Krein–von Neumann Extensions

• Krein–von Neumann extensions are canonical constructions that yield the minimal nonnegative self-adjoint extensions of symmetric operators.
• They are defined using the direct sum of the original domain and the kernel of the adjoint, ensuring the 'softest' boundary conditions.
• Their applications span elliptic PDEs, spectral asymptotics, quantum graphs, and various other boundary value problems.

A Krein–von Neumann extension is a canonical construction in the spectral, extension, and boundary-value theory of (unbounded) symmetric and, most often, nonnegative operators. Given a densely defined, closed, symmetric, semibounded operator, the Krein–von Neumann extension produces the “softest” (that is, minimal in the operator order) nonnegative self-adjoint extension, contrasting with the Friedrichs (“hard” or maximal) extension. It is pivotal in the analysis of elliptic PDEs, operator theory, spectral asymptotics, boundary value problems, and the abstract foundation of extension theory. This article surveys its definitions, parametrizations, boundary formulations, spectral properties, extremal characterization, and applications to both continuous and discrete operators, including modern formulations and technical particulars with specific reference to the technical sources in (Grubb, 2010, Ashbaugh et al., 2012, Mugnolo, 2014, Gesztesy et al., 2014, Ashbaugh et al., 2016, Zemánek et al., 2016, Granovskyi et al., 2017, Fucci et al., 2021, Cho et al., 2021, Gesztesy et al., 16 Apr 2025).

1. Abstract Structure and Parametrization of Krein–von Neumann Extensions

Let $S$ be a densely defined, closed, symmetric, nonnegative operator in a Hilbert space $\mathcal{H}$, with nonzero deficiency indices. Two canonical nonnegative self-adjoint extensions are distinguished:

• The Friedrichs extension $S_F$ (maximal in operator order, with quadratic form the closure of the form defined by $S$).
• The Krein–von Neumann extension $S_K$ (minimal in operator order): $S_K\leq \widetilde S \leq S_F$ for every nonnegative self-adjoint extension $\widetilde S$ of $S$ (Gesztesy et al., 16 Apr 2025, Mugnolo, 2014).

The Krein–von Neumann extension is uniquely defined by the domain

$$\operatorname{dom}(S_K) = \operatorname{dom}(S) \oplus \ker(S^*) \,,$$

and satisfies

$\ker(S_K) = \ker(S^*) \,.$

On the orthogonal complement of its kernel, $S_K$ is strictly positive if $S$ is strictly positive.

A general result (abstract Krein-Višik-Birman theory) gives a one-to-one correspondence between nonnegative self-adjoint extensions of $S$ and nonnegative self-adjoint operators $B$ on closed subspaces $W \subset \ker S^*$. The Friedrichs extension corresponds to $W=\{0\}$, $B$ undefined; the Krein–von Neumann extension corresponds to $W=\ker S^*$, $B=0$ (Gesztesy et al., 16 Apr 2025).

For operators with a finite number of negative squares (i.e., symmetric operators with finite negative index), analogues of the Krein–von Neumann and Friedrichs extensions are constructed by imposing a “minimality” condition on the negative index of the extension (Baidiuk et al., 2014).

2. Domains, Boundary Conditions, and Concrete Realizations

For elliptic differential operators, including higher-order and variable-coefficient cases, and discrete analogs:

• Let $A$ be a strongly elliptic operator of even order $2m$, with minimal (Dirichlet-type) realization $A_{\min}$ and maximal (distributional) realization $A_{\max}$ on a domain $\Omega\subset \mathbb{R}^n$.
• The Krein–like extension $A_a$ is characterized by the domain

$$D(A_a) = \{\,u = v + aA_\gamma^{-1}z + z : v\in D(A_{\min}),\, z\in \ker(A_{\max})\,\}$$

where $A_\gamma$ is typically the Friedrichs extension, and $a\in \mathbb{R}$ is a coupling parameter (Grubb, 2010).

