Asymptotic intertwining by the identity operator and permanence of spectral properties (Q1943507)

Let \(A, B\in B(X)\), \(X\) a Banach space. \(B\) is said to be asymptotically intertwined to \(A\) by the identity operator \(I\in B(X)\) if \(\lim_{n\to\infty}\|\Delta^{n}_{AB}(I)\|^{\frac{1}{n}}=0\), where \(\Delta_{AB}(I)=AI-IB=A-B\). The paper studies certain spectral properties preserved by the above relation. Some interesting results are obtained when \(\Delta_{AB}(I)\) is nilpotent and, more generally, for \(B\) asymptotically intertwined to \(A\) by the identity, but when \(\Delta_{AB}(I)\) is not nilpotent.