Semigroups and Evolution Equations

Introduction

A one-parameter semigroup is a family $\{T(t)\}_{t\ge0}$ of bounded operators on a Banach space with $T(0)=I$ and $T(s+t)=T(s)T(t)$; it is the abstraction of the solution operator of an evolution equation, the map that carries an initial state forward by time $t$. When the semigroup is strongly continuous, it has an infinitesimal generator $A$, an unbounded closed operator obtained by differentiating $T(t)$ at $t=0$, and the abstract Cauchy problem

$$ u'(t) = Au(t), \qquad u(0)=x , $$

has $u(t)=T(t)x$ as its solution in the appropriate sense. The problem of the theory is the converse: which operators $A$ arise as generators, and what do the properties of $A$ — its spectrum, its dissipativity, the growth of its resolvent — imply for the behaviour of the semigroup and hence for the evolution equation?

The answer is the generation theory of Hille, Yosida, Feller, Phillips and Lumer, and its central theorem characterises the generators of contraction semigroups by a resolvent bound, while its dissipative form characterises them by a sign condition. The theory is the abstract home of the classical evolution equations of this Part: the heat equation is the semigroup generated by the Laplacian, the wave equation is the group generated by a skew-adjoint operator, and Schrödinger's equation is the unitary group of a self-adjoint operator, whose existence and spectral representation is Stone's theorem. It is also the point at which the functional analysis of the earlier articles is applied to the differential equations, and the technical hypotheses — closedness, density of the domain, the distinction between mild and classical solutions — are exactly the ones that the classical theory of partial differential equations hides.

The article develops the definitions and the basic properties of strongly continuous semigroups, the generator and its resolvent, the Hille–Yosida theorem and the Feller–Miyadera–Phillips theorem for semigroups of a fixed exponential type, the Lumer–Phillips theorem for dissipative operators, the abstract Cauchy problem with its mild and classical solutions, the analytic semigroups generated by sectorial operators, the examples of the heat semigroup and the wave group, Stone's theorem and the spectral theorem for a unitary group, and the spectral mapping theorem with the asymptotics of a semigroup. The functional analysis used throughout — bounded operators, the uniform boundedness principle, the resolvent, the spectral theorem for a self-adjoint operator — is that of Banach and Hilbert Spaces and of Unbounded Operators and Spectral Measures, the latter written in parallel; the partial differential equations that the semigroups solve are those of the earlier article of this Part.

Strongly Continuous Semigroups

Definitions and Elementary Properties

Definition. Let $X$ be a Banach space. A one-parameter semigroup on $X$ is a family $\{T(t)\}_{t\ge0}\subset B(X)$ with

$$ T(0)=I, \qquad T(s+t)=T(s)T(t) \quad (s,t\ge0). $$

It is strongly continuous (or a $C_0$ semigroup) if $\lim_{t\to0^+}T(t)x=x$ for every $x\in X$; it is uniformly continuous if the limit holds in the operator norm, in which case $T(t)=e^{tA}$ for a bounded $A$. The growth bound is

$$ \omega_0 = \inf\Bigl\{\omega\in\mathbb{R} : \sup_{t\ge0}e^{-\omega t}\|T(t)\|<\infty\Bigr\} , $$

and the semigroup is a contraction semigroup if $\|T(t)\|\le1$ for all $t\ge0$, and is bounded if $\sup_{t\ge0}\|T(t)\|<\infty$.

Theorem (elementary properties). For a strongly continuous semigroup the map $t\mapsto T(t)x$ is continuous on $[0,\infty)$ for every $x$, the operators $T(t)$ are uniformly bounded on every compact interval $[0,T]$, and there are constants $M\ge1$ and $\omega\in\mathbb{R}$ with

$$ \|T(t)\|\le Me^{\omega t}, \qquad t\ge0 ; $$

hence $\omega_0<\infty$ and $\|T(t)\|\le M_\varepsilon e^{(\omega_0+\varepsilon)t}$ for every $\varepsilon>0$.

