On the poles of the resolvent in Calkin algebra

Fredj, Olfa Bel Hadj

doi:10.1090/s0002-9939-07-08733-3

“…The answer is that this happens if and only if R(T n ) and R(T n+1 ) are closed. With the answer to those questions, we retrieve in particular some similar results established in the case of Hilbert spaces in [4].…”

Section: Introductionmentioning

confidence: 65%

Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the calkin algebra

Berkani

¹

,

Zivkovic-Zlatanovic

²

2019

Filomat

4

0

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We define here a pseudo B-Fredholm operator as an operator such that 0 is isolated in its essential spectrum, then we prove that an operator T is pseudo-B-Fredholm if and only if T = R + F where R is a Riesz operator and F is a B-Fredholm operator such that the commutator [R, F ] is compact. Moreover, we prove that 0 is a pole of the resolvent of an operator T in the Calkin algebra if and only if T = K + F , where K is a power compact operator and F is a B-Fredholm operator, such that the commutator [K, F ] is compact. As an application, we characterize the mean convergence in the Calkin algebra.Recall that the class of linear bounded B-Fredholm operators were defined in [5]. If F 0 (X) is the ideal of finite rank operators in L(X) and π : L(X) −→ A is the canonical homomorphism, where A = L(X)/F 0 (X), it is well known by the Atkinson's theorem [3, Theorem 0.2.2, p.4], that T ∈ L(X) is a Fredholm operator if and only if its projection π(T ) in the algebra A is invertible. Similarly, the following result established an Atkinson-type theorem for B-Fredholm operators.

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“…In the article [6], given a separable infinite dimensional Hilbert space H and T ∈ L(H), it was characterized when zero is a pole of the resolvent of π(T ) ∈ C(H) in terms of the operator T ∈ L(H), where C(H) is the Calkin algebra and π : L(H) → C(H) is the quotient map. In fact, zero is a pole of the resolvent of π(T ) ∈ C(H) if and only if there are operators A, B and K ∈ L(H) such that A is nilpotent, B is Fredholm, K is compact and T = A ⊕ B + K (see [6,Theorem 2.2]).…”

Section: Introductionmentioning

confidence: 99%

“…In the article [6], given a separable infinite dimensional Hilbert space H and T ∈ L(H), it was characterized when zero is a pole of the resolvent of π(T ) ∈ C(H) in terms of the operator T ∈ L(H), where C(H) is the Calkin algebra and π : L(H) → C(H) is the quotient map. In fact, zero is a pole of the resolvent of π(T ) ∈ C(H) if and only if there are operators A, B and K ∈ L(H) such that A is nilpotent, B is Fredholm, K is compact and T = A ⊕ B + K (see [6,Theorem 2.2]). It is worth noticing that according to [32,Proposition 1.5] or [10,Theorem 12], given a complex unital Banach algebra A, the set of the poles of the resolvent of a ∈ A coincide with {λ ∈ iso σ(a) : such that a − λ1 is Drazin invertible}, where iso σ(a) denotes the set of isolated points of the spectrum σ(a) and 1 ∈ A denotes the identity of A.…”

Section: Introductionmentioning

confidence: 99%