Friedrichs and Kreĭn type extensions in terms of representing maps

A semibounded operator or relation S in a Hilbert space with lower bound γ∈R has a symmetric extension S_f=S+^({0}×mulS^∗), the weak Friedrichs extension of S, and a selfadjoint extension S_F, the Friedrichs extension of S, that satisfy S⊂S_f⊂S_F. The Friedrichs extension S_F has lower bound γ and it is the largest semibounded selfadjoint extension of S. Likewise, for each c≤γ, the relation S has a weak Kreĭn type extension S_k,c=S+^(ker(S^∗-c)×{0}) and Kreĭn type extension S_K,c of S, that satisfy S⊂S_k,c⊂S_K,c. The Kreĭn type extension S_K,c has lower bound c and it is the smallest semibounded selfadjoint extension of S which is bounded below by c. In this paper these special extensions and, more generally, all extremal extensions of S are constructed via the semibounded sesquilinear form t(S) that is associated with S; the representing map for the form t(S)-c plays an essential role here.