Content-Type: multipart/mixed; boundary="-------------0907120845909" This is a multi-part message in MIME format. ---------------0907120845909 Content-Type: text/plain; name="09-109.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-109.comments" 16 pages ---------------0907120845909 Content-Type: text/plain; name="09-109.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-109.keywords" Krein-von Neumann extension, buckling problem ---------------0907120845909 Content-Type: application/x-tex; name="KreinBuckling.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KreinBuckling.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% @texfile{ %% filename="KreinBucklingn.tex", %% version="1.0", %% date="Feb-2009", %% cdate="20090227", %% filetype="LaTeX2e", %% journal="Math. Nachr. (to appear)", %% copyright="Copyright (C) M. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg, and G. 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\newtheorem{remark}[theorem]{Remark} \begin{document} \title[The Krein--von Neumann Extension]{The Krein--von Neumann Extension and its Connection to an Abstract Buckling Problem} \author[M.\ S.\ Ashbaugh]{Mark S.\ Ashbaugh} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{\mailto{ashbaughm@missouri.edu}} \urladdr{\url{http://www.math.missouri.edu/personnel/faculty/ashbaughm.html}} \author[F.\ Gesztesy]{Fritz Gesztesy} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{\mailto{gesztesyf@missouri.edu}} \urladdr{\url{http://www.math.missouri.edu/personnel/faculty/gesztesyf.html}} \author[M.\ Mitrea]{Marius Mitrea} \address{Department of Mathematics, University of Missouri, Columbia, MO 65211, USA} \email{\mailto{mitream@missouri.edu}} \urladdr{\url{http://www.math.missouri.edu/personnel/faculty/mitream.html}} \author[R.\ Shterenberg]{Roman Shterenberg} \address{Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA} \email{\mailto{shterenb@math.uab.edu}} \author[G. Teschl]{Gerald Teschl} \address{Faculty of Mathematics\\ Nordbergstrasse 15\\ 1090 Wien\\ Austria\\ and International Erwin Schr\"odinger Institute for Mathematical Physics\\ Boltzmanngasse 9\\ 1090 Wien\\ Austria} \email{\mailto{Gerald.Teschl@univie.ac.at}} \urladdr{\url{http://www.mat.univie.ac.at/~gerald/}} \thanks{Based upon work partially supported by the US National Science Foundation under Grant Nos.\ DMS-0400639 and FRG-0456306 and the Austrian Science Fund (FWF) under Grant No.\ Y330.} \dedicatory{Dedicated to the memory of Erhard Schmidt (1876--1959).} \thanks{To appear in {\it Math. Nachrichten.}} \date{\today} %\date{July 20, 2008} \subjclass[2000]{Primary 35J25, 35J40, 47A05; Secondary 47A10, 47F05.} \keywords{Krein--von Neumann extension, buckling problem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. In the concrete case where $S=\overline{-\Delta|_{C_0^\infty(\Omega)}}$ in $L^2(\Omega; d^n x)$ for $\Omega\subset\mathbb{R}^n$ an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian $S_K$ (i.e., the Krein--von Neumann extension of $S$), \[ S_K v = \lambda v, \quad \lambda \neq 0, \] is in one-to-one correspondence with the problem of {\em the buckling of a clamped plate}, \[ (-\Delta)^2u=\lambda (-\Delta) u \, \text{ in } \, \Omega, \quad \lambda \neq 0, \quad u\in H_0^2(\Omega), \] where $u$ and $v$ are related via the pair of formulas \[ u = S_F^{-1} (-\Delta) v, \quad v = \lambda^{-1}(-\Delta) u, \] with $S_F$ the Friedrichs extension of $S$. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \maketitle % {\scriptsize \tableofcontents} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{s1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Suppose that $S$ is a densely defined, symmetric, closed operator with nonzero deficiency indices in a separable complex Hilbert space $\cH$ that satisfies \begin{equation} S\geq \varepsilon I_{\cH} \, \text{ for some $\varepsilon >0$,} \lb{1.1} \end{equation} and denote by $S_K$ and $S_F$ the Krein--von Neumann and Friedrichs extensions of $S$, respectively (with $I_{\cH}$ the identity operator in $\cH$). Then an abstract version of Proposition\ 1 in Grubb \cite{Gr83}, describing an intimate connection between the nonzero eigenvalues of the Krein--von Neumann extension of an appropriate minimal elliptic differential operator of order $2m$, $m\in\bbN$, and nonzero eigenvalues of a suitable higher-order buckling problem (cf.\ Example \ref{e3.6}), to be proved in Lemma \ref{l3.3}, can be summarized as follows: \begin{align} & \text{There exists $0 \neq v \in \dom(S_K)$ satisfying } \, S_K v = \lambda v, \quad \lambda \neq 0, \lb{1.1a} \\ & \text{if and only if} \no \\ & \text{there exists a $0 \neq u \in \dom(S^* S)$ such that } \, S^* S u = \lambda S u, \lb{1.1b} \end{align} and the solutions $v$ of \eqref{1.1a} are in one-to-one correspondence with the solutions $u$ of \eqref{1.1b} given by the pair of formulas \begin{equation} u = (S_F)^{-1} S_K v, \quad v = \lambda^{-1} S u. \lb{1.1c} \end{equation} Next, we will go a step further and describe a unitary equivalence result going beyond the connection between the eigenvalue problems \eqref{1.1a} and \eqref{1.1b}: Given $S$, we introduce the following sesquilinear forms in $\cH$, \begin{align} a(u,v) & = (Su,Sv)_{\cH}, \quad u, v \in \dom(a) = \dom(S), \lb{1.2} \\ b(u,v) & = (u,Sv)_{\cH}, \quad u, v \in \dom(b) = \dom(S). \lb{1.3} \end{align} Then $S$ being densely defined and closed, implies that the sesquilinear form $a$ is also densely defined and closed, and thus one can introduce the Hilbert space \begin{equation} \cW=(\dom(S), (\cdot,\cdot)_{\cW}) \lb{1.4} \end{equation} with associated scalar product \begin{equation} (u,v)_{\cW}=a(u,v) = (Su,Sv)_{\cH}, \quad u, v \in \dom(S). \lb{1.5} \end{equation} Suppressing for simplicity the continuous embedding operator of $\cW$ into $\cH$, we now introduce the following operator $T$ in $\cW$ by \begin{align} (w_1,T w_2)_{\cW} & = a( w_1,T w_2) = b(w_1,w_2) = (w_1,S w_2)_{\cH}, \quad w_1, w_2 \in \cW. \lb{1.8} \end{align} One can prove that $T$ is self-adjoint, nonnegative, and bounded and we will call $T$ the {\it abstract buckling problem operator} associated with the Krein--von Neumann extension $S_K$ of $S$. Next, introducing the Hilbert space $\hatt \cH$ by \begin{equation} \hatt \cH = [\ker (S^*)]^{\bot} = \big[I_{\cH} - P_{\ker(S^*)}\big] \cH = \big[I_{\cH} - P_{\ker(S_K)}\big] \cH = [\ker (S_K)]^{\bot}, \end{equation} where $P_{\cM}$ denotes the orthogonal projection onto the subspace $\cM \subset \cH$, we introduce the operator \begin{equation} \hatt S: \begin{cases} \cW \to \hatt \cH, \\ w \mapsto S w, \end{cases} \lb{1.12} \end{equation} and note that $\hatt S\in\cB(\cW,\hatt \cH)$ maps $\cW$ unitarily onto $\hatt \cH$. Finally, defining the {\it reduced Krein--von Neumann operator} $\hatt S_K$ in $\hatt \cH$ by \begin{equation} \hatt S_K:=S_K|_{[\ker(S_K)]^{\bot}} \, \text{ in $\hatt \cH$,} \label{1.13} \end{equation} we can state the principal unitary equivalence result to be proved in Theorem \ref{t3.3}: The inverse of the reduced Krein--von Neumann operator $\hatt S_K$ in $\hatt \cH$ and the abstract buckling problem operator $T$ in $\cW$ are unitarily equivalent, \begin{equation} \big(\hatt S_K\big)^{-1} = \hatt S T (\hatt S)^{-1}. \lb{1.20} \end{equation} In addition, \begin{equation} \big(\hatt S_K\big)^{-1} = U_S \big[|S|^{-1} S |S|^{-1}\big] (U_S)^{-1}. \lb{1.20a} \end{equation} Here we used the polar decomposition of $S$, \begin{equation} S = U_S |S|, \, \text{ with } \, |S| = (S^* S)^{1/2} \geq \varepsilon I_{\cH}, \; \varepsilon > 0, \, \text{ and } \, U_S \in \cB\big(\cH,\hatt \cH\big) \, \text{ unitary,} \lb{1.19b} \end{equation} and one observes that the operator $|S|^{-1} S |S|^{-1}\in\cB(\cH)$ in \eqref{1.20a} is self-adjoint in $\cH$. As discussed at the end of Section \ref{s3}, one can readily rewrite the abstract linear pencil buckling eigenvalue problem \eqref{1.1b}, $S^* S u = \lambda S u$, $\lambda \neq 0$, in the form of the standard eigenvalue problem $|S|^{-1} S |S|^{-1} w = \lambda^{-1} w$, $\lambda \neq 0$, $w = |S| u$, and hence establish the connection between \eqref{1.1a}, \eqref{1.1b} and \eqref{1.20}, \eqref{1.20a}. As mentioned in the abstract, the concrete case where $S$ is given by $S=\overline{-\Delta|_{C_0^\infty(\Omega)}}$ in $L^2(\Omega; d^n x)$, then yields the spectral equivalence between the inverse of the reduced Krein--von Neumann extension $\hatt S_K$ of $S$ and the problem of the buckling of a clamped plate. More generally, Grubb \cite{Gr83} actually treated the case where $S$ is generated by an appropriate elliptic differential expression of order $2m$, $m\in\bbN$, and also introduced the higher-order analog of the buckling problem; we briefly summarize this in Example \ref{e3.6}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Abstract Krein--von Neumann Extension} \label{s2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% To get started, we briefly elaborate on the notational conventions used throughout this paper and especially throughout this section which collects abstract material on the Krein--von Neumann extension. Let $\cH$ be a separable complex Hilbert space, $(\cdot,\cdot)_{\cH}$ the scalar product in $\cH$ (linear in the second factor), and $I_{\cH}$ the identity operator in $\cH$. Next, let $T$ be a linear operator mapping (a subspace of) a Banach space into another, with $\dom(T)$, $\ran(T)$, and $\ker(T)$ denoting the domain, range, and kernel (i.e., null space) of $T$. The closure of a closable operator $S$ is denoted by $\ol S$. The spectrum, essential spectrum, discrete spectrum, and resolvent set of a closed linear operator in $\cH$ will be denoted by $\sigma(\cdot)$, $\sigma_{\rm ess}(\cdot)$, $\sigma_{\rm d}(\cdot)$, and $\rho(\cdot)$, respectively. The Banach spaces of bounded and compact linear operators in $\cH$ are denoted by $\cB(\cH)$ and $\cB_\infty(\cH)$, respectively. Similarly, the Schatten--von Neumann (trace) ideals will subsequently be denoted by $\cB_p(\cH)$, $p\in (0,\infty)$. Analogous notation $\cB(\cH_1,\cH_2)$, $\cB_\infty (\cH_1,\cH_2)$, etc., will be used for bounded, compact, etc., operators between two Hilbert spaces $\cH_1$ and $\cH_2$. Whenever applicable, we retain the same type of notation in the context of Banach spaces. Moreover, $\cX_1\hookrightarrow \cX_2$ denotes the continuous embedding of the Banach space $\cX_1$ into the Banach space $\cX_2$. $\cX_1 \dotplus \cX_2$ denotes the (not necessarily orthogonal) direct sum of the subspaces $\cX_1$ and $\cX_2$ of $\cX$. A linear operator $S:\dom(S)\subseteq\cH\to\cH$, is called {\it symmetric}, if \begin{equation}\label{Pos-2} (u,Sv)_\cH=(Su,v)_\cH, \quad u,v\in \dom (S). \end{equation} In this manuscript we will be particularly interested in this question within the class of densely defined (i.e., $\ol{\dom(S)}=\cH$), non-negative operators (in fact, in most instances $S$ will even turn out to be strictly positive) and we focus almost exclusively on self-adjoint extensions that are non-negative operators. In the latter scenario, there are two distinguished constructions which we review briefly next. To set the stage, we recall that a linear operator $S:\dom(S)\subseteq\cH\to \cH$ is called {\it non-negative} provided \begin{equation}\label{Pos-1} (u,Su)_\cH\geq 0, \quad u\in \dom(S). \end{equation} (In particular, $S$ is symmetric in this case.) $S$ is called {\it strictly positive}, if for some $\varepsilon >0$, $(u,Su)_\cH\geq \varepsilon \|u\|_{\cH}^2$, $u\in \dom(S)$. Next, we recall that $A \leq B$ for two self-adjoint operators in $\cH$ if \begin{align} \begin{split} & \dom\big(|A|^{1/2}\big) \supseteq \dom\big(|B|^{1/2}\big) \, \text{ and } \\ & \big(|A|^{1/2}u,U_A |A|^{1/2}u\big)_{\cH} \leq \big(|B|^{1/2}u, U_B |B|^{1/2}u\big)_{\cH}, \quad u \in \dom\big(|B|^{1/2}\big). \lb{AleqB} \end{split} \end{align} Here $U_C$ denotes the partial isometry in $\cH$ in the polar decomposition $C = U_C |C|$, $|C|=(C^* C)^{1/2}$, of a densely defined closed operator $C$ in $\cH$. (If $C$ is in addition self-adjoint, then $|C|$ and $U_C$ commute.) We also recall that for $A\geq 0$ self-adjoint, \begin{equation} \ker(A) =\ker\big(A^{1/2}\big) \end{equation} (with $D^{1/2}$ denoting the unique nonnegative square root of a nonnegative self-adjoint operator $D$ in $\cH$). For simplicity we will always adhere to the conventions that $S$ is a linear, unbounded, densely defined, nonnegative (i.e., $S\geq 0$) operator in $\cH$, and that $S$ has nonzero deficiency indices. Since $S$ is bounded from below, the latter are necessarily equal. In particular, \begin{equation} {\rm def} (S) = \dim (\ker(S^*-z I_{\cH})) \in \bbN\cup\{\infty\}, \quad z\in \bbC\backslash [0,\infty), \lb{DEF} \end{equation} is well-known to be independent of $z$. Moreover, since $S$ and its closure $\ol{S}$ have the same self-adjoint extensions in $\cH$, we will without loss of generality assume that $S$ is closed in the remainder of this paper. The following is a fundamental result to be found in M.\ Krein's celebrated 1947 paper \cite{Kr47} (cf.\ also Theorems\ 2 and 5--7 in the English summary on page 492)\footnote{We are particularly indebted to Gerd Grubb for a clarification of the necessary and sufficient nature of the inequalities \eqref{Fr-Sa} (resp.,\ \eqref{Res}) for $\widetilde{S}$ to be a self-adjoint extension of $S$.}: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{T-kkrr} Assume that $S$ is a densely defined, closed, nonnegative operator in $\cH$. Then, among all non-negative self-adjoint extensions of $S$, there exist two distinguished ones, $S_K$ and $S_F$, which are, respectively, the smallest and largest $($in the sense of order between self-adjoint operators, cf.\ \eqref{AleqB}$)$ such extension. Furthermore, a non-negative self-adjoint operator $\widetilde{S}$ is a self-adjoint extension of $S$ if and only if $\widetilde{S}$ satisfies \begin{equation}\label{Fr-Sa} S_K\leq\widetilde{S}\leq S_F. \end{equation} In particular, \eqref{Fr-Sa} determines $S_K$ and $S_F$ uniquely. In addition, if $S\geq \varepsilon I_{\cH}$ for some $\varepsilon >0$, one has $S_F \geq \varepsilon I_{\cH}$, and \begin{align} \dom (S_F) &= \dom (S) \dotplus (S_F)^{-1} \ker (S^*), \lb{SF} \\ \dom (S_K) & = \dom (S) \dotplus \ker (S^*), \lb{SK} \\ \dom (S^*) & = \dom (S) \dotplus (S_F)^{-1} \ker (S^*) \dotplus \ker (S^*) \no \\ & = \dom (S_F) \dotplus \ker (S^*), \lb{S*} \end{align} in particular, \begin{equation} \label{Fr-4Tf} \ker(S_K)= \ker\big((S_K)^{1/2}\big)= \ker(S^*) = \ran(S)^{\bot}. \end{equation} \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We also note that \begin{align} S_F u & = S^* u, \quad u \in \dom(S_F), \\ S_K v & = S^* v, \quad v \in \dom(S_K). \end{align} Here the operator inequalities in \eqref{Fr-Sa} are understood in the sense of \eqref{AleqB} and they can equivalently be written as \begin{equation} (S_F + a I_{\cH})^{-1} \le \big(\wti S + a I_{\cH}\big)^{-1} \le (S_K + a I_{\cH})^{-1} \, \text{ for some (and hence for all) $a > 0$.} \lb{Res} \end{equation} For classical references on the subject of self-adjoint extensions of semibounded operators (not necessarily restricted to the Krein--von Neumann extension) we refer to Birman \cite{Bi56}, \cite{Bi08}, Friedrichs \cite{Fr34}, Freudenthal \cite{Fr36}, Grubb \cite{Gr68}, \cite{Gr70}, Krein \cite{Kr47a}, {\u S}traus \cite{St73}, and Vi{\u s}ik \cite{Vi63} (see also the monographs by Akhiezer and Glazman \cite[Sect. 109]{AG81a}, Faris \cite[Part III]{Fa75}, Fukushima, Oshima, and Takeda \cite[Sect.\ 3.3]{FOT94}, and the recent book by Grubb \cite[Sect.\ 13.2]{Gr09}). We will call the operator $S_K$ the {\it Krein--von Neumann extension} of $S$. See \cite{Kr47} and also the discussion in \cite{AS80} and \cite{AN70}. It should be noted that the Krein--von Neumann extension was first considered by von Neumann \cite{Ne29} in 1929 in the case where $S$ is strictly bounded from below, that is, if $S \geq \varepsilon I_{\cH}$ for some $\varepsilon >0$. (His construction appears in the proof of Theorem 42 on pages 102--103.) However, von Neumann did not isolate the extremal property of this extension as described in \eqref{Fr-Sa} and \eqref{Res}. M.\ Krein \cite{Kr47}, \cite{Kr47a} was the first to systematically treat the general case $S\geq 0$ and to study all nonnegative self-adjoint extensions of $S$, illustrating the special role of the {\it Friedrichs extension} (i.e., the ``hard'' extension) $S_F$ of $S$ and the Krein--von Neumann (i.e., the ``soft'') extension $S_K$ of $S$ as extremal cases when considering all nonnegative extensions of $S$. For a recent exhaustive treatment of self-adjoint extensions of semibounded operators we refer to \cite{AT02}--\cite{AT09}. For convenience of the reader we also mention the following intrinsic description of the Friedrichs extension $S_F$ of $S\geq 0$ ($S$ densely defined and closed in $\cH$) due to Freudenthal \cite{Fr36}, \begin{align} & S_F u:=S^*u, \no \\ & u \in \dom(S_F):=\big\{v\in\dom(S^*)\,\big|\, \mbox{there exists} \, \{v_j\}_{j\in\bbN}\subset \dom(S), \label{Fr-2} \\ & \quad \mbox{with} \, \lim_{j\to\infty}\|v_j-v\|_{\cH}=0 \mbox{ and } ((v_j-v_k),S(v_j-v_k))_\cH\to 0 \mbox{ as } j,k\to\infty\big\}, \no \end{align} and an intrinsic description of the Krein--von Neumann extension $S_K$ of $S\geq 0$ due to Ando and Nishio \cite{AN70}, \begin{align} & S_Ku:=S^*u, \no \\ & u \in \dom(S_K):=\big\{v\in\dom(S^*)\,\big|\,\mbox{there exists} \, \{v_j\}_{j\in\bbN}\subset \dom(S), \label{Fr-2X} \\ & \quad \mbox{with} \, \lim_{j\to\infty} \|Sv_j-S^*v\|_{\cH}=0 \mbox{ and } ((v_j-v_k),S(v_j-v_k)_\cH\to 0 \mbox{ as } j,k\to\infty\big\}. \no \end{align} Throughout the rest of this paper we make the following assumptions: %%%%%%%%%%%%% \begin{hypothesis} \lb{h2.2} Suppose that $S$ is a densely defined, symmetric, closed operator with nonzero deficiency indices in $\cH$ that satisfies \begin{equation} S\geq \varepsilon I_{\cH} \, \text{ for some $\varepsilon >0$.