Asymptotic similarity preserving additive maps on \(\mathcal B(X)\) (Q1426060)

Let \(X\) be an infinite-dimensional complex Banach space and \({\mathcal B}(X)\) be the algebra of all bounded linear operators on \(X\). In this paper, the authors prove that an additive surjection on \({\mathcal B}(X)\) preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar \(c\), an invertible bounded linear or conjugate linear operator \(A\), and an asymptotic similarity invariant additive functional \(\phi\) on \({\mathcal B}(X)\) such that \(\Phi\) has the form \(\Phi(T)= cATA^{-1}+ \phi(T)I\) for all \(T\in{\mathcal B}(X)\) or \(\Phi(T)= cAT^*A^{-1}+ \phi(T)I\) for all \(T\in{\mathcal B}(X)\). In the case that \(X\) has finite multiplicity, especially if \(X\) is an infinite-dimensional Hilbert space, the above asymptotic similarity invariant additive functional \(\phi\) is always zero.