The authors study the relationship between the spectra of the products \(AB\) and \(BA\) and of the block operator matrix \[ T = \left( \begin{matrix} 0 & A \\ B & 0 \end{matrix}\right) \] for closed unbounded operators acting in Banach spaces. Under the condition that the resolvent sets of both products are not empty, they prove for \(\lambda \not = 0\):
(1) \(\lambda\) is in the spectrum of \(AB\) if and only if it is in the spectrum of \(BA\);
(2) if \(\lambda\) is not in the spectrum of both products, then the commutation relation
\(( \overline{A(BA-\lambda)^{-1}B} - \lambda (AB-\lambda)^{-1} = \text{Id})\) holds;
(3) \(\lambda\) is in the spectrum of \(T\) if and only if \(\lambda^{2}\) is in the spectrum of \(AB\).
The authors give nonsymmetric versions of these results and similar results for the non-discrete spectra. They discuss the case of Hilbert spaces and symmetric block operator matrices. Finally, they apply the results to isospectral Schrödinger operators with complex potentials.