Improper Gauges in Symmetric Linear Relations

• Symmetric linear relations with improper gauges extend classical symmetric operator theory by utilizing rigged spaces and gauge subspaces.
• The theory constructs L-resolvent matrices via analytic matrix functions, enabling complete parametrization of self-adjoint extensions and spectral measures.
• By incorporating boundary triples and generalized Nevanlinna families, the framework unifies extension, inverse, and spectral problems in Hilbert and Pontryagin spaces.

The Kreĭn–Saakyan theory addresses the parametrization of all generalized (or exit-space) resolvents for closed symmetric operators and, more generally, symmetric linear relations in Hilbert and Pontryagin spaces. Central to the theory is the construction and interpretation of $\sL$-resolvent matrices, which encode the dependence of resolvents on spectral boundary data captured by a gauge subspace $\sL$. The theory describes how all such resolvents can be parametrized via a single analytic matrix function, leading to a comprehensive understanding of self-adjoint extensions, spectral measures, and associated transformation formulas, with reach into canonical differential systems and indefinite-inner-product settings. The framework has evolved from the original proper-gauge Hilbert space setting (Kreĭn, Saakyan) to improper gauges (Shmul'yan, Tsekanovskii) and, via Langer–Textorius, to arbitrary symmetric linear relations with boundary triples formalism (Derkach, 4 Dec 2025, Derkach, 20 May 2025, Mogilevskii, 2014).

1. Rigged Spaces, Deficiency Indices, and the Gauge Subspace $\sL$

A closed symmetric linear relation $A$ in a separable Hilbert space $\sH$ with finite and equal deficiency indices $p<\infty$ naturally admits a rigged (Gel'fand triple) structure

$\sH_+ \subset \sH \equiv H \subset \sH_-$

where $\sH_+ = \dom A^*$, endowed with the graph norm, and $\sH_-$ its (anti-)dual. The deficiency index $p$ is $\dim \ker(A^* - zI)$ for $z \notin \mathbb{R}$.

A gauge is defined as a $p$-dimensional subspace $\sL \subset \sH_-$ such that, for some $z$ in the resolvent set of $A$ (with respect to $\sL$), the decomposition

$\sH_- = \ran(A - zI) \oplus \sL$

holds. If $\sL \subset \sH$, it is called a proper gauge; if $\sL \not\subset \sH$, but $\sL \subset \sH_-$, it is called an improper gauge. The set of such $z$ is denoted $\rho(A, \sL)$, and the projection onto $\sL$ along $\ran(A-zI)$ is $\Pi_{\sL}^z$ (Derkach, 4 Dec 2025, Derkach, 20 May 2025).

This abstract framework generalizes the classical situation of symmetric operators and sets the stage for extending spectral and resolvent theories to singular and multi-valued relations.

2. $\sL$-Resolvents and Their Parametrization via Matrix Functions

Given a self-adjoint extension $\widetilde{A}$ (possibly acting in an enlarged space), the generalized resolvent

$R_z = P_H (\widetilde{A} - zI)^{-1} |_{H}$

is regularized by setting $R^0 = \frac{1}{2}(R_i + R_{-i})$ and defining $\widehat{R}_z = R_z - R^0$, which can be extended to an operator $\sH_- \to \sH_+$. For an invertible map $L\colon \mathbb{C}^p \to \sL \subset \sH_-$,

$r(z) := L^* \widehat{R}_z L$

is called the $\sL$-resolvent of $A$.

Every such $r(z)$ belongs to the Nevanlinna class $\mathcal{R}^p$: holomorphic in $\mathbb{C} \setminus \mathbb{R}$, with a symmetry condition and a nonnegative kernel. All $\sL$-resolvents are parameterized via a linear-fractional (Möbius-type) transform associated with a $2p \times 2p$ matrix function $W(z)$ (the $\sL$-resolvent matrix):

$$r(z) = T_{W(z)}[\tau(z)] := (w_{11}(z)\tau(z) + w_{12}(z))(w_{21}(z)\tau(z) + w_{22}(z))^{-1}$$

where $\tau(z)$ ranges over all operator-valued Nevanlinna families. The J-positivity

$J_p - W(z) J_p W(z)^* \ge 0$

(with $J_p$ the symplectic form) ensures the correct analytic and positivity properties of the transform (Derkach, 4 Dec 2025, Derkach, 20 May 2025).

