Krein's Resolvent Formula in Operator Theory

• Krein's resolvent formula is a key result defining how the resolvent of one self-adjoint extension is expressed via another using operator corrections.
• It provides an analytic framework for parametrizing solutions in differential operators, moment problems, and boundary value problem contexts.
• Modern adaptations incorporate finite-dimensional corrections and Weyl function techniques, facilitating computations of spectral shift functions and trace formulas.

Krein's resolvent formula is a foundational result in operator theory and spectral analysis, describing how the resolvents of different self-adjoint extensions of a symmetric (or Hermitian) operator are related. In contemporary research, the formula serves as a central tool in the analysis and parametrization of solutions to diverse problems, including the moment problem, extension theory for differential operators, boundary value problems, random and singular perturbations, and spectral invariants. Its modern forms accommodate varying levels of generality, including matrix-valued, nonsmooth, and infinite-dimensional settings, and can be framed in terms of analytic operator functions (Weyl functions or M-operators) as well as explicit finite-rank corrections. The formula encapsulates both algebraic decompositions in terms of boundary data and analytic parametrizations of the family of extensions.

1. Conceptual Structure of Krein's Resolvent Formula

The classical Krein resolvent formula relates the resolvents of two self-adjoint extensions $A_1$ and $A_2$ of a densely defined symmetric operator $A$ in a Hilbert space. It expresses the resolvent of one extension in terms of the other and a finite-rank (or, more generally, compact) correction involving projections onto deficiency spaces and “boundary maps.”

A prototypical abstract form is: $$(A_2 - zI)^{-1} = (A_1 - zI)^{-1} + \gamma(z) \left[ M(z) - \Theta \right]^{-1} \gamma(\bar{z})^*$$ where:

• $(A_1 - zI)^{-1}$ is the resolvent of a "reference" extension, e.g., the Friedrichs extension.
• $\gamma(z)$ is the so-called gamma-field, mapping boundary data into the deficiency space.
• $M(z)$ is the Weyl function or (generalized) Dirichlet-to-Neumann-type map, analytic in $z$.
• $\Theta$ parametrizes the self-adjoint extension (encapsulating the boundary condition or perturbation).

The formula generalizes to non-negative Hermitian operators, Krein-von Neumann extensions, operators on Krein spaces, and more, often using auxiliary Hilbert spaces, operator-valued functions, and functional analytic notions such as M-operators (Donoghue-type).

2. Self-Adjoint Extensions and Operator-Valued Parametrization

A central application is the parametrization of all self-adjoint extensions of a densely defined symmetric (Hermitian) operator, especially in scenarios where the deficiency indices are nonzero. The formula identifies a privileged extension (typically the Friedrichs extension $A_F$ or the so-called “hard” extension, $S_F$) and expresses other (possibly “softer,” such as Krein-von Neumann, $S_K$) extensions as finite-rank or analytic perturbations.

For a non-negative Hermitian operator $A$, the formula in the version of Derkach and Malamud is: $$R_z = (A_F - zI)^{-1} - \gamma(z)[T(z) + M(z) - M(0)]^{-1} \gamma^*(\bar{z})$$ Here, $T(z)$ is an operator-valued function (in a prescribed analytic class) that parametrizes the generalized resolvents and the entire set of self-adjoint extensions; $M(z)$ retains the role of the Weyl function.

This structure underpins the explicit description of the solution set for the matrix Stieltjes moment problem (Zagorodnyuk, 2010), where the induced operator $A$ is constructed from the moment sequence and the set of all solutions is parametrized by $T(z)$ via the Krein formula.

3. Boundary Value Problems and Dirichlet-to-Neumann Maps

For second-order elliptic (or more generally, differential) operators in domains with boundary, Krein-type resolvent formulas allow for explicit comparison of realizations corresponding to different boundary conditions (Dirichlet, Neumann, Robin, mixed, or nonlocal). The resolvent difference,

$$(A_L - \lambda)^{-1} - (A_{\gamma} - \lambda)^{-1} = -K_{\gamma}^{\lambda} (L^{\lambda})^{-1} (K'_{\gamma}(\lambda))^*$$

depends on the Poisson operator $K_{\gamma}^{\lambda}$ and the boundary operator $L^{\lambda}$ (which encodes the deviation from the reference, e.g., Dirichlet problem) (Abels et al., 2010, Grubb, 2011). The trace-class or compactness properties of the correction encode spectral properties and allow sharp estimates for eigenvalue asymptotics, such as Weyl laws for boundary condition-induced spectral differences.

In the case of Sturm-Liouville operators and Schrödinger operators with arbitrary self-adjoint boundary conditions, the formula is recast in terms of boundary data maps (matrix-valued generalizations of Dirichlet-to-Neumann maps) (Clark et al., 2012, Gesztesy et al., 2010): $$[(H_{A',B'}-zI)^{-1} - (H_{A,B}-zI)^{-1}] = - [Y_{A,B}(H_{A,B}-zI)^{-1}]^*\, \Lambda_{A,B}^{A',B'}(z)^{-1} Y_{A,B}(H_{A,B}-zI)^{-1}$$ where $Y_{A,B}$ is a trace operator and $\Lambda_{A,B}^{A',B'}(z)$ encodes the boundary relation.