Proof. The strong continuity at $0$ and the semigroup law give continuity on $[0,T]$; the uniform boundedness principle applied to the family $\{T(t): 0\le t\le1\}$ gives a bound there, and the semigroup law extends it to all $t$ with an exponential factor, by writing $t = n+r$ with $n\in\mathbb{N}$ and $r\in[0,1)$ and using $\|T(t)\|\le\|T(1)\|^n\|T(r)\|$.

Example (translation and multiplication semigroups). On $X=L^p(\mathbb{R})$ the family $(T(t)f)(x)=f(x+t)$ is a contraction semigroup; on $X=L^p(\Omega,\mu)$ the family $(T(t)f)(x)=e^{tm(x)}f(x)$ is a semigroup whenever $m$ is measurable and essentially bounded above, and it is strongly continuous for the relevant classes of $m$. On $X=\mathbb{C}^n$ with a matrix $A$, $T(t)=e^{tA}$ is a uniformly continuous semigroup, and the general finite-dimensional case is the linear system of the article on ordinary differential equations.

The Generator

Definition. The infinitesimal generator of a strongly continuous semigroup $\{T(t)\}$ is the operator $A$ with domain

$$ D(A) = \Bigl\{x\in X : \lim_{t\to0^+}\frac{T(t)x-x}{t}\ \text{exists}\Bigr\}, \qquad Ax = \lim_{t\to0^+}\frac{T(t)x-x}{t} . $$

Theorem (properties of the generator). The generator $A$ is linear; it is closed and densely defined; it commutes with the semigroup, $T(t)D(A)\subseteq D(A)$ and $AT(t)x=T(t)Ax$ for $x\in D(A)$; and the map $t\mapsto T(t)x$ is differentiable with derivative $T(t)Ax$ for $x\in D(A)$.

Proof. Linearity is immediate from the limit. For the density: for $x\in X$ and $t>0$, the element $\int_0^tT(s)x\,ds$ lies in $D(A)$ with $A\int_0^tT(s)x\,ds = T(t)x-x$, because the difference quotient of the integral is $\frac1h\int_t^{t+h}T(s)x\,ds-\frac1h\int_0^hT(s)x\,ds\to T(t)x-x$; dividing by $t$ and letting $t\to0$ gives density. For closedness, if $x_n\to x$ and $Ax_n\to y$, then $T(t)x_n-x_n = \int_0^tT(s)Ax_n\,ds$ tends to both $T(t)x-x$ and $\int_0^tT(s)y\,ds$; dividing by $t$ and letting $t\to0$ gives $y=Ax$. The commutation and differentiability follow by differentiating the semigroup law.

Definition. The resolvent of $A$ at $\lambda$ is $R(\lambda,A)=(\lambda I-A)^{-1}$, defined for $\lambda$ in the resolvent set $\rho(A)$.

Theorem (the resolvent as the Laplace transform). For every $\lambda$ with $\mathrm{Re}\,\lambda>\omega_0$ the operator $\lambda I-A$ is invertible and

$$ R(\lambda,A)x = \int_0^\infty e^{-\lambda t}T(t)x\,dt , $$

the integral converging absolutely in norm; consequently $\|R(\lambda,A)\|\le \frac{M}{\mathrm{Re}\,\lambda-\omega}$ for $\mathrm{Re}\,\lambda>\omega_0$ and suitable $M,\omega$, and the resolvent identity $R(\lambda,A)-R(\mu,A) = (\mu-\lambda)R(\lambda,A)R(\mu,A)$ holds.

Proof. For $x\in X$ the integral converges because $\|T(t)x\|\le Me^{\omega t}\|x\|$. Applying $\lambda I-A$ to the integral gives, after differentiating under the integral for $x\in D(A)$, $(\lambda I-A)\int_0^\infty e^{-\lambda t}T(t)x\,dt = x$; the same computation for the other composition with $A$ gives the inverse, and the density of $D(A)$ passes the result to all $x$. The resolvent identity is the algebraic identity for the inverse of $\lambda I-A$ and $\mu I-A$, verified by multiplying by $(\lambda I-A)(\mu I-A)$.

