Spectral Theory of Compact Operators

• Spectral Theory of Compact Operators is the study of bounded linear operators whose unit ball maps to a relatively compact set, leading to a discrete spectrum with zero as the only accumulation point.
• The theory reveals canonical diagonalization, geometric characterizations of joint spectra, and robust perturbation methods that inform solutions to PDEs and quantum mechanics.
• Extensions to non-selfadjoint, real-linear, and operator tuple contexts broaden its applicability to numerical analysis, inverse spectral problems, and operator algebra research.

A compact operator on a separable Hilbert space is a bounded linear operator whose image of the unit ball is relatively compact. The spectral theory of compact operators investigates the structure of their spectrum, the existence and properties of eigenvalues and eigenvectors, generalizations to operator tuples, perturbation and stability properties, and extensions to modules and non-selfadjoint or real-linear contexts. This theory underpins many advanced developments in functional analysis, operator algebras, mathematical physics, and spectral geometry.

1. Classical Spectral Theory of Compact Operators

Compact self-adjoint operators on separable complex Hilbert spaces enjoy a canonical diagonalization: for $T = T^*$ compact, $T = \sum_{n=1}^\infty \lambda_n P_n$ with $\lambda_n \in \mathbb{R}$, $\lambda_n \to 0$, and $P_n$ orthogonal projections onto corresponding eigenspaces, which have finite multiplicity. The spectrum of $T$ consists of $0$ (possibly the only accumulation point) and countably many eigenvalues of finite multiplicity. The diagonalization generalizes the spectral theorem for matrices and underpins the solution theory of integral and differential equations, as well as quantum mechanics (Mukhamedov et al., 2015).

For compact normal operators, the spectrum is also countable with accumulation only at zero, and the operator admits an orthonormal basis of eigenvectors. In the case of tuples of commuting compact self-adjoint or normal operators, simultaneous diagonalization (i.e., a common orthonormal basis) is key and underpins spectral theory in multivariable settings (Chagouel et al., 2013).

2. Joint Spectrum and Commutativity of Operator Tuples

Chagouel, Stessin, and Zhu introduced a geometric approach to the joint spectrum for tuples $(A_1, ..., A_n)$ of compact operators. The joint point spectrum

$$\Sigma_p(A) = \{ (z_1, ..., z_n) \in \mathbb{C}^n : I + z_1A_1 + \cdots + z_nA_n \text{ is singular} \}$$

provides a complete invariant for the tuple's algebraic structure. The central result is that a tuple $(A_1, ..., A_n)$ of compact self-adjoint operators commutes pairwise if and only if $\Sigma_p(A)$ is a union of countably many, locally finite complex hyperplanes in $\mathbb{C}^n$: $$\Sigma_p(A) = \bigcup_{k=1}^\infty \{ \alpha_{k1}z_1 + \cdots + \alpha_{kn}z_n + 1 = 0 \}.$$ For finite collections of $N \times N$ normal matrices, commutativity is equivalent to the complete reducibility of the characteristic polynomial $\det(I + z_1A_1 + \cdots + z_nA_n)$ into linear factors. This establishes a geometric link between operator-algebraic commutativity and the flat structure (hyperplanes rather than more general analytic varieties) of the joint spectrum. Similar geometric characterizations hold for normality and various commutativity levels, as detailed in the various corollaries (Chagouel et al., 2013).

3. Spectral Theory in Hilbert–Kaplansky and Kaplansky–Hilbert Modules

The spectral theorem for compact operators extends to Hilbert–Kaplansky modules, i.e., modules over commutative von Neumann algebras equipped with an $L^\infty(\Omega)$-valued inner product. In this generalized context, self-adjoint cyclically compact operators admit a measurable diagonalization: $$T = \sum_{k=1}^\infty \sum_{n=1}^\infty \lambda_{k,n}(\omega) (\cdot, \xi_{k,n}(\omega)) \xi_{k,n}(\omega)$$ where the $\lambda_{k,n} \in L^\infty(\Omega)$ tend to zero and $\{\xi_{k,n}\}$ is an orthonormal field over the base measure space $\Omega$ (Mukhamedov et al., 2015). The spectrum of $T(\omega)$ in each fiber $X(\omega)$ is pure point, with possible accumulation only at zero, mirroring the classical situation but with a measurable field of spectral data.

In the case of Kaplansky–Hilbert modules with mixed norm, the spectral theorem for self-adjoint cyclically compact partial integral operators provides a fiberwise Mercer-type decomposition of the kernel and a projector-valued spectral measure $\{E_\lambda\}$, enabling a spectral integral representation and full functional calculus. This generalizes the Mercer theorem and the spectral theorem to this more abstract module context and handles partial integral equations as well as operators arising from Type I von Neumann algebras (Kudaybergenov et al., 19 May 2025).

