C0-semigroup ($C_0$-semigroup)

A C0-semigroup is a family of bounded operators $U(t)$ for $t\ge 0$ satisfying $U(0)=I$, $U(t+s)=U(t)U(s)$, and $U(t)x\to x$ in norm as $t\downarrow 0$ for every $x$.

An analytic semigroup is a strongly continuous semigroup whose bounded-operator-valued extension is holomorphic in a sector about the positive time axis. Its infinitesimal generator of a semigroup $L$ satisfies $S(t)H\subseteq D(L)$ for $t>0$ and, on each bounded time interval, $\Vert LS(t)\Vert \le C/t$. This estimate gives instantaneous spatial smoothing but creates an integrable-singularity question in an inhomogeneous variation-of-constants formula.

A contraction semigroup is a strongly continuous semigroup of bounded linear operators satisfying $\Vert S(t)\Vert \le 1$ for every $t\ge 0$. The Hille-Yosida theorem characterizes its infinitesimal generator of a semigroup by dense domain, closedness, and the resolvent estimates $\Vert (\lambda -A{)}^{-n}\Vert \le {\lambda }^{-n}$ for real $\lambda >0$ and integers $n\ge 1$.

A semigroup is hypercontractive when evolution improves integrability through norm bounds $\Vert P_{t}f{\Vert }_{q(t)}\le \Vert f{\Vert }_p$ with $q(t)>p$ for positive times. For a symmetric Markov semigroup satisfying a logarithmic Sobolev inequality with the joint-energy normalization, one may take $q(t)=1+(p-1)e^{4t/c_{\mathrm{LS}}}$.

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On its maximal domain in ${\ell }^2$, put $g_n=2^n{f}_n$. Then $g_n-g_{n-1}=-2^{-n}(Qf{)}_n$, so $g_n$ has a finite limit ${\ell }_f$. Summation gives $\mathrm{Re}⟨Qf,f⟩=-\frac{1}{2}(|g_0{|}^2+|{\ell }_f{|}^2+{\sum }_{n\ge 1}|g_n-g_{n-1}{|}^2)$. For $\alpha >0$, the resolvent equation is the recursion $g_n=(g_{n-1}+2^{-n}{h}_n)/(1+\alpha 4^{-n})$, which produces a bounded $g_n$ and hence $f\in {\ell }^2$. The Lumer-Phillips theorem proves contraction-semigroup generation.

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On a compact metric space, a conservative Feller semigroup is a strongly continuous semigroup of positive unital operators on $C(X)$. The Riesz-Markov-Kakutani representation theorem associates a Markov kernel to each operator. On noncompact spaces the standard definition uses $C_0(X)$; conservativity then refers to the corresponding kernels having mass one, even when $1\notin C_0(X)$.

A Feller semigroup symmetric in the space associated with an invariant measure has a self-adjoint contraction realization in $L^2(\mu )$. Its two-point kernel measure $\mu (dx)P_t(x,dy)$ is symmetric. This reversibility expresses its joint energy as an integral of products of increments.

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For functions in the form domain, this symmetric bilinear energy equals $-⟨Lf,g⟩$ when $f$ is in the generator domain. Its finite-time approximation is $(2t{)}^{-1}\int (f(x)-f(y))(g(x)-g(y))\mu (dx)P_t(x,dy)$. The quadratic energy ${\mathcal{E}}_{\mu }(f)={\mathcal{E}}_{\mu }(f,f)$ may take the extended value infinity.

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For $a,b\ge 0$ and $q>2$, Cauchy-Schwarz inequality applied to the integral of $(q/2)s^{q/2-1}$ gives $(a^{q/2}-b^{q/2}{)}^2\le q^2(a^{q-1}-b^{q-1})(a-b)/(4(q-1))$. Integrating against the symmetric two-point kernel measure and taking the energy limit proves the displayed inequality.