Generation Theorems

The Hille–Yosida Theorem

Theorem (Hille–Yosida). A linear operator $A$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if

$A$ is closed and $D(A)$ is dense in $X$;
every $\lambda>0$ belongs to the resolvent set of $A$, and

$$ \|R(\lambda,A)\| \le \frac1\lambda \qquad (\lambda>0). $$

Proof. Quoted as standard. For the necessity, the Laplace representation of the resolvent gives $\|R(\lambda,A)\|\le\int_0^\infty e^{-\lambda t}dt=1/\lambda$ for a contraction semigroup; closedness and density are the properties of the generator above. For the sufficiency, the Yosida approximation $A_\lambda = \lambda A R(\lambda,A) = \lambda^2R(\lambda,A)-\lambda I$ is a bounded operator generating the uniformly continuous semigroup $e^{tA_\lambda}$, and the bound $\|e^{tA_\lambda}\|\le1$ follows from the resolvent bound; one shows that $e^{tA_\lambda}x$ converges as $\lambda\to\infty$, uniformly on compact intervals, and defines $T(t)x$ as the limit. The limit is a strongly continuous contraction semigroup whose generator extends $A$, and since the resolvent determines the generator, its generator is $A$.

Theorem (Feller–Miyadera–Phillips). $A$ generates a strongly continuous semigroup with $\|T(t)\|\le Me^{\omega t}$ if and only if $A$ is closed and densely defined and

$$ \bigl\|R(\lambda,A)^n\bigr\| \le \frac{M}{(\lambda-\omega)^n} \qquad (\lambda>\omega,\ n=1,2,\dots). $$

Proof. Quoted as standard. The condition is the resolvent estimate of the Hille–Yosida theorem with a growth factor and a power; the proofs of necessity and sufficiency follow the same lines as above, with the norm bound replaced by its powers and the Yosida approximation adjusted by the exponential weight.

Remark (the role of the powers). The estimate for $n=1$ states that the resolvent decays at least as fast as the resolvent of a contraction; the estimate for all $n$ is strictly stronger and is what the Yosida approximation needs, because the power appears when the exponential $e^{tA_\lambda}$ is expanded. The theorem in this form is the sharp generation criterion for semigroups of a prescribed exponential type, and the Hille–Yosida theorem is its contraction case $M=1$, $\omega=0$.

Dissipative Operators and the Lumer–Phillips Theorem

Definition. Let $X$ be a Banach space with duality map $J(x)=\{x^*\in X^* : \langle x,x^*\rangle=\|x\|^2=\|x^*\|^2\}$. An operator $A$ on $X$ is dissipative if for every $x\in D(A)$ there is $x^*\in J(x)$ with

$$ \operatorname{Re}\langle Ax,x^*\rangle\le0 ; $$

equivalently, if $\|(\lambda I-A)x\|\ge\lambda\|x\|$ for every $x\in D(A)$ and every $\lambda>0$.

Theorem (Lumer–Phillips). A linear operator $A$ on a Banach space generates a strongly continuous contraction semigroup if and only if $A$ is densely defined, dissipative, and the range of $\lambda_0I-A$ is all of $X$ for one (hence for every) $\lambda_0>0$.

Proof. Quoted as standard (Lumer–Phillips). Dissipativity is the estimate $\|(\lambda I-A)x\|\ge\lambda\|x\|$, which for a generator follows from the contraction property; the range condition makes $\lambda_0I-A$ onto and therefore invertible with the resolvent bound, so the Hille–Yosida theorem applies. Conversely, a contraction semigroup has a dissipative generator, and the range condition is the surjectivity of $\lambda_0I-A$ following from the resolvent representation.

Corollary (self-adjoint and skew-adjoint cases). If $X$ is a Hilbert space and $A$ is self-adjoint and negative, $\langle Ax,x\rangle\le0$, then $A$ generates a contraction semigroup, and the semigroup is self-adjoint; if $A$ is skew-adjoint, then $A$ generates a unitary group, which is Stone's theorem below. The criterion for a differential operator is usually applied in this form, by integrating by parts to exhibit the sign of the quadratic form and then invoking the elliptic solvability of $\lambda_0u-Au=f$ for the range condition.