This parametrization encapsulates both proper and improper gauge cases: for improper gauges, the class of Nevanlinna families must be extended to generalized Nevanlinna classes $\mathcal{N}_\kappa$ in Pontryagin/Kreĭn spaces.

3. Boundary Triples and Connection to Linear Relations

The use of boundary triples is a categorical tool for describing the domain and extension theory for $A^*$. For a boundary triple $(\mathbb{C}^p, \Gamma_0, \Gamma_1)$,

$\Gamma_j: A^* \longrightarrow \mathbb{C}^p$

with surjectivity and Green's identity,

$$(f', g) - (f, g') = (\Gamma_1 f) \cdot (\Gamma_0 g) - (\Gamma_0 f) \cdot (\Gamma_1 g)$$

one defines the Weyl function $M(z)$ and the $\gamma$-field

$$M(z)\Gamma_0 f_z = \Gamma_1 f_z \,,\qquad \gamma(z) = (\Gamma_0|_{N_z})^{-1}$$

where $N_z = \ker(A^* - zI)$.

Every generalized resolvent of $A$ arises via the Kreĭn-type formula

$$R_z = R_z^0 - \gamma(z) [M(z) + \tau(z)]^{-1} \gamma(z)^*$$

establishing a bijection between all generalized resolvents $R_z$ and Nevanlinna families $\tau(z)$ (Derkach, 4 Dec 2025, Mogilevskii, 2014).

For symmetric linear relations (possibly unbounded or multi-valued), Langer and Textorius extended the $\sL$-resolvent theory by adapting the boundary triple formalism, introducing concepts like the multi-valued part and the use of directing mappings valued in an ambient Pontryagin space (Derkach, 20 May 2025).

4. Construction and Representation of the $\sL$-Resolvent Matrix

The analytic machinery is encoded in a $2p \times 2p$ preresolvent matrix:

$$\mathfrak{A}(z) = \begin{pmatrix} M(z) & \gamma(z)^* L\ L^* \gamma(z) & L^* R_z^0 L \end{pmatrix}$$

where $M(z)$ and $\gamma(z)$ are as above. For $z$ in the appropriate domain, with invertible off-diagonal blocks, the $\sL$-resolvent matrix is given via a Schur complement construction. Concretely,

$$W(z) = [0\;\;a_{12}; -I_p\;\; a_{22}]^{-1}[I_p\;\; a_{11}; 0\;\; a_{21}]$$

where $a_{ij}(z)$ are blocks of $\mathfrak{A}(z)$. This form is equivalent to a boundary-operator formula,

$W(z) = \left( \Gamma P(z)^* \right)^*$

with $P(z) = L^{-1} \Pi_{\sL}^z$, providing a direct link between the abstract analytic construction and boundary data (Derkach, 4 Dec 2025, Derkach, 20 May 2025).

A crucial property is that $W(z)$ lies in the so-called Kreĭn–Saakyan class $\mathcal{W}_\kappa(J_p)$, which guarantees that the kernel

$$\frac{J_p - W(z) J_p W(\omega)^*}{-i(z - \overline{\omega})}$$

has at most $\kappa$ negative squares, ensuring the well-posedness of the extension problem in indefinite inner product spaces.

