Property (ω) and quasi-class (A, k) operators
In this paper, we prove the following assertions: (i) If T is of quasi-class (A, k), then T is polaroid and reguloid; (ii) If T or T* is an algebraically of quasi-class (A, k) operator, then Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iii) If T* is an algebraically of quasi-class (A, k) operator, then a-Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iv) If T* is algebraically of quasi-class (A, k) then property (ω) holds for T.
Polaroid Operators with Svep and Perturbations of Property (Gaw)
. In this paper we establish for a bounded linear operator deﬁned on a Banach space several suﬃcient and necessary conditions for which property ( gaw ) holds. In this work, we consider commutative perturbations by algebraic operator and quasinilpotent operator for T ∈ B ( X ) such that T ∗ satisﬁes property ( gaw ). We prove that if A is an algebraic and T ∈ PS ( X ) is such that AT = TA, then f ( T ∗ + A ∗ ) satisﬁes property ( gaw ) for every f ∈ H c ( σ ( T + A )) . Moreover, we show that if Q is a quasi-nilpotent operator and T ∈ PS ( X ) is such that TQ = QT, then f ( T ∗ + Q ∗ ) satisﬁes the property ( gaw ) for every f ∈ H c ( σ ( T + Q )) . At the end of this paper, we apply the obtained results to a number of subclasses of PS ( X ) .
On the Isolated Points of the Spectrum of Paranormal Operators
Abstract.For paranormal operator T on a separable complex Hilbert space
$$\mathcal{H},$$ we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π00(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)*.