Friedrichs and Kre{}n type extensions in terms of representing maps

A semibounded operator or relation S in a Hilbert space with lower bound m ∈ R has a symmetric extension Sf=S \, + \, (\0\ × mul\, S^*), the weak Friedrichs extension of S, and a selfadjoint extension SF, the Friedrichs extension of S, that satisfy S ⊂ Sf ⊂ SF. The Friedrichs extension SF 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 Kre{}n type extension S_ k,c=S \, + \, ( (S^*-c) × \0\) and Kre{}n type extension S_ K,c of S, that satisfy S ⊂ S_ k,c ⊂ S_ K,c. The Kre{}n 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 in terms of a representing map for t(S)-c and their properties are being considered.