Spectral Theory of Semigroups

• Spectral Theory of Semigroups is a framework that connects generator properties to semigroup behaviors through mapping theorems and spectral inclusions.
• It employs functional calculus and Laplace transform techniques to transfer spectral information from generators to evolution operators in various application settings.
• Quantitative spectral decompositions and gap inequalities offer practical insights for establishing exponential convergence and stability in dynamical systems.

A $C_0$-semigroup is a one-parameter family of bounded linear operators $(T_t)_{t \ge 0}$ on a topological vector space, usually a Banach or locally convex space, such that $T_0 = I$, $T_{t+s} = T_t T_s$ for all $s, t \ge 0$, and $\lim_{t \to 0^+} T_t x = x$ for all $x$ in the space. The spectral theory of such semigroups connects the properties of the generator $A$ (defined by $A x = \lim_{t \to 0^+} \frac{T_t x - x}{t}$ with domain consisting of those $x$ for which the limit exists) to the long-time, local, and global behavior of the semigroup and the corresponding evolution equations. Modern spectral theory addresses mapping theorems, structure of spectrum, local spectral properties, and asymptotic results in a variety of analytic, geometric, and probabilistic settings.

1. Spectral Inclusion, Mapping Theorems, and Generator Relations

A cornerstone of the subject is the spectral mapping theorem, connecting the spectrum of the semigroup $T(t)$ to the spectrum of its generator $A$. If $\sigma(A)$ denotes the (sometimes point/approximate/residual) spectrum of $A$, then for a $C_0$-semigroup on a Banach or locally convex space, general inclusion relations hold: $e^{t \sigma(A)} \subseteq \sigma(T(t))$ with more refined versions for distinguished parts of the spectrum including

$e^{t \sigma_p(A)} = \sigma_p(T(t)) \setminus \{0\}$

for the point spectrum, cf. (Kruse, 15 Sep 2025), and analogous inclusions for residual, algebraic, and various forms of approximate spectra. The resolvent at $\lambda$ relates to the semigroup via the Laplace transform: $$R(\lambda, A) = \int_0^\infty e^{-\lambda t} T_t\,dt$$ and the explicit formula

$$(\lambda-A)^{-1}x = (e^{\lambda t} - T(t))^{-1} \int_0^t e^{\lambda (t-s)} T(s)x ds$$

enables transference of spectral (non-)invertibility. For scalar type spectral operators, the precise mapping theorem holds: $$\sigma(T(t)) \setminus \{0\} = e^{t \sigma(A)} \setminus \{0\}$$ for each $t \ge 0$ (Markin, 2020).

On Hilbert spaces, refinement provided by extending functional calculus through analytic Besov algebras yields further versions such as

$\sigma(f(A)) \cup \{0\} = f(\sigma(A)) \cup \{0\}$

for $f$ in suitable function spaces (Batty et al., 2019).

In the locally convex setting, provided the semigroup is locally (quasi-)equicontinuous, all principal spectral inclusions for algebraic, point, residual, and (bounded) approximate spectra carry over (Kruse, 15 Sep 2025). This generalization is nontrivial due to the lack of normed structure, requiring new technical approaches.

2. Advanced Local and Global Spectral Properties

Local spectral theory investigates the spectrum that a single vector "feels" under the action of an operator. For $C_0$-semigroups and their generators, if $x \in X$, the local spectrum $\sigma_A(x)$ consists of $\lambda$ for which the equation $(A - \lambda)f(\lambda) = x$ cannot be locally solved using analytic vector-valued functions. There are powerful transference results: if $\lambda \in \sigma_A(x)$, then $e^{\lambda t} \in \sigma_{T(t)}(x)$ for all $t \ge 0$ (Tajmouati et al., 2016).

Subordinated operators are constructed as

$\mathcal{H}_\nu = \int_0^\infty T_t\,d\nu(t)$

for a semigroup $(T_t)_{t \ge 0}$ and a sufficiently regular Borel measure $\nu$. Typically, these operators do not have the Single Valued Extension Property (SVEP): there exist non-zero $x$ with multiple analytic continuations of the local resolvent at some points. In contrast, their adjoints may have the Dunford property ("property C"): all their spectral subspaces associated with closed spectral sets are closed and, for the Cesàro operator or subordinated operator to a hyperbolic composition semigroup, each non-zero vector has maximal local spectrum (Gallardo-Gutiérrez et al., 6 Aug 2025).

In various settings, the local properties of semigroup elements and subordinated operators reflect not only the algebraic generating structure but also geometric aspects (e.g., the Koenigs domain in composition semigroups).

3. Quantitative Structure: Spectral Decomposition, Gaps, and Growth

Generalized Weyl-type theorems show that, under suitable assumptions (splitting of the generator into dissipative and regular parts, or quasi-compactness conditions), the part of the spectrum in a right half-plane consists only of isolated eigenvalues of finite algebraic multiplicity. The semigroup admits a decomposition

$$e^{t\Lambda} = \sum_{j=1}^J e^{t \xi_j} \Pi_j + R(t)$$

with decay estimates on $R(t)$, and the principal eigenprojection $\Pi_{\Lambda, \lambda}$ is rank-one associated to positive eigenfunctions if positivity holds. Explicit spectral gap inequalities ensue, such as

$$\|e^{t \Lambda} f - e^{\lambda t} \Pi_{\Lambda,\lambda} f\|_X \le C_a e^{a t} \|f - \Pi_{\Lambda,\lambda} f\|_X$$

for any $a < \lambda$, showing strict exponential convergence away from the dominant eigenmode (Mischler et al., 2013).

