A factorization of extremal extensions with applications to block operator matrices
Let \(A\) be a densely defined nonnegative operator in a separable Hilbert space \(\mathcal H\) with inner product \(\langle\cdot, \cdot\rangle\). There exist two nonnegative selfadjoint extensions, the Friedrichs extension \(A_{F}\) and the Kreĭn-von Neumann extension \(A_{N}\), with the property that every nonnegative selfadjoint extension \(\widetilde{A}\) of \(A\) satisfies the inequalities \[ \langle(A_{F}+\lambda)^{-1}f,f\rangle\leq \langle(\widetilde{A}+\lambda)^{-1}f,f\rangle \leq \langle(A_{N}+\lambda)^{-1}f,f\rangle\quad \text{for}\quad f\in {\mathcal H},\;\lambda\geq 0. \] A nonnegative selfadjoint extension \(\widetilde{A}\) of \(A\) is called extremal if \(\inf \{\langle\widetilde{A} (\varphi-f),\varphi-f\rangle: f\in{\mathcal D}(A)\}=0\) for all \(\varphi\in{\mathcal D}(\widetilde{A})\). The description of extremal extensions may be based on a special factorization. Define the operators \(Q\) and \(J\) by \(Q : {\mathcal H} \supset {\mathcal D}(A) \to {\mathcal H}_A\), \(f\mapsto Af\), \(J:{\mathcal H}_A \supset {\mathcal R}(A) \to {\mathcal H}\), \(Af \mapsto Af\), where \({\mathcal H}_A = \{ {\mathcal R}(A), \langle\cdot,\cdot\rangle\}\widehat{\;}\) is the completion of the range of \(A\) with respect to the inner product \(\langle Af, Ag\rangle\). In this case, we have the following factorization [see \textit{Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyén} and \textit{H. S. V. de Snoo}, Oper. Theory, Adv. Appl. 127, 41--81 (2001; Zbl 0996.47029)]: \[ A=JQ, \quad A_F=Q^\star Q^{\star\star}, \quad A_N=J^{\star\star}J^\star. \] In the paper under review, the authors consider two Hilbert spaces \(\mathcal H, \mathcal K\) and a nonnegative operator \(A = KC\) in \(\mathcal H\), where \(K\) and \(C\) are acting from \(\mathcal K\) into \(\mathcal H\) and from \(\mathcal H\) into \(\mathcal K\), respectively. It is not assumed that the operators \(A\), \(C\) and \(K\) are closable or densely defined. A~description of all extremal extensions by means of the factors \(K\) and \(C\) is obtained. In particular, a discussion of the Friedrichs and the Kreĭn-von Neumann extension of \(KC\) is included. More precisely, the authors prove that all extremal extensions of \(KC\) can be constructed using the above factorization approach. Applications to systems of differential operators are presented.
scientific article; zbMATH DE number 5857786
scientific article