Semigroups encode time evolution by operators. Their generators are often unbounded, so semigroup theory connects differential equations with resolvents and closed-operator methods.
Strongly continuous semigroups
A semigroup represents deterministic time evolution. Strong continuity says that each initial state moves continuously at time zero.
Definition. A family \((T(t))_{t\ge0}\subset\mathcal L(X)\) on a Banach space is a \(C_0\)-semigroup if \(T(0)=I\), \(T(t+s)=T(t)T(s)\), and \(\lim_{t\downarrow0}\|T(t)x-x\|=0\) for every \(x\in X\).
Definition. The generator \(A\) is \(Ax=\lim_{t\downarrow0}(T(t)x-x)/t\) on the domain \(D(A)\) of vectors for which this norm limit exists.
Exponential semigroups
Bounded operators generate semigroups by the ordinary exponential series. This is the model case before unbounded generators enter.
Theorem. If \(A\in\mathcal L(X)\), then \(T(t)=e^{tA}=\sum_{n=0}^\infty t^nA^n/n!\) is a \(C_0\)-semigroup with generator \(A\).
Proof. The series converges absolutely in \(\mathcal L(X)\) because \(\sum t^n\|A\|^n/n!<\infty\). Multiplication of absolutely convergent operator series gives \(e^{(t+s)A}=e^{tA}e^{sA}\). Norm continuity in \(t\) follows from uniform convergence of the series on compact time intervals. Finally \((e^{tA}x-x)/t\to Ax\) by separating the \(n=1\) term and bounding the remainder by \(O(t)\|x\|\).
Generators are closed and densely defined
The generator is the infinitesimal derivative of the semigroup. It is usually unbounded, but closedness and dense domain keep it analytically meaningful.
Theorem. The generator \(A\) of a \(C_0\)-semigroup is closed and densely defined.
Proof. Closedness: suppose \(x_n\in D(A)\), \(x_n\to x\), and \(Ax_n\to y\). For \(x_n\), the identity \(T(t)x_n-x_n=\int_0^tT(s)Ax_n\,\mathrm{d}s\) holds. Passing to the limit under the boundedness of \(T(s)\) on compact \(s\)-intervals gives \(T(t)x-x=\int_0^tT(s)y\,\mathrm{d}s\). Dividing by \(t\) and letting \(t\downarrow0\) yields \(x\in D(A)\) and \(Ax=y\).
Density follows from Yosida averages \(x_t=t^{-1}\int_0^tT(s)x\,\mathrm{d}s\). Strong continuity gives \(x_t\to x\). One checks \(x_t\in D(A)\) and \(Ax_t=t^{-1}(T(t)x-x)\). Thus \(D(A)\) is dense.
Resolvent as Laplace transform
The resolvent can be recovered from the semigroup by a Laplace transform. This connects long-time evolution with inverse-operator estimates.
Theorem. If \(\|T(t)\|\le Me^{\omega t}\), then for \(\lambda>\omega\), \((\lambda I-A)^{-1}x=\int_0^\infty e^{-\lambda t}T(t)x\,\mathrm{d}t\), and \(\|(\lambda I-A)^{-1}\|\le M/(\lambda-\omega)\).
Proof. The Bochner integral converges because \(\int_0^\infty e^{-\lambda t}\|T(t)x\|\,\mathrm{d}t\le M\|x\|/(\lambda-\omega)\). Let \(R_\lambda x\) denote the integral. Using the semigroup identity and differentiating \(T(s)R_\lambda x\) under the integral gives \(R_\lambda x\in D(A)\) and \((\lambda I-A)R_\lambda x=x\). Conversely, for \(x\in D(A)\), integrate \(e^{-\lambda t}T(t)(\lambda x-Ax)\) by parts to obtain \(x\).
Hille-Yosida contraction theorem
Hille-Yosida characterizes which unbounded operators generate contraction semigroups. It is the semigroup analogue of an existence theorem for differential equations.
Theorem. A densely defined closed operator \(A\) generates a contraction semigroup on \(X\) iff \((0,\infty)\subset\rho(A)\) and \(\|\lambda(\lambda I-A)^{-1}\|\le1\) for every \(\lambda>0\).
Proof. Necessity follows from the Laplace transform resolvent formula with \(M=1,\omega=0\). Then \(\|\lambda R(\lambda,A)\|\le\lambda\int_0^\infty e^{-\lambda t}\|T(t)\|\,\mathrm{d}t\le1\).
For sufficiency, define the Yosida approximants \(A_\lambda=\lambda A(\lambda I-A)^{-1}=\lambda^2(\lambda I-A)^{-1}-\lambda I\), which are bounded. The bounded-operator theorem gives contraction semigroups \(e^{tA_\lambda}\). The resolvent estimates imply these semigroups are uniformly contractive and Cauchy on dense sets as \(\lambda\to\infty\). Extending by contraction gives \(T(t)x=\lim_{\lambda\to\infty}e^{tA_\lambda}x\). The semigroup law and strong continuity pass to the limit, and the generator is \(A\) by resolvent convergence on \(D(A)\).
Variation of constants
Variation of constants is the operator form of solving an inhomogeneous evolution equation by integrating the propagated forcing term.
Proposition. If \(A\) generates \(T(t)\) and \(u'(t)=Au(t)+f(t)\), \(u(0)=x\), then the mild solution is \(u(t)=T(t)x+\int_0^tT(t-s)f(s)\,\mathrm{d}s\).
Proof. For smooth \(f\) and \(x\in D(A)\), differentiate the right side using the generator identity and the fundamental theorem of calculus for Banach-valued integrals. The formula then extends to continuous \(f\) and general \(x\) by density of \(D(A)\), closedness of \(A\), and uniform boundedness of \(T(t)\) on compact intervals.
Heat equation
The heat equation is the basic example: the semigroup smooths and contracts, while its generator is a second-order differential operator.
Example. On \(L^2(\mathbb R^d)\), the Laplacian with domain \(H^2(\mathbb R^d)\) generates the heat semigroup \(T(t)f=G_t*f\), where \(G_t\) is the Gaussian heat kernel. The Fourier transform gives \(\widehat{T(t)f}(\xi)=e^{-t|\xi|^2}\hat f(\xi)\), a contraction semigroup whose generator is multiplication by \(-|\xi|^2\), i.e. \(\Delta\).