Some Remarks on the Product Formula for Defect Numbers of Closed Operators

This largely pedagogical paper recalls some facts on defect numbers of products of closed operators employing results from the theory of semi-Fredholm operators and then applies these facts to positive integer powers of symmetric operators and subsequently to certain minimal Sturm--Liouville and minimal higher even-order ordinary and partial differential operators. We also point out some unexpected missed opportunities when comparing the work of different groups on this subject.