• The Krein–von Neumann extension corresponds to $a=0$:

$$D(A_0) = \{\,u = v + z : v\in D(A_{\min}),\, z\in \ker(A_{\max})\,\} \,.$$

• In the framework of pseudodifferential boundary operator calculus, this ansatz reexpresses as a generalized non-elliptic Neumann-type boundary condition:

$$\gamma_1 u = a(y^{z})^* y^z \gamma_0 u + P^{y,x} \gamma_0 u,$$

where $\gamma_0,\gamma_1$ are trace operators, $y^z$ maps $\ker(A_{\max})$ to boundary data, and $P^{y,x}$ is a PDO.

• On quantum graphs or metric graphs, the Krein–von Neumann extension imposes nonlocal vertex conditions, specifically coupling normal derivatives to function values through a discrete Laplacian (or Dirichlet-to-Neumann-type operator) (Muller et al., 2020).

For regular even order quasi-differential and Sturm–Liouville operators, the Krein–von Neumann extension is characterized by boundary conditions derived from the boundary triplet formalism, using (generalized) trace data and explicit coupling matrices such as Toeplitz blocks or those determined by a basis in $\ker(A^*)$ (Granovskyi et al., 2017, Fucci et al., 2021, Cho et al., 2021).

3. Extremal Properties, Operator Ordering, and the Role in Extension Theory

The Krein–von Neumann extension is extremal in the following sense:

• Among all nonnegative self-adjoint extensions, $S_K$ is minimal (soft), $S_F$ is maximal (hard), and every other nonnegative self-adjoint extension $\widetilde S$ satisfies

$S_K \leq \widetilde S \leq S_F$

(operator order), or, equivalently, for all $a>0$,

$$(S_F + a)^{-1} \leq (\widetilde S + a)^{-1} \leq (S_K + a)^{-1} \,.$$

• This structure persists even when negative index (finite number of negative squares) is allowed, provided one restricts to extensions maintaining the minimal negative index (Baidiuk et al., 2014).
• In the context of relations (possibly multi-valued) with orthogonal domain and range, extremality of the Krein–von Neumann (and Friedrichs) extensions is characterized by the property that their domains and ranges are orthogonal (i.e., numerical range $\{0\}$) (Hassi et al., 2019).

4. Spectral Properties and Asymptotic Analysis

Spectral theory of Krein–von Neumann extensions reveals substantial features:

• On bounded domains, the Krein–von Neumann extension possesses a discrete nonzero spectrum with asymptotics matching the classical Weyl law, i.e.,

$$N^+(t;A_a) = C_A\,t^{n/(2m)} + O \Big( t^{(n-1+\delta)/(2m)} \Big),\quad t\to\infty,$$

for all $\delta>0$, where $C_A$ matches the Dirichlet principal term (Grubb, 2010, Ashbaugh et al., 2012, Gesztesy et al., 2014, Ashbaugh et al., 2016).

• The eigenvalue problem for the Krein–von Neumann extension is unitarily equivalent to a fourth-order (buckling-type) eigenvalue problem. This connection is pivotal in elasticity theory (for buckling of plates) and in obtaining sharp eigenvalue (counting) estimates (Ashbaugh et al., 2012, Gesztesy et al., 2014).
• The operator $A_0$ often admits a substantially larger null space than the Friedrichs extension; e.g., on an interval $(0,1)$, $\ker(A_K)$ is two-dimensional (affine functions), and in certain multidimensional or graph settings, it can be infinite-dimensional (Mugnolo, 2014, Muller et al., 2020).
• For variable coefficient, nonsmooth, and higher-order elliptic operators, precise bounds and asymptotics for the eigenvalue counting function are available via variational/buckling arguments and generalized Fourier analysis (Gesztesy et al., 2014, Ashbaugh et al., 2016).

5. Resolvent Formulas, Boundary Triplets, and Trace Ideals

Krein-type resolvent formulas and the boundary triplet formalism underpin much of the functional analysis for these extensions:

• The resolvent of the Krein–von Neumann extension (restricted to orthogonal complement of its kernel) can be represented in terms of the Friedrichs resolvent via

$$(S_K - zI)^{-1} = [I - P_{\ker S_K}] (S_F - zI)^{-1} [I - P_{\ker S_K}]$$

for $z$ in the resolvent set of $S_F$, and likewise for the reduced Krein extension (Gesztesy et al., 16 Apr 2025).

