Linear maps preserving the generalized spectrum

Abstract

Let 𝐻 be an infinite-dimensional separable complex Hilbert space and 𝛣 (𝐻) the algebra of all bounded linear operators on 𝐻. For an operator 𝑇 in 𝛣(𝐻), let σɡ(𝑇) denote the generalized spectrum of T. In this paper, we prove that if ø: 𝛣(𝐻) → 𝛣(𝐻) is a surjective linear map, then ø preserves the generalized spectrum (i.e. øɡ(ø (𝑇)) = ø (𝑇) for every 𝑇 ∊ 𝛣(𝐻)) if and only if there is A ∊ 𝛣(𝐻) invertible such that either ø(𝑇) = A𝑇A‾¹ for every 𝑇 ∊𝛣(𝐻), or ø(𝑇) = A𝑇trA‾¹ for every 𝑇 ∊ 𝛣(H). Also, we prove that γ(ø(𝑇)) = γ(𝑇) for every 𝑇 ∊ 𝛣(𝐻) if and only if there is U ∊ 𝛣(𝐻) unitary such that either ø(𝑇) = U𝑇U* for every 𝑇 ∊ 𝛣(𝐻) or ø(𝑇) = U𝑇trU¤ for every 𝑇 ∊ 𝛣(𝐻ɡ). Here γ(𝑇) is the reduced minimum modulus of 𝑇.