Additive maps preserving the ascent and descent of operators (Q732062)

Let \(X\) and \(Y\) be infinite-dimensional complex Banach spaces and let \({\mathcal L}(X)\) and \({\mathcal L}(Y)\) be the algebras of all bounded linear operators on \(X\) and \(Y\), respectively. Assume that \(\varphi:{\mathcal L}(X)\to {\mathcal L}(Y)\) is a surjective additive map. Then \(\varphi\) preserves the ascent of operators, which is defined as \(a(T)=\inf\{n\geq 0:\text{ker}(T^n)=\text{ker}(T^{n+1})\}\), if and only if there exists a nonzero number \(\lambda\) and an invertible bounded linear or conjugate linear operator \(A:X\to Y\) such that \(\varphi(T)=\lambda ATA^{-1}\) for all \(T\in {\mathcal L}(X)\). If \(\varphi:{\mathcal L}(X)\to {\mathcal L}(Y)\) is a surjective additive map that preserves the descent of operators, i.e., the quantity \(\delta(T)=\inf\{n\geq 0:\text{range}(T^n)=\text{range}(T^{n+1})\}\), then there exists a nonzero number \(\lambda\) such that either (i) there exists an invertible bounded linear or conjugate linear operator \(A:X\to Y\) so that \(\varphi(T)=\lambda ATA^{-1}\) for all \(T\in{\mathcal L}(X)\), or (ii) there exists an invertible bounded linear or conjugate linear operator \(A:X^*\to Y\) so that \(\varphi(T)=\lambda AT^*A^{-1}\) for all \(T\in{\mathcal L}(X)\). The results of the paper and the techniques used in the proofs are standard.