Fredholm Theorem for Robbin–Salamon Operators

• The paper introduces a decomposition of the linearized operator into a continuous dominant term and a compact lower-order term, ensuring Fredholm properties.
• It leverages Sobolev space analysis and weak Hessian fields to rigorously define operator continuity and apply Rabier’s abstract Fredholm theorem.
• The index computation via spectral flow and its topological correspondence with the Maslov index establish a robust framework for applications in Floer theory and Hamiltonian dynamics.

The Fredholm theorem for Robbin–Salamon operators formalizes the analytic foundation underlying the linearization of variational problems in infinite-dimensional symplectic geometry, particularly in the context of Floer theory and Hamiltonian systems. These operators, which typically take the form $\mathcal{D}_u = \partial_s + A^u$ where $A^u$ is a family of bounded symmetric operators acting on suitable Sobolev spaces, arise from the linearization of the (perturbed) area or action functional along connecting paths between nondegenerate critical points. Central to the theorem is the identification of conditions under which $\mathcal{D}_u$ is Fredholm, in generality extending to operators with noncontinuous Hessian coefficients, and the precise computation of its index, which encodes the topological information relevant to Morse theory, Floer homology, and the spectral flow.

1. Analytic Structure and Weak Hessian Fields

Robbin–Salamon operators act between Sobolev spaces defined over a Hilbert space triple $(H_0, H_1, H_2)$, with

$$H_0 = L^2(S^1, \mathbb{R}^{2n}),\quad H_1 = W^{1,2}(S^1, \mathbb{R}^{2n}),\quad H_2 = W^{2,2}(S^1, \mathbb{R}^{2n}).$$

A path $u(s)$ in a suitable configuration space specifies at each parameter value a bounded symmetric operator $A^u \colon H_1 \to H_0$, which may also be restricted to $A_2^u : H_2 \to H_1$. The paper (Frauenfelder et al., 23 Oct 2025) introduces the critical notion of a weak Hessian field: a map $u \mapsto A^u$ that is continuous as a map $U_1 \to \mathcal{L}(H_1,H_0)$ and $U_2 \to \mathcal{L}(H_2,H_1)$, satisfying symmetry with respect to $\langle \cdot, \cdot \rangle_{H_0}$ and the Fredholm property for both $A^u$ and $A_2^u$.

Almost extendability is defined by a decomposition

$A^u = F^u + C^u,$

where $F^u$ is continuous in both Sobolev levels and $C^u$ is lower-order, satisfying a scale-Lipschitz compactness estimate over $H_1$ and $H_2$. Precisely, $$C^u \in C^0(U_1, (H_r,H_0)) \cap C^0(U_2, (H_1,H_1))$$ for $r \in [0,1)$, with a Lipschitz-type estimate controlling differences: $$\|C^v - C^w\|_{H_1} \leq \kappa \left(\|v-w\|_{H_2} + \min\{\|v\|_{H_2}, \|w\|_{H_2}\} \|v-w\|_{H_1}\right).$$

2. Fredholmness Criteria for $\mathcal{D}_u = \partial_s + A^u$

For a connecting path $u(s)$ between asymptotically nondegenerate critical points $u_-,u_+$, with $A^{u_-}$ and $A^{u_+}$ invertible, the operator

$$\mathcal{D}_u \colon W^{1,2}_{H_0} \cap L^2_{H_1} \to L^2_{H_0}$$

is Fredholm if $A^u$ is almost extendable. The principal result [(Frauenfelder et al., 23 Oct 2025), Theorem~{main}] is that decomposing $A^u = F^u + C^u$ with $F^u$ continuous (in the strong Sobolev topology) permits application of the abstract Fredholm theorem (due to Rabier) to $\partial_s + F^u$. The "lower-order" term $C^u$ induces a multiplication operator compact on $W^{1,2}_{H_0} \cap L^2_{H_1} \to L^2_{H_0}$ due to the compactness of the inclusion $H_2 \hookrightarrow H_1$, ensuring that the original Robbin–Salamon operator is Fredholm via invariance under compact perturbations: $$\mathrm{Fredholm}(\partial_s + F^u) \ \text{and} \ C^u\ \text{compact} \implies \mathrm{Fredholm}(\partial_s + F^u + C^u).$$

On higher Sobolev levels, identical reasoning applies for

$$\mathcal{D}_u^2 = \partial_s + A_2^u \colon W^{1,2}_{H_1} \cap L^2_{H_2} \to L^2_{H_1}.$$

3. Index Computation, Spectral Flow, and Boundary Conditions

The Fredholm index of Robbin–Salamon operators is computed via the spectral flow of the family $\{A^u(s)\}_{s \in \mathbb{R}}$. Analogous results for finite intervals are established in (Frauenfelder et al., 20 Dec 2024), where a differential operator $D_A = \partial_s + A(s)$ is rendered Fredholm by imposing boundary conditions determined by spectral projections on an interpolation/fractional Sobolev space $H_{1/2} = (H_0, H_1)_{1/2}$. The augmented operator

$$\mathfrak{D}_A : P_1(I) \to P_0(I) \times H_{1/2}^+(A(-T)) \times H_{1/2}^-(A(T)), \qquad \xi \mapsto (\partial_s \xi + A(s)\xi,\, \pi_+^{A(-T)}\xi(-T),\, \pi_-^{A(T)}\xi(T))$$

becomes Fredholm, and its index is

$\mathrm{index}(\mathfrak{D}_A) = \varsigma(A),$

where $\varsigma(A)$ is the net spectral flow (the signed count of eigenvalues of $A(s)$ crossing zero as $s$ increases).