4. Quantitative Spectral Stability and Perturbation Theory

The spectral stability of compact operators under small perturbations is fundamental in applications to PDEs, geometry, and numerical analysis. For families of compact self-adjoint operators $T(\epsilon)$ with $T(0) = T_0$, under mild regularity conditions, eigenvalues admit asymptotic expansions: $$\mu_n(\epsilon) = \mu_n(0) + \epsilon a_n + o(\epsilon),$$ with $a_n$ computable in terms of the perturbed form or domain. The main methodology involves variational principles and resolvent estimates, yielding explicit first-order correction terms (Bisterzo et al., 30 Jul 2024).

For general (not necessarily self-adjoint) compact operators, quantitative perturbation theory stratifies operators by the speed of singular value decay (using compactness classes $K(s)$). Explicit upper bounds are derived for the norm of the resolvent and for Hausdorff distances between spectra of two nearby operators, with the explicit dependency on the perturbation size and the singular value profile. For example, for $s_n \sim n^{-p}$, spectral deviations from finite-rank approximations are $O(|\log\|A-B\||^{-1/p})$ (Guven et al., 2020). This provides precise convergence rates for numerical eigenvalue computations and insights into spectral stability in practical settings.

5. Extensions: Non-selfadjoint, Real-linear, and Hankel Operators

Non-selfadjoint Compact Operators

For classes of non-selfadjoint operators $W = T + A$ with compact real parts, sectorial form-bounds, and compact resolvent, the spectral theory adapts the self-adjoint machinery. Asymptotic spectral properties (Weyl laws for eigenvalues) are governed by the self-adjoint part, and the completeness of the root vector system is guaranteed under angular constraints on the numerical range. Schatten–von Neumann class membership for the resolvent is similarly determined by the order of the real part (Kukushkin, 2018).

Real-linear Compact Operators

Real-linear compact operators, parametrized as $R = C + A$ (complex-linear and antilinear parts), exhibit finer spectral structure. The spectrum is determined by the vanishing of a real-analytic bivariate characteristic polynomial $p(\lambda, \overline{\lambda})$. For finite-rank operators, $p$ admits a sum-of-squares decomposition, providing geometric insight into the spectrum; for trace-class operators, a Fredholm–Carleman determinant extension exists. Notably, nonzero compact antilinear operators always have a nontrivial complex-linear invariant subspace (antilinear Lomonosov theorem), but general real-linear compact operators may lack such invariant complex-linear subspaces (Ruotsalainen, 2012).

Hankel Operators and Inverse Spectral Problems

The inverse spectral problem for compact Hankel operators is characterized sharply: given strictly interlaced sequences of (singular) values ($\lambda_j$ and $\mu_j$), there exists a unique real (or complex, up to phase) symbol $c$ producing $\Gamma_c$ and its shifted companion with the prescribed spectrum. The explicit reconstruction of $c$ relies on Hardy space or Cauchy integral formulas and generalizes classical results by Adamyan–Arov–Krein. This result reveals the necessity to simultaneously prescribe the spectrum of both a Hankel operator and its shift to ensure uniqueness, embodying an infinite-dimensional action-angle variable framework (Gerard et al., 2012).

Rank-One Perturbations of Normal Compact Operators

The spectral theory of rank-one perturbations of cyclic compact normal operators is described via a functional model in Cauchy–de Branges spaces of entire functions. The spectrum, completeness of the (generalized) eigenfunctions, conditions for spectral synthesis, and ordering of invariant subspaces are all characterized in terms of the analytic properties and zero sets of associated model functions. The model elucidates how small perturbations can drastically affect completeness or synthesis properties, and supplies new results on the rigid ordering of certain subclasses of invariant subspaces, even in the self-adjoint context (Baranov, 2018).

6. Operator-Algebraic and Noncommutative Generalizations

Compact operators relative to Type I von Neumann algebras, viewed via Kaplansky–Hilbert modules, admit a spectral decomposition indexed by central decompositions and abelian projections. The spectral structure mirrors the classical compact case but is enriched by measurable or central data (functions in $L^\infty(\Omega)$ or central elements of the von Neumann algebra), with fiberwise multiplicity and spectral ordering managed by measurable partitions of unity. The functional calculus, kernel expansions, and projection-valued spectral measures are all formulated so as to be compatible with the underlying module structure (Mukhamedov et al., 2015, Kudaybergenov et al., 19 May 2025).

7. Applications and Research Directions

The spectral theory of compact operators underlies inverse spectral theory, stability analysis of PDEs, spectral geometry under domain and metric perturbations, non-selfadjoint spectral theory, and the general structure theory of operator tuples. Quantitative analysis has become increasingly vital for numerical analysis, convergence rates of eigenvalue approximations, and the study of perturbed physical systems.

Future developments include higher-order asymptotic expansions for eigenvalues under perturbation, analysis of eigenvalue splitting for cluster spectra, extension to more general operator classes (beyond compact or self-adjoint), the role of random and statistical perturbations, and exploration of the interplay between spectral geometry and noncommutative analysis (Bisterzo et al., 30 Jul 2024, Guven et al., 2020).

This theoretical framework is essential across analysis, geometry, mathematical physics, and operator algebras, anchoring deeper research in spectral asymptotics, synthesis and completeness phenomena, and the geometric understanding of operator algebras.