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. Under 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 commutation relation . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .