Consistent invertibility and Weyl's theorem (Q982591)

One says that Weyl's theorem holds for an operator \(T\) if the set \(\sigma(T)\setminus\{\lambda\in\mathbb{C}\mid A-\lambda I\) is Fredholm of index zero\} is equal to the set of isolated eigenvalues of finite multiplicity. A Banach space operator \(T \in B(X)\) is said to be consistently invertible provided that \(TS\) is invertible if and only if so is \(ST\). The authors investigate Weyl's theorem by offering some conditions involving a subset of the spectrum derived from ``consistent invertibility''.