• For self-adjoint extensions $S_1$, $S_2$ which are relatively prime (i.e., have minimal intersection of domains), resolvent differences are expressed using Donoghue-type $M$-operators (energy-dependent Dirichlet-to-Neumann-type maps):

$(S_2 - z I)^{-1} - (S_1 - z I)^{-1}$

involves boundary mappings and $M$-operators on the deficiency subspaces (Gesztesy et al., 16 Apr 2025).

• For boundary triplets $(\mathcal{G}, \Gamma_0, \Gamma_1)$ associated to $S^*$, Krein–von Neumann extension corresponds to $B=0$ (in the parametrization $A_{C,D}$ by $D\Gamma_1 f - C\Gamma_0f=0$), and the boundary condition $\Gamma_1 f = M(0)\Gamma_0 f$ (Granovskyi et al., 2017).
• Schatten and trace ideal properties transfer: if $(S_F - zI)^{-1}$ belongs to $\mathfrak{B}_p$ for some $p>0$, so does the reduced Krein–von Neumann resolvent (Gesztesy et al., 16 Apr 2025).

6. Applications and Notable Instantiations

Krein–von Neumann extensions and their relatives have found applications in various settings:

• Elliptic Differential Operators: “Krein–like” extensions provide canonical realizations of minimal growth under non-elliptic or coupled boundary conditions; the Krein–von Neumann extension is uniquely singled out by minimal growth in the null-space (i.e., softest boundary) (Grubb, 2010, Cho et al., 2021).
• Quantum and Discrete Graphs: Krein–von Neumann extensions define nonlocal vertex conditions or modified Kirchhoff/Neumann flows. In metric graphs, the corresponding extension produces a coupling of vertex values and normal derivatives by the weighted discrete Laplacian (Mugnolo, 2014, Muller et al., 2020).
• Difference and Discrete Symplectic Systems: The Krein–von Neumann extension is realized as a minimal subspace extension, adding the (potentially multidimensional) kernel of the maximal relation; boundary conditions are parametrized via explicit matrices or unitary parameters (Zemánek et al., 2016).
• In elasticity theory, spectral geometry, and quantum mechanics, these extensions model problems where “soft” or regenerative boundary conditions, maximal null-space, or minimal extension play a structural role; e.g., buckling of plates, models with PT-symmetric Hamiltonians (via C-symmetry construction), or degenerate diffusions (Ashbaugh et al., 2012, Kamuda et al., 2018).

7. Generalizations and Recent Developments

Recent literature broadens the Krein–von Neumann extension to more abstract settings:

• Relations and anti-dual pairs: The extension is constructed in settings without Hilbert space structure, using anti-duality pairs and derived Hilbert spaces on the range of the operator (Tarcsay et al., 2018).
• Operators with indefinite quadratic forms (finite negative index): The Krein–von Neumann theory generalizes to quasi-nonnegative operators, provided the minimality condition on the negative index is imposed (Baidiuk et al., 2014).
• Factorization and extension in non-densely defined operators: Modern proofs make use of factorization through auxiliary “energy spaces” (Hilbert spaces), applicable even to non-densely defined symmetric or positive operators (Sebestyén et al., 2022).
• Extension theory is unified by moving from boundary triples (requiring equal deficiency indices) to boundary quadruples (always available), allowing parametrization of all self-adjoint (and even dissipative) extensions by contractions or unitaries between “boundary” Hilbert spaces (Arendt et al., 2022).
• The Krein–von Neumann extension appears as the extreme limit (minimal extension) in parametrized families, often as $B=0$ in operator-valued boundary conditions.

In summary, the Krein–von Neumann extension is a fundamental construction for parameterizing self-adjoint extensions of semibounded operators, providing minimality properties, concrete spectral and boundary characterizations, and a robust architecture for applications across analysis, PDEs, operator algebras, and mathematical physics.