ACTA issues

A Kreĭn space approach to representation theorems and generalized Friedrichs extensions

Andreas Fleige, Seppo Hassi, Henk de Snoo

Acta Sci. Math. (Szeged) 66:3-4(2000), 633-650

Abstract. Let ${\eufm t}[\cdot,\cdot ]$ be a densely defined symmetric sesquilinear form in a Hilbert space ${\eufm H}$ with inner product $(\cdot,\cdot )$. Assume that for some $\lambda\in {\msbm R}$ the form ${\eufm t}[\cdot,\cdot ]-\lambda(\cdot,\cdot )$ induces a Kreĭn space structure on $\mathop{\rm dom}{\eufm t}$, which can be continuously embedded in ${\eufm H}$. Then there exists a unique selfadjoint operator $T_{\eufm t}$ in ${\eufm H}$ such that $\mathop{\rm dom}T_{\eufm t}\subset\mathop{\rm dom}{\eufm t}$ and ${\eufm t}[f,g]=(T_{\eufm t}f,g)$, $f \in\mathop{\rm dom}T_{\eufm t}$, $g \in\mathop{\rm dom}{\eufm t}$. This generalizes the first representation theorem in T. Kato [Kato] to a non-semibounded situation. Based on the theory of definitizable operators in Kreĭn spaces an analog of the second representation theorem in [Kato] will be given. These results provide an approach to generalized Friedrichs extensions for a class of non-semibounded symmetric operators with defect numbers $(1,1)$, which is analogous to the classical theory in the semibounded case.

AMS Subject Classification (1991): 46C20, 47A67, 47B50; 47B25

Keyword(s): Sesquilinear form, representation theorem, Kreĭn space, singular critical point, generalized Friedrichs extension

Received May 3, 1999. (Registered under 2757/2009.)