Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

Abstract

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we get some results regarding boundaries and connected hulls of the corresponding spectra. In particular, we prove that a bounded linear operator T acting on a Banach space, having topological uniform descent, is a BR operator if and only if 0 is not an accumulation point of the associated spectrum σR(T ) = {λ ∈ C : T − λI / ∈ R}, where R denote any of the following classes: upper semi-Weyl operators, Weyl operators, upper semi-Fredholm operators, Fredholm operators, operators with finite (essential) descent and BR the B-regularity associated to R as in [6]. Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a BR operator for much larger families of sets R.2010 Mathematics subject classification: 47A53, 47A10.

Generalized Drazin-Meromorphic Invertible Operators and Browder's Type Theorems

“…Using the similar argument as in (i) we get λ 0 ∈ ∂σ bb (T ). Using [16,Theorem 3.8] and [9, Theorem 4] we get λ 0 ∈ σ bb (T ).…”

On Koliha-Drazin invertible operators and Browder type theorems

“…(i) ⇐⇒ (iv). Observe that σ b (T ) = σ w (T ) ∪ int σ ap (T ) by Theorem 2.1, part (viii), and[18, Corollary 2.18 (5)].…”

Quasi-Fredholm Spectrum and Compact Perturbations

“…Therefore, ∂σ a (T ) ∩ acc σ a (T ) = S 1 . By [11,Corollary 3.6] we have σ qf (T ) = σ usbb (T ) = S 1 . Hence, T has SVEP at every point of ρ qf (T ) but ρ qf (T ) is not connected.…”