On the spectrum of the product of closed operators

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Unde r the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the co mmutation relation <(A(BA - lambda)(-1) B)over bar>-lambda(AB-lambda)(-1) = 1 for lambda is an element of rho(AB)\{0}. Further, we prove statements co ncerning the relationship between the spectra of the operator AB and the bl ock operator matrix ((0)(A)(B)(0)).