On the adjoint of a closed transformation
Arlen Brown · Proceedings of the American Mathematical Society · 1964
The main purpose of this note is to give a new proof of a known theorem. The basic alternative relations between an operator and its adjoint, of which Theorem 2 represents one possible formulation, are well known for bounded operators and offer no difficulty for closed operators if the spaces are reflexive. More recently the general case has been handled by several authors [1; 3; 4; 5].2 The central idea of the present treatment is, by judicious normalization, to reduce Theorem 2 to Theorem 1, a result of some interest in its own right. The following notation is employed: E and F are Banach spaces and T is a closed linear transformation from E toiF with dense domain O; E* and F* are the dual spaces and O* is the domain of the adjoint transformation T*; the ranges and null spaces of T and T* are 3R, iR*, N, N* respectively. It is well known (Closed Graph Theorem; see, e.g., [2, p. 41 and Theorem 2.12.3]) that T is bounded if and only if O==E. However it seems to have been overlooked that the following straightforward dual assertion is also valid.