Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems
Abstract
We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.
Related
- p and rp-Spectral Element Methods for Two-dimensional Elliptic Boundary Layer Problems
- A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems
- Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
- H2(Ω) A Priori Error Bounds for Boundary-Adapted PINNs Solving Elliptic Dirichlet Problems
- Method of Spectral Mappings for Nonselfadjoint Boundary-Value Problems Associated with Resonance Scattering by Open Structures
BibTeX
@misc{thomas2026spectral,
title = {{Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems}},
author = {Mathew Thomas and V. B. Kiran Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13385},
eprint = {2610.13385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral theorem for compact self-adjoint linear relations with an application to elliptic boundary value problems
AU - Mathew Thomas
AU - V. B. Kiran Kumar
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13385
UR - https://arxiv.org/abs/2610.13385
AB - We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.
ER -