5. Spectral and Pseudo-Spectral Functions from $\sL$-Resolvent Matrices

The $\sL$-resolvent matrix $W(z)$ fully determines the spectral (and generalized Fourier) theory of the corresponding symmetric relation or operator. Given the directing mapping

$f \mapsto (z) f = L^{-1} \Pi_{\sL}^z f \in \mathbb{C}^p$

one defines an LT-spectral function $\sigma$ as a non-decreasing, left-continuous matrix function such that

$\int [(\lambda) f]^*\, d\sigma(\lambda)\, (\lambda) f \le \langle f, f \rangle, \qquad \text{equality on } \dom A.$

This encapsulates the contractive (and isometric on $\dom A$) nature of the associated generalized Fourier transform.

Every $\sigma$ arises from an $\sL$-resolvent $r(z) = L^* \widehat{R}_z L$ by a Stieltjes-type integral:

$$\int \left( (\lambda - z)^{-1} - \frac{\lambda}{1 + \lambda^2} \right) d\sigma(\lambda) = r(z) + K$$

with $K$ a boundary matrix. The function $\sigma$ is pseudo-spectral if the corresponding Fourier transform is a partial isometry with kernel $\mathrm{mul}\;A$; if $\mathrm{mul}\;A=0$, $\sigma$ becomes a spectral function and the transform is onto $L^2(\sigma)$ (Derkach, 4 Dec 2025).

6. Application to Canonical Systems

On an interval $(a, b)$, consider the first-order canonical system

$f'(t) + F(t) f(t) = z H(t) f(t)$

where $H(t) \ge 0$ and $\mathrm{tr}\, H = 1$. In the weighted Hilbert space $L^2_H$, minimal and maximal relations $S_{\min}$ and $S_{\max}$ are constructed. A standard boundary triple is

$$\Gamma_0(f, g) = \frac{f(a) + f(b)}{\sqrt{2}}, \qquad \Gamma_1(f, g) = \frac{-f(a) + f(b)}{\sqrt{2}}$$

with Weyl function and $\gamma$-field expressible through the monodromy matrix $U(b, z)$.

For a gauge of the form $\sL = \mathrm{span}\{\delta_a \otimes u : u \in \mathbb{C}^{2p}\}$ in the distributional dual, the associated mapping and resolvent matrix can be explicitly computed:

$P(z) = L^{-1} \Pi_{\sL}^z,\quad Q(z) = L^* \widehat{R}_z (I - L P(z)),$

$$W(z) = \begin{pmatrix} - \Gamma_0 Q(z)^* & \Gamma_0 P(z)^* \ - \Gamma_1 Q(z)^* & \Gamma_1 P(z)^* \end{pmatrix}^*$$

yielding a rational $2p \times 2p$ matrix in $U(b, z)$. Every self-adjoint boundary condition at $a$ and $b$ can thus be parametrized, and spectral measures recovered, through the general $\sL$-resolvent construction (Derkach, 4 Dec 2025, Derkach, 20 May 2025, Mogilevskii, 2014).

7. Extensions and Generalizations: Pontryagin Spaces and First-Order Systems

The Kreĭn–Saakyan framework extends to Pontryagin spaces (finite negative index), exploiting the same analytic machinery but tracking the signature of the underlying indefinite structure in all resolvent and spectral function constructions. The use of generalized Nevanlinna functions and negative-square kernels allows a faithful description of spectral and boundary data even when the operator is highly singular or non-densely defined (Derkach, 20 May 2025).

For first-order symmetric systems (not necessarily Hamiltonian), the $\sL$-resolvent (or characteristic) matrix $\Omega(\lambda)$ is constructed directly from fundamental solutions and boundary data, and admits both block-matrix and Kreĭn-type sum representations. This encompasses the classical Titchmarsh–Weyl $m$-function and generalizes all classical extension and spectral formulas to the analytic, multi-parameter context required by modern boundary value theory (Mogilevskii, 2014).

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The Kreĭn–Saakyan theory of $\sL$-resolvent matrices thus provides a unifying analytic framework for extension, spectral, and inverse problems for symmetric operators and relations in Hilbert and Pontryagin spaces, with a deep interplay between functional models, boundary triples, rigged spaces, and matrix-valued analytic functions encoding the entirety of the extension and spectral information.