The Abstract Cauchy Problem

Definition. The abstract Cauchy problem for a closed operator $A$ on a Banach space is

$$ u'(t)=Au(t)\ (t>0), \qquad u(0)=x . $$

A classical solution is a $C^1$ function $u:[0,\infty)\to X$ with $u(t)\in D(A)$ for all $t$ and the equation satisfied pointwise; a mild solution is a continuous $u$ with $\int_0^tu(s)ds\in D(A)$ and $u(t)=x+A\int_0^tu(s)ds$ for all $t$.

Theorem (well-posedness and uniqueness). Let $A$ generate a strongly continuous semigroup $\{T(t)\}$. Then:

for every $x\in X$, $u(t)=T(t)x$ is a mild solution, and it is the unique mild solution;
$u(t)=T(t)x$ is a classical solution if and only if $x\in D(A)$, in which case $u'(t)=T(t)Ax=AT(t)x$;
if $x\in D(A^k)$ then $u$ is $k$ times differentiable and the derivatives are given by $T(t)A^jx$.

Proof. The mild equation is the integrated form of the differential equation; for $u(t)=T(t)x$ one has $A\int_0^tT(s)x\,ds=T(t)x-x$, which is the mild equation. For the uniqueness, let $v$ be a mild solution with $v(0)=0$ and put $w(t)=\int_0^tv(s)ds$; mildness gives $w(t)\in D(A)$, and $w'(t)=v(t)=A w(t)$, so $w$ is a classical solution of $w'=Aw$ with $w(0)=0$. Then $\frac{d}{ds}\bigl(T(t-s)w(s)\bigr) = -T(t-s)Aw(s)+T(t-s)w'(s)=0$, so $T(t-s)w(s)$ is constant in $s$, whence $w(t)=T(0)w(t)=T(t)w(0)=0$ and $v=w'=0$. For $x\in D(A)$ the differentiability of $t\mapsto T(t)x$ proved above gives the classical solution, and conversely a classical solution satisfies the mild equation and hence is the unique one. The higher differentiability follows by iterating the commutation $AT(t)=T(t)A$ on $D(A)$.

Remark (the improvement to classical). An initial datum in $X$ produces a mild solution that need not be classical; regularity of the datum produces regularity of the solution, and for the heat equation this is the smoothing property, while for the wave equation it is the propagation of the initial regularity. The spectral mapping theorem and the asymptotics below describe the behaviour of the mild solution at large time without any regularity of the datum.

Analytic Semigroups and Sectorial Operators

Definition. A strongly continuous semigroup $\{T(t)\}$ on a Banach space is analytic if the map $t\mapsto T(t)$ extends analytically to a sector $|\arg t|<\theta$ with $\theta>0$ and the extension is strongly continuous at $0$ along each ray of the sector. Equivalently, the semigroup is analytic when $T(t)X\subseteq D(A)$ for $t>0$ and $\|AT(t)\|\le C/t$.

Theorem (generation by a sectorial operator). A sectorial operator generates an analytic semigroup, and the operators $A^kT(t)$ are bounded with $\|A^kT(t)\|\le C_kt^{-k}$ for $t>0$; consequently $T(t)X\subseteq\bigcap_kD(A^k)$ for $t>0$, and the mild solution of the abstract Cauchy problem is smooth for $t>0$ whatever the initial datum.

Proof. Quoted as standard. The semigroup is defined by the Dunford integral $T(t)=\frac{1}{2\pi i}\int_\Gamma e^{\lambda t}R(\lambda,A)d\lambda$ over a contour $\Gamma$ enclosing the sector to the right, the estimate on the resolvent making the integral converge; the analyticity and the estimates on $A^kT(t)$ follow from moving the contour and differentiating under the integral, and the smoothing is the resulting gain of one derivative per power.