The analytic and topological perspectives on the index are bridged via the symbol calculus in C*-algebras, as elucidated in (Inoue et al., 2017), where the index is shown to correspond to a winding number of the symbol (scattering matrix) in operator algebra quotient spaces: $$\operatorname{Wind}(T^{-}_{m,k;1,0}) = -\operatorname{Index}(W_{m,k;1,0}).$$ This principle aligns with the computation of the Maslov index and spectral flow in the Robbin–Salamon framework.

4. Robustness Under Perturbations and Extensions

Fredholm properties and index values are stable under bounded and compact perturbations. In the quaternionic Hilbert space setting (Muraleetharan et al., 2018), this is formalized by demonstrating that for compact $K$ or small $\|S\|$,

$\operatorname{ind}(A + K) = \operatorname{ind}(A),$

ensured also for general operator ideals as in (Hamdan et al., 2 Jan 2024). In more general, possibly non-Fredholm cases (pseudo B-Fredholm), the index is extended by formalizing an idempotent-based reduction in quotient algebras, and shown to coincide with the classical index for genuine Fredholm operators: $$\operatorname{ind}(T) = \lim_{\lambda \to 0} \operatorname{ind}(T - \lambda I).$$ Such robustness makes the Fredholm theorem applicable to operators arising in Floer theory with noncontinuous Hessians, delayed equations, or non-selfadjoint Hamiltonians.

5. Topological and Physical Implications

The spectral flow—Fredholm index correspondence provides powerful topological invariants in Floer theory, Hamiltonian bifurcation, and the classification of topological phases of matter (Bourne et al., 2019). In the real Hilbert space and Clifford module context, the index takes values in KO-theory and detects phenomena such as phase transitions and bifurcations, as in

$$\mathrm{sff}_{r,s+2}(A(\cdot)) = \operatorname{ind}_{r,s+2}(D),$$

with the operator $D$ expressing the Robbin–Salamon “suspension.” For Hamiltonian systems, the index bundle construction (Skiba et al., 2020) captures bifurcation points via the (possibly nontrivial) KO-theoretic class

$$\mathrm{ind}(L) = [E^s(+\infty)] - [E^s(-\infty)] \in KO(A),$$

showing that spectral and topological invariants persist—even for selfadjoint Fredholm families with zero classical index—when bundled parametrically.

6. Connections to Maslov Index, C*-Algebraic, and Operator Theory

In multidimensional spectral theory, the Evans function, defined as a modified Fredholm determinant of boundary-to-boundary operators, generalizes the classical Evans function and, in selfadjoint settings, coincides (up to analytic factors) with the Cayley transform of the Neumann-to-Dirichlet map, thus matching the Maslov index interpretation of spectral flow (Cox et al., 2022). The approach of relating analytic indices to winding numbers in C*-algebra quotients (e.g., for pseudo-differential operators with various periodicities as in (Inoue et al., 2017)) reveals deep structural parallels to the Robbin–Salamon spectral flow and Maslov index, thereby unifying analytic and topological methods.

7. Summary of Core Principles

• The analytic Fredholm property of Robbin–Salamon operators $\mathcal{D}_u = \partial_s + A^u$ hinges on the decomposition of $A^u = F^u + C^u$ with $F^u$ continuous (for Rabier’s theorem) and $C^u$ compact (lower order).
• The Fredholm index is robust under compact/finite-rank perturbations and extends naturally to more general operator classes.
• Boundary conditions formulated in interpolation spaces and induced by spectral projections are essential to obtain Fredholmness on finite intervals or in noncompact domains.
• There is a precise correspondence between analytic index (Fredholm index) and topological invariants (spectral flow, Maslov index, winding numbers), with applications transcending finite and infinite dimensions, and extending to topological field theories and condensed matter systems.
• Symbolic and C*-algebraic frameworks provide a calculable bridge between operator algebra theory and symplectic/variational analysis, enabling explicit computation of spectral invariants in concrete and physically motivated models.

The Fredholm theorem for Robbin–Salamon operators thus provides a flexible, robust analytic and topological framework, applicable to both classical and modern problems in analysis, topology, and mathematical physics, and underpinned by fundamental stability, symbol-theoretic, and spectral flow principles.