Friedrichs and Kreĭn type extensions in terms of representing maps

Abstract

A semibounded operator or relation S in a Hilbert space with lower bound $$\gamma \in {{\mathbb {R}}}$$ γ ∈ R has a symmetric extension $$S_\textrm{f}=S \, \widehat{+} \,(\{0\} \times \mathrm{mul\,}S^*)$$ S f = S + ^ ( { 0 } × mul S ∗ ) , the weak Friedrichs extension of S, and a selfadjoint extension $$S_{\textrm{F}}$$ S F , the Friedrichs extension of S, that satisfy $$S \subset S_{\textrm{f}} \subset S_\textrm{F}$$ S ⊂ S f ⊂ S F . The Friedrichs extension $$S_{\textrm{F}}$$ S F has lower bound $$\gamma $$ γ and it is the largest semibounded selfadjoint extension of S. Likewise, for each $$c \le \gamma $$ c ≤ γ , the relation S has a weak Kreĭn type extension $$S_{\textrm{k},c}=S \, \widehat{+} \,(\mathrm{ker\,}(S^*-c) \times \{0\})$$ S k , c = S + ^ ( ker ( S ∗ - c ) × { 0 } ) and Kreĭn type extension $$S_{\textrm{K},c}$$ S K , c of S, that satisfy $$S \subset S_{\textrm{k},c} \subset S_{\textrm{K},c}$$ S ⊂ S k , c ⊂ S K , c . The Kreĭn type extension $$S_{\textrm{K},c}$$ 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 $${{\mathfrak {t}}}(S)$$ t ( S ) that is associated with S; the representing map for the form $${{\mathfrak {t}}}(S)-c$$ t ( S ) - c plays an essential role here.