Extended Weyl Type Theorems and Perturbations

Property (gb) and perturbations

“…Property (gw) extends property (w) to the context of B-Fredholm theory, and it is proved in [11] that an operator satisfies property (gw) satisfies property (w) but the converse is not true in general. According to [19], an operator T ∈ L(X) is said to satisfy property (gb) if g a (T ) = π(T ), and is said to possess property (b) if a (T ) = π 0 (T ). It is shown [19,Theorem 2.3] that an operator satisfies property (gb) satisfies property (b) but the converse is not true in general.…”

“…According to [19], an operator T ∈ L(X) is said to satisfy property (gb) if g a (T ) = π(T ), and is said to possess property (b) if a (T ) = π 0 (T ). It is shown [19,Theorem 2.3] that an operator satisfies property (gb) satisfies property (b) but the converse is not true in general.…”

Browder-type Theorems and SVEP

“…Assume that T satisfies generalized a-Browder's theorem, then from [12, Theorem 3.8], T satisfies generalized Browder's theorem. Conversely, assume that T satisfies generalized Browder's theorem and ind(T − λI) = 0 for all λ ∈ Δ g a (T ), then by [13,Theorem 2.12], T possesses property (gb). As mentioned above, we have T satisfies generalized a-Browder's theorem.…”

“…Following [13], we say that property (b) holds for T ∈ L(X) if Δ a (T ) = Π 0 (T ) and that property (gb) holds for T if Δ g a (T ) = Π(T ). Property (gb) extends property (b) to the context of B-Fredholm theory.…”

“…Property (gb) extends property (b) to the context of B-Fredholm theory. It is known [13,Theorem 2.5] that an operator possessing property (b) satisfies Browder's theorem, but the converse does not hold in general and it is proved in [13,Theorem 2.3] that an operator possessing property (gb) possesses property (b), but not conversely and under the assumption Π(T ) = Π a (T ), it is proved in [13, Theorem 2.10 ] that the two properties are equivalent.…”