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 γ ∈ R has a symmetric extension Sf=S \, + \,(\0\ × mul\,S^*) S f = S + ^ ( { 0 } × mul S ∗ ) , the weak Friedrichs extension of S , and a selfadjoint extension SF S F , the Friedrichs extension of S , that satisfy S ⊂ Sf ⊂ SF S ⊂ S f ⊂ S F . The Friedrichs extension SF S F has lower bound γ γ and it is the largest semibounded selfadjoint extension of S . Likewise, for each c ≤ γ c ≤ γ , the relation S has a weak Kreĭn type extension Sk,c=S \, + \,(ker\,(S^*-c) × \0\) S k , c = S + ^ ( ker ( S ∗ - c ) × { 0 } ) and Kreĭn type extension SK,c S K , c of S , that satisfy S ⊂ Sk,c ⊂ SK,c S ⊂ S k , c ⊂ S K , c . The Kreĭn type extension SK,c 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) t ( S ) that is associated with S ; the representing map for the form t(S)-c t ( S ) - c plays an essential role here.