} \lb{3.1} \end{equation} \end{hypothesis} %%%%%%%%%%%%% We recall that the {\it reduced Krein--von Neumann operator} $\hatt S_K$ in the Hilbert space $\hatt \cH$ (cf.\ \eqref{Fr-4Tf}), \begin{equation} \hatt \cH = [\ker (S^*)]^{\bot} = \big[I_{\cH} - P_{\ker(S^*)}\big] \cH = \big[I_{\cH} - P_{\ker(S_K)}\big] \cH = [\ker (S_K)]^{\bot}, \lb{hattH} \end{equation} is given by \begin{align} \label{Barr-4} \hatt S_K:&=S_K|_{[\ker(S_K)]^{\bot}} \\ \begin{split} & = S_K[I_{\cH} - P_{\ker(S_K)}] \, \text{ in $\hatt \cH$} \lb{SKP} \\ &= [I_{\cH} - P_{\ker(S_K)}]S_K[I_{\cH} - P_{\ker(S_K)}] \, \text{ in $\hatt \cH$}, \end{split} \end{align} where $P_{\cM}$ denotes the orthogonal projection onto the subspace $\cM \subset \cH$, and we are alluding to the orthogonal direct sum decomposition of $\cH$ into \begin{equation} \cH = P_{\ker(S_K)}\cH \oplus \hatt \cH = \ker(S_K) \oplus [\ker(S_K)]^\bot. \end{equation} We continue with the following elementary observation: %%%%%%%%%%% \begin{lemma} \lb{l3.1a} Assume Hypothesis \ref{h2.2} and let $v\in\dom(S_K)$. Then the decomposition, $\dom(S_K)= \dom(S) \dotplus \ker(S^*)$ $($cf.\ \eqref{SK}$)$, leads to the following decomposition of $v$, \begin{equation} v= (S_F)^{-1} S_K v + w, \, \text{ where $(S_F)^{-1} S_K v \in \dom(S)$ and $w \in \ker(S^*)$.} \lb{3.1aa} \end{equation} As a consequence, \begin{equation} \big(\hatt S_K\big)^{-1} = [I_{\cH} - P_{\ker(S_K)}] (S_F)^{-1} [I_{\cH} - P_{\ker(S_K)}]. \lb{SKinv} \end{equation} \end{lemma} %%%%%%%%%%% \begin{proof} Let $v = u + w$, with $u \in \dom(S)$ and $w \in \ker(S^*)$. Then \begin{align} v & = u + w = (S_F)^{-1} S_F u + w = (S_F)^{-1} S u + w \no \\ & = (S_F)^{-1} S_K u +w = (S_F)^{-1} S_K (u + w) +w \no \\ & = (S_F)^{-1} S_K v +w \lb{3.1a} \end{align} proves \eqref{3.1aa}. Given $v\in\dom(S_K)$, one infers \begin{equation} S_K v = S_K (P_{\ker(S_K)} + P_{\hatt\cH})v = S_K P_{\hatt\cH} v, \end{equation} since $S_K P_{\ker(S_K)}=0$. In particular, \begin{equation} P_{\hatt \cH} v \in \dom (S_K) \, \text{ whenever } \, v\in\dom(S_K). \end{equation} Applying $P_{\hatt \cH}$ to \eqref{3.1aa} then yields \begin{align} P_{\hatt \cH} v & = P_{\hatt \cH} (S_F)^{-1} S_K [P_{\hatt \cH} + P_{\ker(S_K)}] v = P_{\hatt \cH} (S_F)^{-1} S_K P_{\hatt \cH} v = P_{\hatt \cH} (S_F)^{-1} \hatt S_K P_{\hatt \cH} v \no \\ & = P_{\hatt \cH} (S_F)^{-1} P_{\hatt \cH} \hatt S_K P_{\hatt \cH} v, \quad v\in\dom(S_K). \end{align} Thus, \begin{equation} \big(\hatt S_K\big)^{-1} \big( \hatt S_K P_{\hatt \cH} v\big) = P_{\hatt \cH} (S_F)^{-1} P_{\hatt \cH} \big(\hatt S_K P_{\hatt \cH} v\big), \quad v\in\dom(S_K). \lb{3.1bb} \end{equation} Since $\ran\big(\hatt S_K\big) = \hatt \cH$, \eqref{3.1bb} proves \eqref{SKinv}. \end{proof} %%%%%%%%%%% We note that equation \eqref{SKinv} was proved by Krein in his seminal paper \cite{Kr47} (cf.\ the proof of Theorem\ 26 in \cite{Kr47}). Next, we consider a self-adjoint operator \begin{equation} \label{Barr-1} T:\dom(T)\subseteq \cH\to\cH,\quad T=T^*, \end{equation} which is bounded from below, that is, there exists $\alpha\in\bbR$ such that \begin{equation} \label{Barr-2} T\geq \alpha I_{\cH}. \end{equation} We denote by $\{E_T(\lambda)\}_{\lambda\in\bbR}$ the family of strongly right-continuous spectral projections of $T$, and introduce, as usual, $E_T((a,b))=E_T(b_-) - E_T(a)$, $E_T(b_-) = \slim_{\varepsilon\downarrow 0}E_T(b-\varepsilon)$, $-\infty \leq a < b$. In addition, we set \begin{equation} \label{Barr-3} \mu_{T,j}:=\inf\,\bigl\{\lambda\in\bbR\,|\, \dim (\ran (E_T((-\infty,\lambda)))) \geq j\bigr\},\quad j\in\bbN. \end{equation} Then, for fixed $k\in\bbN$, either: \\ $(i)$ $\mu_{T,k}$ is the $k$th eigenvalue of $T$ counting multiplicity below the bottom of the essential spectrum, $\sigma_{\rm ess}(T)$, of $T$, \\ or \\ $(ii)$ $\mu_{T,k}$ is the bottom of the essential spectrum of $T$, \begin{equation} \mu_{T,k} = \inf \{\lambda \in \bbR \,|\, \lambda \in \sigma_{\rm ess}(T)\}, \end{equation} and in that case $\mu_{T,k+\ell} = \mu_{T,k}$, $\ell\in\bbN$, and there are at most $k-1$ eigenvalues (counting multiplicity) of $T$ below $\mu_{T,k}$. We now record the following basic result: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem} \lb{tKAS} Assume Hypothesis \ref{h2.2}. Then, \begin{equation}\label{Barr-5} \varepsilon \leq \mu_{S_F,j} \leq \mu_{\hatt S_K,j}, \quad j\in\bbN. \end{equation} In particular, if the Friedrichs extension $S_F$ of $S$ has purely discrete spectrum, then, except possibly for $\lambda=0$, the Krein--von Neumann extension $S_K$ of $S$ also has purely discrete spectrum in $(0,\infty)$, that is, \begin{equation} \sigma_{\rm ess}(S_F) = \emptyset \, \text{ implies } \, \sigma_{\rm ess}(S_K) \backslash\{0\} = \emptyset. \lb{ESSK} \end{equation} In addition, let $p\in (0,\infty)\cup\{\infty\}$, then \begin{align} \begin{split} & (S_F - z_0 I_{\cH})^{-1} \in \cB_p(\cH) \, \text{ for some $z_0\in \bbC\backslash [\varepsilon,\infty)$} \\ & \text{implies } \, (S_K - zI_{\cH})^{-1}[I_{\cH} - P_{\ker(S_K)}] \in \cB_p(\cH) \, \text{ for all $z\in \bbC\backslash [\varepsilon,\infty)$}. \lb{CPK} \end{split} \end{align} In fact, the $\ell^p(\bbN)$-based trace ideals $\cB_p(\cH)$ of $\cB(\cH)$ can be replaced by any two-sided symmetrically normed ideals of $\cB(\cH)$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proof} Denote by $\cM_j$ subspaces of $\cH$ of dimension $j\in\bbN$, and similarly, $\hatt \cM_j$ subspaces of $\hatt \cH$ of dimension $j\in\bbN$. Then the inequalities \eqref{Barr-5} follow from $S_F \geq \varepsilon I_{\cH}$, \eqref{SKinv}, and the minimax (better, maximin) theorem as follows: First we note that (cf., e.g., \cite[Theorem\ 5.28]{GS06}, \cite[Sect.