Strong variations of Weyl and Browder type theorems for direct sums and restrictions

Abstract

In this paper, we study the stability under direct sums and restrictions of some strong variations of Weyl and Browder type theorems recently introduced in Rashid and Prasad (

Perturbation Theory for Property (VE) and Tensor Product

“…Proof. Since property (V E ) implies the equality between upper semi-Weyl and Browder spectra (see [3], Theorem 2.27), the direct sense is immediate from [37] (Theorem 3.5).…”

“…Proof. It is well known that properties (Sb) and (V Π ) are equivalent (see [3] (Corollary 2.5)). In addition, property (V Π ) implies the equality of the Browder spectrum and the upper semi B-Weyl spectrum (see [3] (Theorem 2.27)).…”

“…It is well known that properties (Sb) and (V Π ) are equivalent (see [3] (Corollary 2.5)). In addition, property (V Π ) implies the equality of the Browder spectrum and the upper semi B-Weyl spectrum (see [3] (Theorem 2.27)). Thus, the proof follows from the identity σ b (T ⊗ S) = σ(T )σ b (S) ∪ σ b (T )σ(S) (see [37] (Theorem 4.2(a))).…”

“…Recently, other properties have been introduced and studied involving the different spectra of the Fredholm and B-Fredholm theories (started by M. Berkani), which together with the classical properties are known today as Weyl-type theorems. The stability of strong variations of Weyl-type theorems under direct sums and restrictions has been studied, as well as the transmission of spectral properties between a Drazin invertible operator and its Drazin inverse; for example, see [3,4]. In addition, the study of Weyl-type theorems under commuting perturbations has been considered by several authors, among which we can mention Oudghiri [5,6], Berkani et al [7,8], Aiena and Triolo [9].…”

“…Subsequently, studies in this direction have been expanded by Duggal [11], Rashid [12] and Rashid and Prasad [13], involving new Weyl-type theorems. This article follows the same line of research as the works referenced above, but now we consider a strong variation of the Weyl-type theorems that was introduced by Sanabria et al [3,14], namely property (V E ). According to [14], if an operator T satisfies property (V E ), then T satisfies equivalently another forty-four spectral properties, among which are Weyl-type theorems such as the properties (V Π ) and (gaz) recently studied in [15,16], respectively.…”