The Radius of the Essential Spectrum

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Mathematics (CMC)

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In this paper we define an operator measure s on L(X) into R⁺ u {0} satisfying suitable conditions. Then letting I = s−1(0), we consider the quotient algebra L(X)I, instead of Calkin algebra and define α1(T) = {λ∈CΠ2(λ−T) is not invertible in L(X)I where Π2: L(X) →L(X)I is the natural homomorphism, and r1 (T) =sup{λ:λ ϵ σ1 (T)}. After proving the fact that σ1(T) is equal to the essential spectrum of T and replacing standard measure of noncompactness with suitably defined s-measures we obtain that the radius re(T) of the essential spectrum is equal to limn(s(Tu))1n. We also construct examples of such operator measures.

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© 1987 Elsevier Inc.