Unbounded operators, Friedrichs’ extension theorem

Explicit naming of the domain of an unbounded operator is often suppressed, instead writing T1 ⊂ T2 when T2 is an extension of T1, in the sense that the domain of T2 contains that of T1, and the restriction of T2 to the domain of T1 agrees with T1. An operator T ′, D′ is a sub-adjoint to an operator T,D when 〈Tv,w〉 = 〈v, T ′w〉 (for v ∈ D, w ∈ D′) For D dense, for given D′ there is at most one T ′ meeting the adjointness condition. The adjoint T ∗ is the unique maximal element, in terms of domain, among all sub-adjoints to T . That there is a unique maximal sub-adjoint requires proof, given below.