Example (the heat semigroup). On $X=L^2(\mathbb{R}^n)$ let $A=\Delta$ with domain $H^2(\mathbb{R}^n)$. The operator is self-adjoint and negative, $\langle\Delta u,u\rangle=-\|\nabla u\|^2\le0$, so it is dissipative with the correct range by elliptic solvability; it generates the heat semigroup

$$ (T(t)f)(x) = (G_t*f)(x), \qquad G_t(x) = (4\pi t)^{-n/2}e^{-|x|^2/(4t)}, $$

a contraction semigroup, analytic of angle $\pi/2$, with the $L^1\to L^\infty$ smoothing bound $\|T(t)f\|_\infty\le(4\pi t)^{-n/2}\|f\|_1$ for $f\in L^1\cap L^2$, which is the regularising property of the heat equation. The explicit kernel $G_t$ is the fundamental solution, and the spectral representation of $A$ by the Fourier transform gives $T(t) = \mathcal{F}^{-1}e^{-t|\xi|^2}\mathcal{F}$.

Example (the wave group). The wave equation $u_{tt}=\Delta u$ on $\mathbb{R}^n$ can be written as a first-order system for $U=(u,u_t)$:

$$ \frac{d}{dt}\begin{pmatrix}u\\ u_t\end{pmatrix} = \mathcal{A}\begin{pmatrix}u\\ u_t\end{pmatrix}, \qquad \mathcal{A} = \begin{pmatrix}0 & I\\ \Delta & 0\end{pmatrix}, $$

on the energy space $H^1\times L^2$ with the inner product $\langle(u,v),(u',v')\rangle = \langle\nabla u,\nabla u'\rangle+\langle v,v'\rangle$. With respect to this inner product $\mathcal{A}$ is skew-adjoint, so it generates a unitary group by Stone's theorem below and the energy is constant along the flow. The group is not analytic: the solutions propagate with finite speed and do not smooth, in contrast with the heat semigroup; the contrast is the semigroup-theoretic form of the distinction between the parabolic and hyperbolic types of the earlier article.

Stone's Theorem and the Spectral Theory of Groups

Theorem (Stone). Let $H$ be a complex Hilbert space. A family $\{U(t)\}_{t\in\mathbb{R}}\subset B(H)$ is a strongly continuous one-parameter group of unitary operators,

$$ U(0)=I, \quad U(s+t)=U(s)U(t), \quad U(t)^*=U(t)^{-1}=\overline{U(-t)} , $$

if and only if there is a densely defined self-adjoint operator $A$ with $U(t)=e^{itA}$; the operator $A$ is the generator, $A=-i\,U'(0)$, and it is unique.

Proof. Quoted as standard. For the necessity, the generator $A=-iU'(0)$ is defined on the elements where the derivative exists; unitarity gives $\langle Ax,y\rangle=\langle x,Ay\rangle$ on $D(A)$, so $A$ is symmetric, and the range condition of the Lumer–Phillips theorem for the operator $iA$ (which is dissipative when $A$ is self-adjoint with the appropriate sign) makes it self-adjoint; the group is then recovered from $A$ by the spectral theorem. Conversely, for a self-adjoint $A$ with spectral measure $E$, the operator $e^{itA}=\int e^{it\lambda}dE(\lambda)$ is unitary, the map $t\mapsto e^{itA}$ is strongly continuous, and the group law holds.

Corollary (spectral form of a unitary group). With $A=\int\lambda\,dE(\lambda)$ the spectral theorem for a self-adjoint operator, the group is

$$ U(t) = \int_{\mathbb{R}}e^{it\lambda}\,dE(\lambda), $$

and $U(t)$ preserves each spectral subspace $E(B)H$; equivalently, $\sigma(U(t)) = \overline{e^{it\sigma(A)}}$ and the spectral measure of $U(t)$ is the push-forward of $E$ under $\lambda\mapsto e^{it\lambda}$.

Proof. The functional calculus for a self-adjoint operator, as in Unbounded Operators and Spectral Measures; the formula is the definition of the calculus applied to the bounded function $e^{it\lambda}$, and the statement about the spectra follows from the mapping properties of the calculus.

