Anti-G-Hermiticity Preserving Linear Map That Preserves Strongly the Invertibility of Calkin Algebra Elements
A linear map ψ : X → Y of algebras X and Y preserves strongly invertibility if ψ(x−1) = ψ(x)−1 for all x ∈ X−1, where X−1 denotes the set of invertible elements of X. Let B(H) be the Banach algebra of all bounded linear operators on a separable complex Hilbert space H with dim H = ∞. A Calkin algebra C(H) is the quotient of B(H) by K(H), the ideal of compact operators on H. An element A + K(H) ∈ C(H) is said to be anti-G-Hermitian if (A + K(H))# = −A + K(H), where the #-operation is an involution on C(H). A linear map 𝜏 : C(H) → C(H) preserves anti-G-Hermiticity if 𝜏 (A + K(H))# = −𝜏 (A + K(H)) for all anti-G-Hermitian element A + K(H) ∈ C(H). In this paper, we characterize the continuous unital linear map 𝜏 : C(H) → C(H) induced by the essentially anti-G-Hermiticity preserving linear map φ : B(H) → B(H) that preserves strongly the invertibility of operators on H. We also take a look at the linear preserving properties of this induced map and other linear preservers on C(H). The discussion is in the context of G-operators, that is, linear operators on H with respect to a fixed but arbitrary positive definite Hermitian operator G ∈ B(H)−1.
MJS 10 (2017), Pp 136-146 (796.8 KiB)