\ 32]{He86}) \begin{equation} \f{1}{\mu_{S_F,j}} = \sup_{\cM_j \subset \cH} \min_{\substack{u \in \cM_j\\ \|u\|_{\cH}=1}} \big(u, (S_F)^{-1} u\big)_{\cH}, \quad j\in\bbN. \end{equation} As a consequence, \begin{equation} \f{1}{\mu_{S_F,j}} \geq \min_{u \in \cM_j \subset \cH} \big(u, (S_F)^{-1} u\big)_{\cH}, \quad j\in\bbN, \end{equation} for any subspace $\cM_j$ of $\cH$ of dimension $j\in\bbN$. In particular, \begin{align} \f{1}{\mu_{S_F,j}} &\geq \min_{\substack{v \in \hatt \cM_j \subset \hatt\cH\\ \|v\|_{\hatt \cH}=1}} \big(v, (S_F)^{-1} v\big)_{\hatt \cH} \no \\ & = \min_{\substack{v \in \hatt \cM_j \subset \hatt\cH\\ \|v\|_{\hatt \cH}=1}} \big(v, P_{\hatt \cH} (S_F)^{-1} P_{\hatt \cH}v\big)_{\hatt \cH}, \quad j\in\bbN, \end{align} for any subspace $\hatt \cM_j$ of $\hatt \cH$ of dimension $j\in\bbN$. Thus, one concludes \begin{align} \f{1}{\mu_{S_F,j}} & \geq \sup_{\hatt \cM_j \subset \hatt \cH} \; \min_{\substack{v \in \hatt \cM_j\\ \|v\|_{\hatt \cH}=1}} \big(v, P_{\hatt \cH} (S_F)^{-1} P_{\hatt \cH}v\big)_{\hatt \cH} \no \\ & = \sup_{\hatt \cM_j \subset \hatt \cH} \; \min_{\substack{v \in \hatt \cM_j\\ \|v\|_{\hatt \cH}=1}} \big(v, \big(\hatt S_K\big)^{-1} v\big)_{\hatt \cH} \no \\ & = \f{1}{\mu_{\hatt S_K,j}}, \quad j\in\bbN. \end{align} Next, let $\cJ(\cH)$ be a two-sided symmetrically normed ideal of $\cB(\cH)$. Temporarily, we will identify operators of the type $P_{\hatt\cH} TP_{\hatt\cH}$ in $\hatt \cH$ for $T \in \cB(\cH)$, with $2 \times 2$ block operators of the type \begin{equation} \begin{pmatrix} 0 & 0 \\ 0 & P_{\hatt\cH} TP_{\hatt\cH}|_{\hatt \cH} \end{pmatrix} \, \text{ in } \, \cH = (\ker(S_K))^{\bot} \oplus \hatt \cH. \end{equation} By \eqref{SKinv}, and since $P_{\hatt\cH}$ is bounded, one concludes that $(S_F)^{-1} \in \cJ(\cH)$ implies $\big(\hatt S_K)^{-1} = \nlim_{z\to 0}(S_K - zI_{\cH})^{-1}[I_{\cH} - P_{\ker(S_K)}] \in \cJ(\cH)$. The (first) resolvent equation applied to $S_F$, and subsequently, applied to $S_K$, then proves \eqref{CPK}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We note that \eqref{ESSK} is a classical result of Krein \cite{Kr47}, the more general fact \eqref{Barr-5} has not been mentioned explicitly in Krein's paper \cite{Kr47}, although it immediately follows from the minimax principle and Krein's formula \eqref{SKinv}. On the other hand, in the special case ${\rm def}(S)<\infty$, Krein states an extension of \eqref{Barr-5} in his Remark 8.1 in the sense that he also considers self-adjoint extensions different from the Krein extension. Apparently, \eqref{Barr-5} has first been proven by Alonso and Simon \cite{AS80} by a somewhat different method. Concluding this section, we point out that a great variety of additional results for the Krein--von Neumann extension can be found in the very extensive list of references in \cite{AT09} and \cite{AGMT09}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Krein--von Neumann Extension and its Unitary Equivalence to an Abstract Buckling Problem} \label{s3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we prove our principal result, the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, operator $S$ satisfying $S\geq \varepsilon I_{\cH}$ for some $\varepsilon >0$, in a complex separable Hilbert space $\cH$ to an abstract buckling problem operator. We start by introducing an abstract version of Proposition\ 1 in Grubb's paper \cite{Gr83} devoted to Krein--von Neumann extensions of even order elliptic differential operators on bounded domains: %%%%%%%%%%% \begin{lemma} \lb{l3.3} Assume Hypothesis \ref{h2.2} and let $\lambda \neq 0$. Then there exists $0 \neq v \in \dom(S_K)$ with \begin{equation} S_K v = \lambda v \lb{3.1b} \end{equation} if and only if there exists $0 \neq u \in \dom(S^* S)$ such that \begin{equation} S^* S u = \lambda S u. \lb{3.1c} \end{equation} In particular, the solutions $v$ of \eqref{3.1b} are in one-to-one correspondence with the solutions $u$ of \eqref{3.1c} given by the formulas \begin{align} u & = (S_F)^{-1} S_K v, \lb{3.1d} \\ v & = \lambda^{-1} S u. \lb{3.1e} \end{align} Of course, since $S_K \geq 0$, any $\lambda \neq 0$ in \eqref{3.1b} and \eqref{3.1c} necessarily satisfies $\lambda > 0$. \end{lemma} %%%%%%%%%%% \begin{proof} Let $S_K v = \lambda v$, $v\in\dom(S_K)$, $\lambda \neq 0$, and $v = u + w$, with $u \in \dom(S)$ and $w \in \ker(S^*)$. Then, \begin{equation} S_K v = \lambda v \Longleftrightarrow v = \lambda^{-1} S_K v = \lambda^{-1} S_K u = \lambda^{-1} S u. \end{equation} Moreover, $u =0$ implies $v = 0$ and clearly $v=0$ implies $u=w=0$, hence $v \neq 0$ if and only if $u \neq 0$. In addition, $u = (S_F)^{-1} S_K v$ by \eqref{3.1aa}. Finally, \begin{align} \begin{split} & \lambda w = S u - \lambda u \in \ker(S^*) \, \text{ implies} \\ & 0 = \lambda S^* w = S^*(S u - \lambda u) = S^* S u - \lambda S^* u = S^* S u - \lambda S u. \end{split} \end{align} Conversely, suppose $u \in \dom(S^* S)$ and $S^* S u = \lambda S u$, $\lambda \neq 0$. Introducing $v = \lambda^{-1} S u$, then $v \in \dom(S^*)$ and \begin{equation} S^* v = \lambda^{-1} S^* S u = S u = \lambda v. \end{equation} Noticing that \begin{equation} S^* S u = \lambda S u = \lambda S^* u \, \text{ implies } \, S^*(S-\lambda I_{\cH}) u =0, \end{equation} and hence $(S - \lambda I_{\cH}) u \in \ker(S^*)$, rewriting $v$ as \begin{equation} v = u + \lambda^{-1} (S - \lambda I_{\cH}) u \end{equation} then proves that also $v \in \dom(S_K)$, using \eqref{SK} again. \end{proof} %%%%%%%%%%% Due to Example \ref{e3.6} and Remark \ref{r3.7} at the end of this section, we will call the linear pencil eigenvalue problem $S^* Su = \lambda S u$ in \eqref{3.1c} the {\it abstract buckling problem} associated with the Krein--von Neumann extension $S_K$ of $S$. Next, we turn to a variational formulation of the correspondence between the inverse of the reduced Krein extension $\hatt S_K$ and the abstract buckling problem in terms of appropriate sesquilinear forms by following the treatment of Kozlov \cite{Ko79}--\cite{Ko84} in the context of elliptic partial differential operators. This will then lead to an even stronger connection between the Krein--von Neumann extension $S_K$ of $S$ and the associated abstract buckling eigenvalue problem \eqref{3.1c}, culminating in a unitary equivalence result in Theorem \ref{t3.3}. Given the operator $S$, we introduce the following sesquilinear forms in $\cH$, \begin{align} a(u,v) & = (Su,Sv)_{\cH}, \quad u, v \in \dom(a) = \dom(S), \lb{3.2} \\ b(u,v) & = (u,Sv)_{\cH}, \quad u, v \in \dom(b) = \dom(S). \lb{3.3} \end{align} Then $S$ being densely defined and closed implies that the sesquilinear form $a$ shares these properties and \eqref{3.1} implies its boundedness from below, \begin{equation} a(u,u) \geq \varepsilon^2 \|u\|_{\cH}^2, \quad u \in \dom(S). \lb{3.4} \end{equation} Thus, one can introduce the Hilbert space $\cW=(\dom(S), (\cdot,\cdot)_{\cW})$ with associated scalar product \begin{equation} (u,v)_{\cW}=a(u,v) = (Su,Sv)_{\cH}, \quad u, v \in \dom(S). \lb{3.5} \end{equation} In addition, we denote by $\iota_{\cW}$ the continuous embedding operator of $\cW$ into $\cH$, \begin{equation} \iota_{\cW} : \cW \hookrightarrow \cH. \lb{3.6} \end{equation} Hence we will use the notation \begin{equation} (w_1,w_2)_{\cW} =a(\iota_{\cW} w_1,\iota_{\cW} w_2) = (S\iota_{\cW} w_1, S\iota_{\cW} w_2)_{\cH}, \quad w_1, w_2 \in \cW, \lb{3.7} \end{equation} in the following. Given the sesquilinear forms $a$ and $b$ and the Hilbert space $\cW$, we next define the operator $T$ in $\cW$ by \begin{align} \begin{split} (w_1,T w_2)_{\cW} & = a(\iota_{\cW} w_1,\iota_{\cW} T w_2) = (S \iota_{\cW} w_1,S\iota_{\cW} T w_2)_{\cH} \\ & = b(\iota_{\cW} w_1,\iota_{\cW} w_2) = (\iota_{\cW} w_1,S \iota_{\cW} w_2)_{\cH}, \quad w_1, w_2 \in \cW. \lb{3.8} \end{split} \end{align} (In contrast to the informality of our introduction, we now explicitly write the embedding operator $\iota_{\cW}$.) One verifies that $T$ is well-defined and that \begin{equation} |(w_1,T w_2)_{\cW}| \leq \|\iota_{\cW} w_1\|_{\cH} \|S \iota_{\cW} w_2\|_{\cH} \leq \varepsilon^{-1} \|w_1\|_{\cW} \|w_2\|_{\cW}, \quad w_1, w_2 \in \cW, \lb{3.9} \end{equation} and hence that \begin{equation} 0 \leq T = T^* \in \cB(\cW), \quad \|T\|_{\cB(\cW)} \leq \varepsilon^{-1}. \lb{3.10} \end{equation} For reasons to become clear at the end of this section, we will call $T$ the {\it abstract buckling problem operator} associated with the Krein--von Neumann extension $S_K$ of $S$. Next, recalling the notation $\hatt \cH = [\ker (S^*)]^{\bot} = \big[I_{\cH} - P_{\ker(S^*)}\big] \cH$ (cf.\ \eqref{hattH}), we introduce the operator \begin{equation} \hatt S: \begin{cases} \cW \to \hatt \cH, \\ w \mapsto S \iota_{\cW} w, \end{cases} \lb{3.12} \end{equation} and note that \begin{equation} \ran\big(\hatt S\big) = \ran (S) = \hatt \cH, \lb{3.12aa} \end{equation} since $S\geq \varepsilon I_{\cH}$ for some $\varepsilon > 0$ and $S$ is closed in $\cH$ (see, e.g., \cite[Theorem\ 5.32]{We80}). In fact, one has the following result: %%%%%%%%%%%%% \begin{lemma} \lb{l3.1} Assume Hypothesis \ref{h2.2}. Then $\hatt S\in\cB(\cW,\hatt \cH)$ maps $\cW$ unitarily onto $\hatt \cH$. \end{lemma} %%%%%%%%%%%%% \begin{proof} Clearly $\hatt S$ is an isometry since \begin{equation} \big\|\hatt S w\big\|_{\hatt\cH} = \|S \iota_{\cW} w\big\|_{\cH} = \|w\|_{\cW}, \quad w \in \cW. \lb{3.13} \end{equation} Since $\ran\big(\hatt S\big) = \hatt \cH$ by \eqref{3.12aa}, $\hatt S$ is unitary. \end{proof} %%%%%%%%%%%%% Next we recall the definition of the reduced Krein--von Neumann operator $\hatt S_K$ in $\hatt \cH$ defined in \eqref{SKP}, the fact that $\ker(S^*) = \ker(S_K)$ by \eqref{Fr-4Tf}, and state the following auxiliary result: %%%%%%%%%%%%% \begin{lemma} \lb{l3.2} Assume Hypothesis \ref{h2.2}. Then the map \begin{equation} \big[I_{\cH} - P_{\ker(S^*)}\big] : \dom(S) \to \dom \big(\hatt S_K\big) \lb{3.15} \end{equation} is a bijection. In addition, we note that \begin{align} \begin{split} & \big[I_{\cH} - P_{\ker(S^*)}\big] S_K u = S_K \big[I_{\cH} - P_{\ker(S^*)}\big] u = \hatt S_K \big[I_{\cH} - P_{\ker(S^*)}\big] u \\ & \quad = \big[I_{\cH} - P_{\ker(S^*)}\big] S u = Su \in \hatt \cH, \quad u \in \dom(S). \lb{3.16} \end{split} \end{align} \end{lemma} %%%%%%%%%%%%% \begin{proof} Let $u\in\dom(S)$, then $\ker(S^*) = \ker(S_K)$ implies that $\big[I_{\cH} - P_{\ker(S^*)}\big] u \in \dom(S_K)$ and of course $\big[I_{\cH} - P_{\ker(S^*)}\big] u \in \dom\big(\hatt S_K\big)$. To prove injectivity of the map \eqref{3.15} it suffices to assume $v \in \dom(S)$ and $\big[I_{\cH} - P_{\ker(S^*)}\big] v =0$. Then $\dom(S) \ni v = P_{\ker(S^*)} v \in \ker(S^*)$ yields $v=0$ as $\dom(S) \cap \ker(S^*) = \{0\}$. To prove surjectivity of the map \eqref{3.15} we suppose $u \in \dom\big(\hatt S_K)$. The decomposition, $u = f +g$ with $f \in \dom(S)$ and $g \in \ker(S^*)$, then yields \begin{equation} u = \big[I_{\cH} - P_{\ker(S^*)}\big] u = \big[I_{\cH} - P_{\ker(S^*)}\big] f \in \big[I_{\cH} - P_{\ker(S^*)}\big] \dom(S) \lb{3.18} \end{equation} and hence proves surjectivity of \eqref{3.15}. Equation \eqref{3.16} is clear from \begin{equation} S_K \big[I_{\cH} - P_{\ker(S^*)}\big] = \big[I_{\cH} - P_{\ker(S^*)}\big] S_K = \big[I_{\cH} - P_{\ker(S^*)}\big] S_K \big[I_{\cH} - P_{\ker(S^*)}\big]. \lb{3.19} \end{equation} \end{proof} %%%%%%%%%%%%% Continuing, we briefly recall the polar decomposition of $S$, \begin{equation} S = U_S |S|, \lb{3.19a} \end{equation} with \begin{equation} |S| = (S^* S)^{1/2} \geq \varepsilon I_{\cH}, \; \varepsilon > 0, \quad U_S \in \cB\big(\cH,\hatt \cH\big) \, \text{ is unitary.