Integral Equations and Operator Theory (Integral Equations Operator Theory) (20190101), 91, no.~2, Paper No 9, 27~pp. ISSN: 0378-620X (print).eISSN: 1420-8989.
Subject
47 Operator theory -- 47A General theory of linear operators 47A20 Dilations, extensions, compressions 47A56 Functions whose values are linear operators
47 Operator theory -- 47B Special classes of linear operators 47B25 Symmetric and selfadjoint operators
In the paper under review the author extends and completes certain recent results concerning the compression of self-adjoint extensions of symmetric linear relations. Let $A$ be a closed symmetric linear relation on a Hilbert space $\germ{H}$ with equal deficiency indices $n_+(A)=n_-(A)\leq \infty$. An extension $\widetilde{A}\supset A$ on $\widetilde{\germ{H}}$ is called an exit space extension of $A$ in the case $$ \overline{\text{span}}\{\germ{H},(\widetilde{A}-\lambda I_{\widetilde{\germ{H}}})\germ{H} : \lambda\in\Bbb{C}\backslash \Bbb{R}\}=\widetilde{\germ{H}}. $$ The set of self-adjoint exit space extensions of $A$ is denoted by $\text{Self}(A)$ and $\text{Self}_0(A)$ denotes the exit space extensions of $A$ that are operators. Let $A_0$ be the basic extension of $A$ obtained from Krein's formula. Krein's formula for generalized resolvents also provides a bijective correspondence between all relation-valued Nevanlinna functions $\tau$ and the extensions $\widetilde{A}=\widetilde{A}_\tau$ in $\text{Self}(A)$. For an operator-valued Nevanlinna function $\tau$ so that $\widetilde{A}_\tau\in \text{Self}_0(A)$, let $C(\widetilde{A}_\tau)$ be the compression of $\widetilde{A}_\tau$ and define $\Cal{B}_\tau=\lim_{y\to\infty} \frac{1}{i y} \tau(i y)$, which is a positive operator ($\Cal{B}_\tau \geq 0$). In the recent paper [A. Dijksma and H. Langer, Integral Equations Operator Theory {\bf 90} (2018), no.~4, Art. 41; MR3812875], for $A$ being a densely defined, closed symmetric operator with finite deficiency indices, $n_+(A)=n_-(A)< \infty$, necessary conditions are given for when any of \roster \item "(i)" $C(\widetilde{A}_\tau)\subset A_0$, \item "(ii)" $C(\widetilde{A}_\tau)= A$, \item "(iii)" $C(\widetilde{A}_\tau)= A_0$ \endroster occurs in terms of the function $\tau$ and the operator $\Cal{B}_\tau$; the condition given for (ii) is also sufficient. The main contribution of the paper under review is that these results are extended in two ways: namely, by extending to the case where $A$ is a closed symmetric linear relation whose deficiency indices may be infinite, and by providing necessary and sufficient conditions for when any of (i)--(iii) occurs. Among other conditions, the author also provides necessary and sufficient conditions for when the compression $C(\widetilde{A}_\tau)$ is self-adjoint and for when it is self-adjoint and transversal with $A_0$.