On Linear Maps Preserving Spectrum, Spectral Radius and Essential Spectral Radius

Abstract

In this thesis, we focus our study on a part of the linear maps on algebras of operators that preserve certain properties of operators. We study linear maps preserving spectrum. Let X and Y be Banach spaces. We show that a spectrum preserving surjective linear map ϕ from B(X) to B(Y) is either of the form ϕ(T) = ATA-1 for an isomorphism A of X onto Y or the form ϕ(T) = BT×B-1 for an isomorphism B of X/ onto Y. After this, we study linear maps preserving the spectral radius. Let X be a complex Banach space. We show that if ϕ: B(X) ⟶ B(X) is a surjective linear map such that T and ϕ(T) have the same spectral radius for every T ∈ B(X), then ϕ = cθ where θ is either an algebra-automorphism or an antiautomorphism of B(X) and c is a complex constant such that |c|=1. Finally, we characterize linear maps from B(H) onto itself that preserve the essential spectral radius. Where B(H) is the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H.