Essential descent spectrum and commuting compact perturbations

Abstract

Let ℒ(𝑋) be the algebra of all bounded operators acting on an infinite- dimensional complex Banach space. For an operator 𝑇 𝜖 ℒ(𝑋), write ¾(𝑇) for its spectrum and ½(𝑇) for its resolvent. The range and the kernel of T are denoted respectively by 𝑅(𝑇) and 𝑁(𝑇). The operator 𝑇 is called upper semi-Fredholm if dim 𝑁 (𝑇) isfinite and 𝑅 (𝑇) is closed, while 𝑇 is called lower semi-Fredholm if codim 𝑅 (𝑇) is finite, and in this case the closedness of the range follows immediately.