4. Spectral Theory, Trace Formulas, and Perturbation Determinants

Krein's formula's finite-rank nature allows explicit expressions for the trace of resolvent differences, a fact exploited in the computation of spectral shift functions, regularized determinants, and trace formulas (Hillairet et al., 2010, Gesztesy et al., 2010, Clark et al., 2012, Chattopadhyay et al., 2014). For instance, for two self-adjoint extensions $A_F$ and $A_L$,

$$\operatorname{Tr}[(A_L - \lambda)^{-1} - (A_F - \lambda)^{-1}] = - \frac{d}{d\lambda} \log \det(P + Q S(\lambda))$$

where $S(\lambda)$ is the scattering matrix or S-matrix, and $P, Q$ encode the boundary condition. This relation further extends to the comparison of zeta-regularized determinants and spectral shift functions, with applications to global geometric invariants for operators on singular manifolds.

The symmetrized perturbation determinant and the trace formula in the context of boundary data maps further compress the infinite-dimensional spectral problem to finite-dimensional determinants for one-dimensional Schrödinger operators (Gesztesy et al., 2010).

5. Singular Perturbations, Random Operators, and Renormalization

Krein's resolvent formula provides a foundational framework for singular perturbations and random or disordered systems. In the paper of self-adjoint realizations of formal Hamiltonians such as $H + A^* + A$ (modeled on QFT interactions), explicit Krein-type formulas relate the resolvent of the singular (interacting) operator to a regular (cutoff) family plus a correction term (Posilicano, 2020, Posilicano, 2023). The difference is written as,

$$(-H_{\text{int}} + z)^{-1} - (-H + z)^{-1} = -G_z (\Theta + M_z)^{-1} G_z^*$$

where $G_z$ and $M_z$ are built from the free resolvent and the (possibly singular) interaction. Norm-resolvent convergence theorems underpin the construction of renormalized, physically meaningful Hamiltonians.

For random Schrödinger operators with point interactions, Krein's formula enables a reduction to a lattice model where the resolvent is a sum over random-walk paths, each term expressed via matrix elements of a principal matrix evolving under disorder averages (Kaminaga, 18 Aug 2025). A crucial technical device involves contour deformation in the coupling constant plane to achieve uniform bounds, ensuring analyticity of the disorder-averaged resolvent and density of states.

6. Role of M-Operators, Weyl Functions, and Modern Generalizations

The role of the M-operator (an operator-valued Nevanlinna–Herglotz function, often called the Weyl function) is central to modern Krein-type formulas. For two relatively prime self-adjoint extensions $S_1$, $S_2$ of a symmetric positive operator $S$, the formula

$$(S_2 - zI)^{-1} = (S_1 - zI)^{-1} + (S_1 - iI)(S_1 - zI)^{-1} P_{1,2}(z) (S_1 + iI)(S_1 - zI)^{-1}$$

with $$P_{1,2}(z)^{-1} = \tan\theta_{1,2} - M_{S_1,\mathcal{N}_+}(z)$$ utilizes a Donoghue-type $M$-operator. This M-operator generalizes the Dirichlet-to-Neumann map and encapsulates boundary and spectral data (Gesztesy et al., 16 Apr 2025).

A summary table organizing typical objects appearing in Krein-type formulas:

Object	Description	Role in Formula
$A_F$, $S_F$	Friedrichs ("hard") extension	Reference resolvent
$A_K$, $S_K$	Krein-von Neumann ("soft") extension	Comparison resolvent
$\gamma(z)$	Gamma-field (solution operator)	Couples boundary–defect
$M(z)$	Weyl function / M-operator	Boundary response
$T(z)$, $\Theta$	Extension parameter/operator	Parameterizes extensions

7. Extensions to Nonsmooth, Matrix, and General Operator Theoretic Settings

Krein's formula has been rigorously developed in domains with low-regularity boundaries, operators with minimal smoothness assumptions, and in the context of matrix-valued (or operator-valued) moment problems (Abels et al., 2010, Zagorodnyuk, 2010). Pseudodifferential boundary operator calculus, symbol smoothing, and order-reducing techniques facilitate these generalizations, ensuring the existence of Poisson and boundary trace operators, and yielding explicit resolvent formulas even in low-regularity settings.

Further, recent works have extended Krein-type resolvent techniques to cases involving block-matrix operator pencils (e.g., for abstract Klein–Gordon equations on Krein spaces (Georgescu et al., 2012)), and to the analysis of invariant subspaces and the structure of J-unitary operators in spaces with indefinite metric (Choroszavin, 2013).

Summary

Krein's resolvent formula provides a unifying analytic and algebraic framework that expresses the resolvent of one self-adjoint extension (or perturbed operator) in terms of another and a finite-rank or analytic correction involving explicit operator functions (gamma-fields, Weyl/M-operators, boundary data maps). Through these relations, the formula enables:

• Complete parametrization of solutions to classic problems (e.g., matrix Stieltjes moment problem),
• Direct comparison of resolvents for operators with differing boundary conditions (including spectral and trace formulas),
• Explicit construction and analysis of singular and random operators,
• Control over analytic properties such as trace ideal membership, regularity, and analyticity of spectral data,
• Broad applicability to settings with singularities, low regularity, or infinite-dimensional features.

Modern forms incorporate operator-valued function theory and functional analytic extension theory, making Krein's resolvent formula central for contemporary mathematical analysis in spectral theory, boundary value problems, and mathematical physics.