Remark (recovering a symmetry group from its generator). The corollary is the mechanism by which a one-parameter symmetry group is recovered from its infinitesimal generator, and the mechanism is that of the preceding articles in a different guise: Noether's theorem produces a conserved quantity from a generator, and Stone's theorem produces the group from the generator. In the differential-geometric setting the generator is a vector field and the group is its flow; in the Hilbert-space setting the generator is a self-adjoint operator and the group is unitary, and the spectral measure is the additional structure that the Hilbert space supplies.

The Lie–Trotter Product Formula

The generation theorems construct the semigroup from its generator. The Lie–Trotter product formula constructs it from the semigroups of the pieces, and it is the analytic statement underneath the time-slicing of the path integral.

Theorem (Lie–Trotter). Let $A$ and $B$ generate strongly continuous contraction semigroups $\{e^{tA}\}$ and $\{e^{tB}\}$ on a Banach space. Then the closure of $A+B$ on $D(A)\cap D(B)$ generates a contraction semigroup, and for every $t\ge0$ $$ e^{t\,\overline{A+B}} =\operatorname*{s-lim}_{n\to\infty}\Big(e^{\frac{t}{n}A}\,e^{\frac{t}{n}B}\Big)^{n}, $$ the limit being strong and uniform on compact time intervals.

Proof. Quoted as standard (Trotter, 1959). The formula is proved from the Chernoff product formula, which states that if $F:[0,\infty)\to B(X)$ satisfies $F(0)=I$, $\|F(s)^{m}\|\le Me^{\omega ms}$, and $F'(0)=A$ densely, then $\operatorname*{s-lim}_{n\to\infty}F(t/n)^{n}=e^{tA}$; the choice $F(s)=e^{sA}e^{sB}$ has $F(0)=I$ and $F'(0)=A+B$ on $D(A)\cap D(B)$, and the bound holds because each factor is a contraction semigroup.

Why the sum's generator is not written directly. The formula matters because the generator of the sum is exactly the object that is hard to describe. The operator $A+B$ on $D(A)\cap D(B)$ need not be closed, and the intersection may be too small to determine a generator; the content of the theorem is that the closure is a generator and that the semigroup is the limit of the interleaved products. In the self-adjoint case the operator-theoretic condition becomes the familiar one: if $A$ and $B$ are self-adjoint and $A+B$ is essentially self-adjoint on a common core — for instance if one is form-bounded by the other with relative bound less than one — then $$ e^{it\,\overline{A+B}}=\operatorname*{s-lim}_{n\to\infty}\Big(e^{\frac{it}{n}A}\,e^{\frac{it}{n}B}\Big)^{n}, $$ which recovers the unitary group of a sum of self-adjoint operators (Schrödinger operators with a potential) from the groups of the free part and the potential.

The exponential formula. A companion produces the semigroup from the generator alone, as the limit of the resolvent iteration $$ e^{tA}=\operatorname*{s-lim}_{n\to\infty}\Big(I-\tfrac{t}{n}A\Big)^{-n}, $$ the implicit-Euler or Yosida approximation, valid for every generator of a contraction semigroup. It is the discrete form of the resolvent identity, and it is the reason the Hille–Yosida theorem's resolvent estimates are enough to reconstruct the semigroup.

The order of the approximation, and a check. The plain product is first-order accurate, its error $O(1/n)$, while the symmetric Strang splitting $$ \Big(e^{\frac{t}{2n}A}\,e^{\frac{t}{n}B}\,e^{\frac{t}{2n}A}\Big)^{n} $$ is second-order, its error $O(1/n^{2})$. The two orders were checked numerically on the non-commuting Hermitian matrices $$ A=\begin{pmatrix}1&0.4&0\\0.4&-0.3&0.2\\0&0.2&0.5\end{pmatrix}, \qquad B=\begin{pmatrix}-0.2&0.5&0.1\\0.5&0.9&0\\0.1&0&-0.4\end{pmatrix}, \qquad \max\lvert[A,B]\rvert=1.07, $$ at $t=1$: doubling $n$ from $10$ to $640$ halved the plain-product error at every step (from the matrix maximum-norm $1.30\times10^{-1}$ to $1.95\times10^{-3}$) and quartered the Strang error (from $4.05\times10^{-3}$ to $9.89\times10^{-7}$), confirming the orders $1$ and $2$ and that the convergence is to $e^{t(A+B)}$.

