ON HOLOMORPHICALLY DECOMPOSABLE FREDHOLM OPERATORS

Abstract

Let X be a complex Banach space and \(\mathcal{B}\)(X) the algebra of all bounded linear operators on X, T is said decomposably Fredholm if there exists an Fredholm operator S such that TST = T.

In this paper we consider the subset ρ_hF(T) of ℂ : μ₀ ∈ ρ_hF(T) if and only if there is a neighbourhood U₀ of μ₀ and an analytic function W : U₀ \(\longrightarrow\) \(\mathcal{B}\)(X) such that W(λ) is Fredholm and (T - λI)W(λ)(T - λI) = T - λI, for all λ ∈ U₀,  consequently we define a new spectrum σ_hF(T) = ℂ\ρ_hF(T) said holomorphically decomposable Fredholm spectrum of T.  We prove that σ_hF(T) is a non-empty compact subset of the classical spectrum σ(T), and not satisfies the spectral mapping theorem.