} \lb{3.19b} \end{equation} At this point we are in position to state our principal unitary equivalence result: %%%%%%%%%%%%% \begin{theorem} \lb{t3.3} Assume Hypothesis \ref{h2.2}. Then the inverse of the reduced Krein--von Neumann extension $\hatt S_K$ in $\hatt \cH = \big[I_{\cH} - P_{\ker(S^*)}\big] \cH$ and the abstract buckling problem operator $T$ in $\cW$ are unitarily equivalent, in particular, \begin{equation} \big(\hatt S_K\big)^{-1} = \hatt S T (\hatt S)^{-1}. \lb{3.20} \end{equation} Moreover, one has \begin{equation} \big(\hatt S_K\big)^{-1} = U_S \big[|S|^{-1} S |S|^{-1}\big] (U_S)^{-1}, \lb{3.20a} \end{equation} where $U_S\in \cB\big(\cH,\hatt \cH\big)$ is the unitary operator in the polar decomposition \eqref{3.19a} of $S$ and the operator $|S|^{-1} S |S|^{-1}\in\cB(\cH)$ is self-adjoint in $\cH$. \end{theorem} %%%%%%%%%%%%% \begin{proof} Let $w_1, w_2 \in \cW$. Then, \begin{align} & \big(w_1,\big(\hatt S\big)^{-1} \big(\hatt S_K\big)^{-1} \hatt S w_2\big)_{\cW} = \big(\hatt S w_1, \big(\hatt S_K\big)^{-1} \hatt S w_2\big)_{\hatt \cH} \no \\ & \quad = \big( \big(\hatt S_K\big)^{-1} \hatt S w_1, \hatt S w_2\big)_{\hatt \cH} = \big( \big(\hatt S_K\big)^{-1} S \iota_{\cW} w_1, \hatt S w_2\big)_{\hatt \cH} \no \\ & \quad = \big( \big(\hatt S_K\big)^{-1} \big[I_{\cH} - P_{\ker(S^*)}\big] S \iota_{\cW} w_1, \hatt S w_2\big)_{\hatt \cH} \quad \text{by \eqref{3.16}} \no \\ & \quad = \big( \big(\hatt S_K\big)^{-1} \hatt S_K \big[I_{\cH} - P_{\ker(S^*)}\big] \iota_{\cW} w_1, \hatt S w_2\big)_{\hatt \cH} \quad \text{again by \eqref{3.16}} \no \\ & \quad = \big(\big[I_{\cH} - P_{\ker(S^*)}\big] \iota_{\cW} w_1, \hatt S w_2\big)_{\hatt \cH} \no \\ & \quad = \big(\iota_{\cW} w_1, S \iota_{\cW} w_2\big)_{\cH} \no \\ & \quad = \big(w_1, T w_2\big)_{\cW} \quad \text{by definition of $T$ in \eqref{3.8},} \lb{3.22} \end{align} yields \eqref{3.20}. In addition one verifies that \begin{align} \big(\hatt S w_1, \big(\hatt S_K\big)^{-1} \hatt S w_2\big)_{\hatt \cH} & = \big(w_1, T w_2\big)_{\cW} \no \\ & = \big(\iota_{\cW} w_1, S \iota_{\cW} w_2\big)_{\cH} \no \\ & = \big(|S|^{-1} |S| \iota_{\cW} w_1, S |S|^{-1} |S| \iota_{\cW} w_2\big)_{\cH} \no \\ & = \big(|S| \iota_{\cW} w_1, \big[|S|^{-1} S |S|^{-1}\big] |S| \iota_{\cW} w_2\big)_{\cH} \no \\ & = \big((U_S)^* S \iota_{\cW} w_1, \big[|S|^{-1} S |S|^{-1}\big] (U_S)^* S \iota_{\cW} w_2\big)_{\cH} \no \\ & = \big(S \iota_{\cW} w_1, U_S \big[|S|^{-1} S |S|^{-1}\big] (U_S)^* S \iota_{\cW} w_2\big)_{\cH} \no \\ & = \big(\hatt S w_1, U_S \big[|S|^{-1} S |S|^{-1}\big] (U_S)^* \hatt S w_2\big)_{\hatt \cH} \, , \lb{3.37} \end{align} where we used $|S|=(U_S)^* S$. \end{proof} %%%%%%%%%%%%% Equation \eqref{3.20a} is of course motivated by rewriting the abstract linear pencil buckling eigenvalue problem \eqref{3.1c}, $S^* S u = \lambda S u$, $\lambda \neq 0$, in the form \begin{equation} \lambda^{-1} S^* S u = \lambda^{-1} (S^* S)^{1/2} \big[(S^* S)^{1/2} u\big] = S (S^* S)^{-1/2} \big[(S^* S)^{1/2} u\big] \lb{3.38} \end{equation} and hence in the form of a standard eigenvalue problem \begin{equation} |S|^{-1} S |S|^{-1} w = \lambda^{-1} w, \quad \lambda \neq 0, \quad w = |S| u. \lb{3.39} \end{equation} We conclude this section with a concrete example discussed explicitly in Grubb \cite{Gr83} (see also \cite{Gr68}--\cite{Gr71} for necessary background) and make the explicit connection with the buckling problem. It was this example which greatly motivated the abstract results in this note: %%%%%%%%%%%%% \begin{example} \lb{e3.6} $($\cite{Gr83}.$)$ Let $\cH = L^2(\Om; d^n x)$, with $\Om \subset\bbR^n$, $n \geq 2$, open and bounded, with a smooth boundary $\partial\Om$, and consider the minimal operator realization $S$ of the differential expression $\mathscr{S}$ in $L^{2}(\Om; d^n x)$, defined by \begin{align} & S u = \mathscr{S} u, \lb{3.40} \\ & u \in \dom(S) = H_0^{2m}(\Om) = \big\{v \in H^{2m}(\Om) \, \big| \, \gamma_{k} v =0, \, 0 \leq k \leq 2m-1\big\}, \quad m \in\bbN, \no \end{align} where \begin{align} & \mathscr{S} = \sum_{0 \leq |\alpha| \leq 2m} a_{\alpha}(\cdot) D^{\alpha}, \lb{3.41} \\ & D^{\alpha} = (-i \partial/\partial x_1)^{\alpha_1} \cdots (-i\partial/\partial x_n)^{\alpha_n}, \quad \alpha =(\alpha_1,\dots,\alpha_n) \in \bbN_0^n, \lb{3.42} \\ & a_{\alpha} (\cdot) \in C^\infty(\ol \Om), \quad C^\infty(\ol \Om) = \bigcap_{k\in\bbN_0} C^k(\ol \Om), \lb{3.43} \end{align} and the coefficients $a_\alpha$ are chosen such that $S$ is symmetric in $L^2(\bbR^n; d^n x)$, that is, the differential expression $\mathscr{S}$ is formally self-adjoint, \begin{equation} (\mathscr{S} u, v)_{L^2(\bbR^n; d^n x)} = (u, \mathscr{S} v)_{L^2(\bbR^n; d^n x)}, \quad u, v \in C_0^\infty(\Om), \lb{3.44} \end{equation} and $\mathscr{S}$ is strongly elliptic, that is, for some $c>0$, \begin{equation} \Re\bigg(\sum_{|\alpha|=2m} a_{\alpha} (x) \xi^{\alpha}\bigg) \geq c |\xi|^{2m}, \quad x \in \ol \Om, \; \xi \in\bbR^n. \lb{3.45} \end{equation} In addition we assume that $S\geq \varepsilon I_{L^{2}(\Om; d^n x)}$ for some $\varepsilon >0$.