For Markov and non-local semigroups (e.g., generalized Jacobi or Laguerre semigroups), spectral expansions can be made explicit in terms of generalized polynomials or polynomials intertwined with Lebesgue exponential functionals, with the convergence rate determined by the explicit spectral gap (Cheridito et al., 2019, Patie et al., 2015).

Spectral gap results further provide the key mechanism for establishing exponential mixing and contraction phenomena in stochastic processes and branching systems, with the normalized or $h$-transformed semigroup often exhibiting enhanced stability (Moral et al., 2021).

4. Eventual, Asymptotic, and Local Positivity: Implications for the Peripheral Spectrum

Many semigroups do not act positively for all time but become positive (eventually or asymptotically) after a sufficient interval. If $(e^{tA})_{t \ge 0}$ is individually or uniformly eventually positive, the spectral bound $s(A)$ is structurally dominant: under relatively weak compactness of orbits and eventual positivity, the only eigenvalue of $A$ on the line $\text{Re }\lambda = s(A)$ is $s(A)$ itself; the peripheral spectrum is cyclic if the semigroup is persistently irreducible (Daners et al., 2015, Arora et al., 2020, Arora, 25 May 2024). Ultrapower and bidual techniques are fundamental to lifting approximate eigenvalues to genuine ones in these settings (Arora, 25 May 2024).

If the semigroup is only locally eventually positive, orbits become asymptotically positive (their distance to the positive cone decays to zero as $t \to \infty$), and under relatively compact orbits, the peripheral point spectrum is typically trivial except at $s(A)$ (Mui, 2022).

In all these cases, the spectral projection $P$ corresponding to $s(A)$ plays the central role; for eventual positivity, $P$ is strongly positive, and the semigroup converges strongly to $P$ under boundedness and compactness assumptions (Daners et al., 2015).

5. Applications: Matrix Semigroups, Functional Calculus, Quantum and Non-Commutative Cases

Matrix semigroups of constant spectral radius are classified structurally by irreducibility and the ability to simultaneously conjugate all nonsingular elements to orthogonal matrices; they are central to uniform geometric transformations, combinatorics, and algorithms for linear switching systems. For irreducible families, the constant spectral radius property is polynomial-time decidable, whereas it is undecidable for reducible semigroups (Protasov et al., 2014).

In function spaces, very general forms of the spectral mapping theorem are derived by extending the functional calculus to analytic Besov classes, producing equalities of the form

$\sigma(f(A)) \cup \{0\} = f(\sigma(A)) \cup \{0\}$

accommodating Laplace transforms with Rajchman measures and sharply improving earlier inclusions (Batty et al., 2019).

In quantum settings, the spectrum of the induced generator of a Gaussian quantum Markov semigroup coincides with the additive semigroup generated by the eigenvalues of the drift matrix; a quasi-derivation property allows construction of higher-order eigenvectors recursively and ensures the spectrum is discrete and fully characterized in the case of a spectral gap (Fagnola et al., 16 Apr 2025).

Spectral theory for $C_0$-semigroups on locally convex spaces inherits, with suitable modifications, most classical spectral inclusion and mapping results, with careful attention to topology, set-theoretic spectrum definitions, and continuity structure (Kruse, 15 Sep 2025).

6. Special Constructions: Subordinated Operators, Cesàro Operator, and Composition Semigroups

Given a $C_0$-semigroup $(T_t)$ and a measure $\nu$, the subordinated operator

$\mathcal{H}_\nu = \int_0^\infty T_t\,d\nu(t)$

produces integral transformations with spectral properties closely related to the generator. The Laplace transform of $\nu$ translates the spectrum of $A$ to the subordinated operator, encompassing resolvent-type averages for measures $\nu_\lambda$ with $d\nu_\lambda(t) = e^{-\lambda t}\,dt$. Subordinated operators often lack SVEP, leading to dense glocal spectral subspaces and "maximal" local spectra for nonzero vectors, yet their adjoints may have all spectral subspaces closed (the Dunford property) (Gallardo-Gutiérrez et al., 6 Aug 2025).

For the Cesàro operator $\mathcal{C}$ on Hardy spaces $H^p$, local spectral theory shows the local spectrum at any $f \neq 0$ matches $\sigma(\mathcal{C})$, and all nontrivial invariant subspaces have full spectrum. For composition semigroup generators with hyperbolic Koenigs domain, the local spectral picture for subordinated operators can often be described in geometric terms, determined by the Koenigs domain width and shape.

7. Structural, Algorithmic, and Functional Implications

A unifying theme across these developments is the transfer of spectral information from generators to semigroups via exponential and Laplace transforms, as well as by functional calculus in extended function algebras. Quantitative versions for regular, Fredholm, and B-Fredholm spectra are established via regularity-preserving identities, e.g.,

$e^{t \sigma_{v}(A)} \subseteq \sigma_{v}(T(t))$

for several regularity classes $v$, and precise closure properties are maintained in spectral subspaces.

Algorithmic recognition of spectral invariants can be tractable or undecidable according to structural properties (irreducibility, compactness), while generalized spectral theory on locally convex spaces opens significant directions for PDE and functional analysis beyond the Banach space framework.

The subject continues to interface with stability analysis, probabilistic convergence, ergodic theory, non-self-adjoint harmonic analysis, and representations in quantum and non-commutative probability. The spectral theory of semigroups remains a principal analytic and structural tool for the paper of evolution equations and dynamical systems across mathematical domains.