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 ) .
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).