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For a Markov process generator and suitable real functions, this bilinear expression records the local quadratic energy. Positivity of $\Gamma (f,f)$ follows by differentiating $P_t(f^2)\ge (P_{t}f{)}^2$. For $L=D^2-xD$ it is $f^{\prime }{g}^{\prime }$, giving the Gaussian Dirichlet energy after integration against the invariant measure.

If real $f$ and $f^2$ belong to the generator domain of a Feller semigroup, Jensen inequality gives $P_t(f^2)-(P_{t}f{)}^2\ge 0$. This expression is zero at $t=0$, so its right derivative is nonnegative, proving the inequality. It is equivalent to positivity of the carré du champ.

The one-dimensional semigroup associated with the standard Ornstein-Uhlenbeck process has invariant standard Gaussian measure and generator $Lf=f^{\prime \prime }-x{f}^{\prime }$. Its Mehler formula for the Ornstein-Uhlenbeck semigroup makes positivity and invariance explicit. The normalized Probabilists' Hermite polynomials diagonalize it, with $P_t{h}_n=e^{-nt}{h}_n$. It has spectral gap one and satisfies the sharp Gaussian Poincaré inequality.

Differentiating the Mehler formula for the Ornstein-Uhlenbeck semigroup proves the identity for continuously differentiable functions with bounded derivative. Density extends it to the Gaussian Sobolev form domain. Together with a weighted Cauchy-Schwarz inequality, it yields the Ornstein-Uhlenbeck entropy dissipation identity proof of the Gaussian logarithmic Sobolev inequality.

On polynomials in $L^2$ of standard Gaussian measure, Gaussian integration by parts makes $D$ and $x-D$ adjoints, with commutator $[a^{-},a^{+}]=I$. The Probabilists' Hermite polynomials are obtained by applying $a^{+}$ repeatedly to $1$, and $a^{-}{\mathrm{He}}_n=n{\mathrm{He}}_{n-1}$. These give a Gaussian-space realization of creation and annihilation operators.

The normalized Probabilists' Hermite polynomials diagonalize the Gaussian number operator with eigenvalues $n\ge 0$. Its self-adjoint closure has domain $\sum n^2|c_n{|}^2<\infty$. Polynomial truncations are dense for its graph norm. The semigroup $e^{-tN}$ is the Ornstein-Uhlenbeck semigroup and its energy form is the Gaussian Dirichlet energy.

The energy form of the Gaussian number operator is the displayed squared derivative integral. Its form domain requires $\sum n|c_n{|}^2<\infty$, while the full operator domain requires $\sum n^2|c_n{|}^2<\infty$. It can be defined by ${\lim }_{t\downarrow 0}{t}^{-1}(\Vert f{\Vert }_2^2-⟨f,P_{t}f⟩)$ even when $Nf$ does not exist in $L^2$. This energy occurs in the Gaussian Poincaré inequality and Gaussian logarithmic Sobolev inequality.

Here $Z$ has the standard normal distribution. Applied to the generating function $e^{sx-s^2/2}$, the Gaussian moment-generating function replaces $s$ by $e^{-t}s$, proving the Hermite eigenvalue identity. Positivity and preservation of Gaussian measure then give the $L^2$ contraction and identify the full Ornstein-Uhlenbeck semigroup by polynomial density.

A nonnegative continuous multiplier $k$ defines a contraction semigroup on the space of continuous functions vanishing at infinity and on $L^2(\mu )$ by $P_{t}f=e^{-tk}f$. Strong continuity follows from uniform convergence on compact sets and small tails in $C_0$, or the dominated convergence theorem in $L^2$. The multiplier itself may be unbounded.

The infinitesimal generator of a semigroup for $P_{t}f=e^{-tk}f$ is multiplication by $-k$ on its maximal domain. The bound $|(1-e^{-tk})/t|\le k$ proves the norm derivative when $kf$ is integrable; in $C_0$ use a compact-small-tail argument on $kf$. Conversely norm convergence identifies the pointwise derivative as $-kf$. Limits of graph pairs satisfy the same multiplication identity, proving it is a closed operator.