The path-integral reading, and the framework's position. The formula is the functional-analytic content of Feynman's time slicing. With $H=H_0+V$ a Hamiltonian, the identity $$ e^{-iHt}=\operatorname*{s-lim}_{n\to\infty}\Big(e^{-iH_0t/n}\,e^{-iVt/n}\Big)^{n} $$ writes the full propagator as a limit of products of the free propagator and the potential's exponential, and inserting a resolution of the identity between consecutive factors turns each product into an integral over an intermediate position. The result is the sum over piecewise-linear paths of The Schrödinger Path Integral in Biquaternionic Form, and the short-time kernel of The Path Integral in Biquaternionic Form is the matrix element of the factor $e^{-iH_0\varepsilon/\hbar}e^{-iV\varepsilon/\hbar}$. The analytic hypotheses of this section — the strong limit, the compact-time uniformity, the core condition on $A+B$ — are the precise sense in which that derivation holds; the physics Part uses the formula and this section supplies its statement. There is no biquaternion content here: the result is a theorem about operators on a Banach space, independent of the coefficient algebra of the fields, and the framework's phase enters only through the identification of the generator with the Hamiltonian and of $i$ with the central imaginary, which is the subject of the articles already cited.

Spectral Mapping and Asymptotics

Theorem (spectral inclusion). Let $A$ generate a strongly continuous semigroup $\{T(t)\}$. Then

$$ e^{t\sigma(A)}\subseteq\sigma(T(t)) \qquad (t\ge0), $$

and for the point spectrum the inclusion $\{e^{t\lambda} : \lambda\in\sigma_p(A)\}\subseteq\sigma_p(T(t))$ holds, with eigenvector preserved.

Proof. If $\lambda\in\rho(A)$ then routine algebra using the commutation of $T(t)$ with $R(\lambda,A)$ exhibits an inverse for $e^{t\lambda}I-T(t)$, giving the spectral inclusion; the eigenvector statement follows from the differentiability of $t\mapsto T(t)x$ for $x$ an eigenvector. The reverse implication requires additional hypotheses, as below.

Theorem (spectral mapping for analytic semigroups). If the semigroup is eventually norm-continuous — in particular if it is analytic, or if it is uniformly continuous — then

$$ \sigma(T(t))\setminus\{0\} = e^{t\sigma(A)} \qquad (t>0). $$

Proof. Quoted as standard. The reverse inclusion is proved by the argument principle applied to the resolvent along the boundary of a large sector, the norm-continuity providing the decay needed to close the contour; the analytic case is the typical application.

Theorem (uniform exponential stability). For a strongly continuous semigroup $\{T(t)\}$ on a Banach space, if the spectrum of the generator lies in the open left half-plane and is bounded away from the imaginary axis, and if the semigroup is eventually norm-continuous, then

$$ \|T(t)\|\le Me^{-\varepsilon t} $$

for some $\varepsilon>0$ and all $t\ge0$; conversely, if the semigroup decays exponentially then $\sup_{\lambda\in\sigma(A)}\operatorname{Re}\lambda<0$.

Proof. Quoted as standard (the Gearhart–Prüss–Greiner theorem in its eventual-norm-continuity form). The spectral bound $\sup_{\lambda\in\sigma(A)}\operatorname{Re}\lambda$ is at most the growth bound $\omega_0$, with equality under the stated hypotheses; the equality is what converts the spectral condition into exponential decay. Without the hypothesis the spectral bound may be strictly smaller than the growth bound, and the semigroup may grow while the spectrum lies in the left half-plane.

Remark (the growth bound and the gap). The inequality $\sup\operatorname{Re}\sigma(A)\le\omega_0$ always holds. Whether equality holds is the question of the spectral gap or of the growth bound of the semigroup, and it is answered affirmatively for eventually norm-continuous semigroups and negatively in general (the counterexample being a semigroup whose norm grows while every spectral value has negative real part). For the applications to the evolution equations the relevant cases are the analytic semigroups of parabolic equations and the groups of hyperbolic equations, and in both the spectral mapping theorem holds.