\ The trace operators $\gamma_k$ are defined as follows: Consider \begin{equation} \mathring{\gamma}_k : \begin{cases} C^\infty(\ol \Om) \to C^\infty(\partial \Om) \\ u \mapsto (\partial^k_n u)|_{\partial\Om}, \end{cases} \lb{3.46} \end{equation} with $\partial_n$ denoting the interior normal derivative. The map $\mathring{\gamma}$ then extends by continuity to a bounded operator \begin{equation} \gamma_k : H^s(\Om) \to H^{s-k - (1/2)}(\partial\Om), \quad s > k + (1/2), \lb{3.47} \end{equation} in addition, the map \begin{equation} \gamma^{(r)} = (\gamma_0,\dots,\gamma_r) : H^s(\Om) \to \prod_{k=0}^r H^{s-k-(1/2)}(\partial\Om), \quad s > r +(1/2), \lb{3.48} \end{equation} satisfies \begin{equation} \ker\big(\gamma^{(r)}\big) = H_0^s(\Om), \quad \ran\big(\gamma^{(r)}\big) = \prod_{k=0}^r H^{s-k-(1/2)}(\partial\Om). \lb{3.49} \end{equation} Then $S^*$, the maximal operator realization of $\mathscr{S}$ in $L^{2}(\Om; d^n x)$, is given by \begin{equation} S^* u = \mathscr{S} u, \quad u \in \dom(S^*) = \big\{v\in L^{2}(\Om; d^n x) \,\big|\, \mathscr{S} v\in L^{2}(\Om; d^n x)\big\}, \lb{3.50} \end{equation} and $S_F$ is characterized by \begin{equation} S_F u = \mathscr{S} u, \quad u \in \dom(S_F) = \big\{v \in H^{2m}(\Om) \, \big| \, \gamma_k v =0, \, 0 \leq k \leq m-1\big\}. \lb{3.51} \end{equation} The Krein--von Neumann extension $S_K$ of $S$ then has the domain \begin{equation} \dom(S_K) = H_0^{2m}(\Om) \dotplus \ker(S^*), \quad \dim(\ker(S^*)) = \infty, \lb{3.52} \end{equation} and elements $u \in \dom(S_K)$ satisfy the nonlocal boundary condition \begin{align} & \gamma_N u - P_{\gamma_D,\gamma_N} \gamma_D u =0, \lb{3.53} \\ & \gamma_D u = (\gamma_0 u,\dots,\gamma_{m-1} u), \quad \gamma_N u = (\gamma_m u,\dots,\gamma_{2m-1} u), \quad u \in \dom(S_K), \lb{3.54} \end{align} where \begin{align} \begin{split} & P_{\gamma_D, \gamma_N} = \gamma_N \gamma_Z^{-1} : \prod_{k=0}^{m-1} H^{s-k-(1/2)} (\partial\Om) \to \prod_{j=m}^{2m-1} H^{s-j-(1/2)} (\partial\Om) \\ & \quad \text{continuously for all $s\in\bbR$,} \end{split} \end{align} and $\gamma_Z^{-1}$ denotes the inverse of the isomorphism $\gamma_Z$ given by \begin{align} &\gamma_D : Z_{\mathscr{S}}^s \to \prod_{k=0}^{m-1} H^{s-k-(1/2)} (\partial\Om), \\ & Z_{\mathscr{S}}^s = \big\{u\in H^s(\Om) \, \big| \, \mathscr{S} u =0 \, \text{in $\Om$ in the sense of distributions in $\cD^\prime(\Om)$}\big\}, \quad s\in\bbR. \end{align} Moreover one has \begin{equation} \big(\hatt S\big)^{-1} = \iota_{\cW} [I_{\cH} - P_{\gamma_D, \gamma_N} \gamma_D] \big(\hatt S_K\big)^{-1}, \end{equation} since $[I_{\cH} - P_{\gamma_D, \gamma_N} \gamma_D] \dom(S_K) \subseteq \dom(S)$ and $S [I_{\cH} - P_{\gamma_D, \gamma_N} \gamma_D] v = \lambda v$, $v\in\dom(S_K)$. As discussed in detail in Grubb \cite{Gr83}, \begin{equation} \sigma_{\rm ess} (S_K) = \{0\}, \quad \sigma(S_K) \cap (0,\infty) = \sigma_{\rm d} (S_K) \lb{3.55} \end{equation} and the nonzero $($and hence discrete$)$ eigenvalues of $S_K$ satisfy a Weyl-type asymptotics. The connection to a higher-order buckling eigenvalue problem established by Grubb then reads \begin{align} & \text{There exists $0 \neq v \in S_K$ satisfying } \, \mathscr{S} v = \lambda v \, \text{ in } \, \Om, \quad \lambda \neq 0 \lb{3.57} \\ & \text{if and only if} \no \\ & \text{there exists $0 \neq u \in C^\infty (\ol \Om)$ such that } \, \begin{cases} \mathscr{S}^2 u = \lambda \, \mathscr{S} u \, \text{ in } \, \Om, \quad \lambda \neq 0, \\ \gamma_k u = 0, \; 0 \leq k \leq 2m-1, \end{cases} \lb{3.58} \end{align} where the solutions $v$ of \eqref{3.57} are in one-to-one correspondence with the solutions $u$ of \eqref{3.58} via \begin{equation} u = S_F^{-1} \mathscr{S} v, \quad v = \lambda^{-1} \mathscr{S} u. \lb{3.59} \end{equation} \end{example} %%%%%%%%%%%%% Since $S_F$ has purely discrete spectrum in Example \ref{e3.6}, we note that Theorem \ref{tKAS} applies in this case. %%%%%%%%%%%%% \begin{remark} \lb{r3.7} In the particular case $m=1$ and $\mathscr{S} = -\Delta$, the linear pencil eigenvalue problem \eqref{3.58} (i.e., the concrete analog of the abstract buckling eigenvalue problem $S^* S u = \lambda S u$, $\lambda \neq 0$, in \eqref{3.1c}), then yields the {\it buckling of a clamped plate problem}, \begin{equation} (-\Delta)^2u=\lambda (-\Delta) u \,\text{ in } \,\Omega, \quad \lambda \neq 0, \; u\in H^2_0(\Omega), \lb{3.60} \end{equation} as distributions in $H^{-2}(\Om)$. Here we used the fact that for any nonempty bounded open set $\Om\subset\bbR^n$, $n\in\bbN$, $n\geq 2$, $(-\Delta)^m\in \cB\big(H^k(\Om), H^{k-2m}(\Om)\big)$, $k\in\bbZ$, $m\in\bbN$. In addition, if $\Om$ is a Lipschitz domain, then one has that $-\Delta\colon H^1_0(\Om) \to H^{-1}(\Om)$ is an isomorphism and similarly, $(-\Delta)^2\colon H^2_0(\Om) \to H^{-2}(\Om)$ is an isomorphism. (For the natural norms on $H^k(\Om)$, $k\in\bbZ$, see, e.g., \cite[p.\ 73--75]{Mc00}.) We refer, for instance, to \cite[Sect.\ 4.3B]{Be77} for a derivation of \eqref{3.60} from the fourth-order system of quasilinear von K{\'a}rm{\'a}n partial differential equations. To be precise, \eqref{3.60} should also be considered in the special case $n=2$. \end{remark} %%%%%%%%%%%%% %%%%%%%%%%%%% \begin{remark} \lb{r3.8} We emphasize that the smoothness hypotheses on $\partial\Om$ can be relaxed in the special case of the second-order Schr\"odinger operator associated with the differential expression $-\Delta + V$, where $V\in L^\infty(\Om; d^nx)$ is real-valued: Following the treatment of self-adjoint extensions of $S = \ol{(-\Delta + V)|_{C_0^\infty(\Om)}}$ on quasi-convex domains $\Om$ first introduced in \cite{GM09}, the case of the Krein--von Neumann extension $S_K$ of $S$ on such quasi-convex domains (which are close to minimally smooth) is treated in great detail in \cite{AGMT09}. In particular, a Weyl-type asymptotics of the associated (nonzero) eigenvalues of $S_K$ has been proven in \cite{AGMT09}. 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