On linear maps compressing or depressing certain subspaces (Q1947158)

For a bounded linear operator \(T\) on a complex Banach space \(X\), let \(N(T)\) and \(R(T)\) be its kernel and range, respectively. The intersection \(R^{\infty}(T)=\cap_{n=1}^{\infty} R(T)\) is called the hyper-range of \(T\) and \(K(T)=\{ x\in X\mid\exists~a>0,~\exists~\{ x_n\}_{n=0}^{\infty}\subset X,\;\|x_n\|\leq a^n\| x\|,\;x_0=x,\;Tx_{n+1}=x_n\}\) is called the analytic core of \(T\). The main results of the paper describe those surjective linear maps on \({\mathcal L}(X)\) which compress or depress ranges, kernels, hyper-ranges, and analytic cores. For instance, if \(\phi:{\mathcal L}(X)\to {\mathcal L}(X)\) is a surjective linear map and \(\phi(I)\) is invertible, then the following are equivalent: (i) \(R(\phi(T))\subseteq R(T)\) (for all \(T\)); (ii) \(R(\phi(T))\supseteq R(T)\) (for all \(T\)); (iii) \(\phi(T)\subseteq T\phi(I)\) (for all \(T\)). The same equivalences hold if ranges are replaced by kernels. For hyper-ranges and analytic cores, one has similar equivalences with the additional information that \(\phi(I)\) is a nonzero scalar multiple of \(I\).