Elementary wellknown
examples show that the sum of two closed operators need not even have a closed
extension; the same is true for products, as one can see by taking the composition of
maps f → f′ followed by f → f(0) defined on the obvious domains in C[0,1]. The
natural question which then arises concerns the complexity of operators which might
arise by taking repeated sums and products, starting with the closed operators.
Somewhat unexpectedly, the answer is very simple: all can be reduced to products of
two closed operators. Because of this, we shall distinguish this latter class by the
name “semiclosed”.
