On Closed Operators with Closed Range1

It is well-known that if T is an everywhere defined bounded operator on a Banach space X to a Banach space Y and T* is its adjoint, then the range R(T*) of T* is closed in X* if and only if the range R(T) of T is closed in Y (cf. [2, pp. 487-489]). The object of this note is to establish this result for closed but possibly unbounded operators. This result, for the unbounded case, is of great utility in the study of differential operators and has been considered by F. E. Browder,2 I. C. Gohberg and M. G. Kreïn, and by G. C. Rota. In the sequel, we shall have occasion to consider a set under several different topologies. We shall use the following convention: ïî A is a linear set and B is a set of linear functionals on A, then the set A with the weak topology induced by the elements of B will be denoted by (A, B). Thus, the assertion that a set C is closed (dense, etc.) in (.4, B) shall mean that C is a subset of A which is closed (dense, etc.) in this weak topology. If A is a Banach space, then the assertion that a set C is closed (dense, etc.) in A shall refer to the norm topology of A. Henceforth, X and Y will be Banach spaces. If F is a closed operator with domain D(T) in X and range R(T) in Y, then it was noted by Sz.-Nagy [5], that D(T) becomes a Banach space under the norm |x|:r=|x|-|-|Fx| (which we shall call the F-norm) and F is a bounded operator on this space. If in addition, D(T) is dense in X, it is well known that T has a uniquely defined adjoint T* with domain D(T*) dense in (Y*, Y), and that T* is a closed operator.