Abstract
Property (UW {scriptstyle Pi }), introduced in Berkani and Kachad (Bull Korean Math Soc 49:1027–1040, 2015) and studied more recently in Aiena and Kachad (Acta Sci Math (Szeged) 84:555–571, 2018) may be thought as a variant of Browder’s theorem, or Weyl’s theorem, for bounded linear operators acting on Banach spaces. In this article we study the stability of this property under some commuting perturbations, as quasi-nilpotent perturbation and, more in general, under Riesz commuting perturbations. We also study the transmission of property (UW {scriptstyle Pi }) from T to f(T), where f is an analytic function defined on a neighborhood of the spectrum of T. Furthermore, it is shown that this property is transferred from a Drazin invertible operator T to its Drazin inverse S.
Highlights
Introduction and preliminariesLet T ∈ L(X ) be a bounded linear operator on an infinite-dimensional complex Banach spaces X, and denote by α(T ) and β(T ) the dimension of the kernel N (T ) and the codimension of the range R(T ) = T (X ), respectively
Proof Suppose that T satisfies property (U W ), i.e. σa(T ) = σuw(T ) part (i) and part (iii) of Theorem 2.4 we have (T )
Property (U W ) is transmitted to T + K if we assume that the approximatepoint spectrum σa(T ) has no isolated points: Theorem 3.3 Suppose that T, K ∈ L(X ) commute, and K n is finite-dimensional for some n ∈ N
Summary
The following assertions are equivalent: (i) T satisfies property (U W ); (ii) (iii) satisfies a-Browder’s theorem, ∗ has SVEP at every λ ∈ a(T Proof Suppose that T satisfies property (U W ), i.e. σa(T ) = σuw(T ) part (i) and part (iii) of Theorem 2.4 we have (T ).
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