# Anti-*G*-Hermiticity Preserving Linear Map That Preserves Strongly the Invertibility of Calkin Algebra Elements

**Abstract**

A linear map *ψ* : *X → Y* of algebras *X* and *Y* preserves strongly invertibility if *ψ(x ^{−1}) = ψ(x)^{−1}* for all

*x*∈

*, where*

^{X−1}*X*denotes the set of invertible elements of X. Let

^{−1}*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)