Remarks on commutators with operators having either finite nullity of finite deficiency (Q1336406)

One gives a characterization for commutators with a left invertible bounded operator \(\Delta\) on a Banach space \(X\), whose range has finite codimension. One assumes that there exists \(L\) such that \(L\Delta= I\), \(\text{ker}(L)\oplus\text{range}(\Delta)= X\), and \(\bigcup\text{ker}(L^ n)\) is dense. Then one characterizes bounded operators \(A\) such that \(A\Delta= \Delta A\), \(A\text{ ker}(L)\subset \text{ker}(L)\). Moreover, one gives a characterization for commutators of a perturbation of \(\Delta\).