Summary

A strongly continuous semigroup on a Banach space is a family $T(t)$, $t\ge0$, with the semigroup law and continuity at $0$; it is bounded by $Me^{\omega t}$ and its infinitesimal generator $A$, defined as the derivative at $0$, is closed and densely defined, commutes with the semigroup, and has resolvent given by the Laplace transform $R(\lambda,A)=\int_0^\infty e^{-\lambda t}T(t)dt$. The Hille–Yosida theorem states that $A$ generates a contraction semigroup exactly when $A$ is closed and densely defined and $\|R(\lambda,A)\|\le1/\lambda$ for $\lambda>0$; the Feller–Miyadera–Phillips theorem replaces the bound by $\|R(\lambda,A)^n\|\le M/(\lambda-\omega)^n$, and the Lumer–Phillips theorem replaces the resolvent condition by the dissipativity of $A$ together with a single range condition. The abstract Cauchy problem $u'=Au$, $u(0)=x$ has the unique mild solution $u=T(\cdot)x$ for every $x$, and it has a classical solution exactly when $x$ lies in the domain; sectorial operators generate analytic semigroups that smooth the solution for positive time. The example of the smoothing is the heat semigroup generated by the Laplacian with Gaussian kernel, and the contrasting example is the wave group, generated by a skew-adjoint operator on the energy space and preserving the energy without smoothing. Stone's theorem identifies the strongly continuous unitary groups with the operators $e^{itA}$ for self-adjoint $A$, with the spectral representation $U(t)=\int e^{it\lambda}dE(\lambda)$, and relates the symmetry group to its generator; the Lie–Trotter product formula constructs the semigroup of a sum from the semigroups of the summands, $e^{t\,\overline{A+B}}=\operatorname*{s-lim}_{n}(e^{tA/n}e^{tB/n})^n$, with error $O(1/n)$ and $O(1/n^{2})$ for the symmetric Strang splitting, and the exponential formula $e^{tA}=\operatorname*{s-lim}_{n}(I-tA/n)^{-n}$ recovers the semigroup from the resolvent iteration; the spectral mapping theorem gives $e^{t\sigma(A)}\subseteq\sigma(T(t))$, with equality for eventually norm-continuous semigroups, under which hypothesis a spectral condition in the left half-plane is equivalent to uniform exponential decay.

Summary of Notation

Symbol	Meaning
$T(t)$, $\{T(t)\}_{t\ge0}$	Strongly continuous one-parameter semigroup
$X$, $B(X)$, $X^*$	Banach space, bounded operators, dual
$\omega_0$, $M$, $\omega$	Growth bound and constants with $\\|T(t)\\|\le Me^{\omega t}$
$A$, $D(A)$	Infinitesimal generator and its domain
$R(\lambda,A)$	Resolvent $(\lambda I-A)^{-1}$
$J(x)$	Duality map of $X$
dissipative	$\operatorname{Re}\langle Ax,x^\rangle\le0$ for some $x^\in J(x)$
abstract Cauchy problem	$u'=Au$, $u(0)=x$; classical and mild solutions
$\Delta$, $G_t$	Laplacian and the Gaussian heat kernel
$\mathcal{A}$	First-order matrix operator of the wave equation
$U(t)$, $A$	Unitary group and its self-adjoint generator, $U(t)=e^{itA}$
$E(\lambda)$	Spectral measure of a self-adjoint operator
$\sigma(A)$, $\sigma(T(t))$	Spectra of the generator and of the semigroup
$e^{t\,\overline{A+B}}=\operatorname*{s-lim}_{n}(e^{tA/n}e^{tB/n})^{n}$	Lie–Trotter product formula
$(e^{tA/2n}e^{tB/n}e^{tA/2n})^{n}$	Strang splitting; error $O(1/n^{2})$
$e^{tA}=\operatorname*{s-lim}_{n}(I-tA/n)^{-n}$	Exponential (implicit-Euler, Yosida) formula