Property \((w)\) under compact or Riesz commuting perturbations (Q2910026)

Let \(X\) be a Banach space and \(L(X)\) denote the space of all bounded linear operators on \(X\) into itself. The property (w) is a variant of Weyl's theorem for \(T\in L(X)\). In the first part of the paper, the authors prove that property (w) is preserved under commuting compact perturbations and Riesz commuting perturbations. In the second part, they consider a strong version of property (w), the so-called generalized property (gw) recently introduced by \textit{M. Amouch} and \textit{M. Berkani} [Mediterr. J. Math. 5, No.~3, 371--378 (2008; Zbl 1188.47011); Acta Sci. Math. 74, No.~3--4, 769--781 (2008; Zbl 1199.47068)]. They show that, if \(T\in L(X)\) is left polaroid, then property (w), generalized property (gw), a-Weyl's theorem and generalized a-Weyl's theorem are equivalent. They prove also that, if \(T\in L(X)\) is a-polaroid, property (gw) is preserved under commuting nilpotent perturbations. At the end, preservation under commuting algebraic perturbations is shown.NEWLINENEWLINEEditorial remark. The authors have published another paper under the same title in [Acta Sci. Math. 76, No.~ 1--2, 135--153 (2010; Zbl 1261.47008)].