Operators