A bounded linear operator A on a Banach space X is said to be ?polynomially
Riesz?, if there exists a nonzero complex polynomial p such that the image
p(A) is Riesz. In this paper we give some characterizations of these
operators.
In this paper, we characterize matrix transformations between the sequence space bv p (1 < p < ∞) and certain BK spaces. Furthermore, we apply the Hausdorff measure of noncompactness to give necessary and sufficient conditions for a linear operator between these spaces to be compact.
We introduce the class of "almost essentially Ruston elements" with respect to a homomorphism between two Banach algebras, a class intermediate between Ruston and Fredholm elements.
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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