The semigroup property is $U(t+s)=U(t)U(s)$ for nonnegative times, together with $U(0)=I$. It expresses that evolving for two consecutive intervals is equivalent to evolving for their total duration.

An operator family $U(t)$ is strongly continuous when $t\mapsto U(t)x$ is norm-continuous for every vector $x$. This is continuity in the strong operator topology.

The infinitesimal generator is the generally unbounded operatoron the vectors for which the norm limit exists.

For a strongly continuous semigroup, apply the bounded operator $T_t$ to the difference quotient defining its semigroup generator. The limit proves domain invariance and commutation on the generator domain. This is not an assertion that the unbounded generator acts on every vector.

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The Yosida averaging of a semigroup gives $x_h=h^{-1}{\int }_0^{h}S(t)xdt\in D(A)$ with $x_h\to x$, proving that the generator domain is dense. For $x\in D(A)$, the Bochner integral identity $S(t)x-x={\int }_0^{t}S(s)Axds$ holds. If $x_n\to x$ and $A{x}_n\to y$, pass to the limit in this identity and divide by $t$. The strongly continuous semigroup property makes the resulting limit $y$, so $x\in D(A)$ and $Ax=y$. This proves that the infinitesimal generator of a semigroup is a closed operator.

The generator domain consists of the vectors whose semigroup orbit is right-differentiable at zero. It is a dense linear subspace, and the generator is closed.

Equip $D(A)$ with its graph norm. The restrictions $U(t){|}_{D(A)}$ form a C0-semigroup on this Banach space; its generator is $A$ restricted to $D(A^2)$.

The Hille-Yosida theorem characterizes generators of exponentially bounded C0-semigroups by closedness, dense domain, a resolvent half-line, and uniform bounds on every positive resolvent power.

If $A$ generates a C0-semigroup satisfying $\Vert U(t)\Vert \le M{e}^{\omega t}$, then for $\mathrm{Re}z>\omega$ its resolvent operator isRepeated differentiation gives $n!(zI-A{)}^{-(n+1)}x={\int }_0^{\infty }{t}^n{e}^{-zt}U(t)xdt$.

For $x$ in the underlying Banach space,belongs to the generator domain, satisfies $A{x}_t=[U(t)x-x]/t$, and converges to $x$ as $t\downarrow 0$.

If $U(t)$ has generator $A$, then $\hat{U}(t)=e^{-\omega t}U(t)$ has generator $A-\omega I$. This shift converts a growth bound $M{e}^{\omega t}$ into the uniform bound $M$.

A time-indexed family $A(t)$ is stable with constants $M,\omega$ when products of its frozen semigroups obeyfor all nonnegative $s_j$. This condition controls products uniformly as the number of time slices grows.

On a Hilbert space, an operator $A$ is dissipative when $\mathrm{Re}(Ax,x)\le 0$ for every $x\in D(A)$. This is the infinitesimal form of norm contraction.

For a closed densely defined dissipative operator, the estimate $\Vert (\alpha I-Z)f\Vert \ge \alpha \Vert f\Vert$ makes its range closed and its kernel trivial. Thus a positive spectral value means this range is proper, which is equivalent to a nonzero vector in $\ker (\alpha I-Z^{\ast })$. Dissipativity is essential: a real multiplication operator may have positive continuous spectrum with no adjoint eigenvector.

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If $\mathrm{Re}⟨Lv,v⟩\le 0$ in a finite-dimensional Hilbert space, $I-hL/2$ is invertible for every positive $h$. The relation $v_{+}-v_{-}=(h/2)L(v_{+}+v_{-})$ gives $\Vert v_{+}{\Vert }^2-\Vert v_{-}{\Vert }^2=(h/2)\mathrm{Re}⟨L(v_{+}+v_{-}),v_{+}+v_{-}⟩\le 0$. Thus the Crank-Nicolson method contracts for such a linear system even when $L$ is a non-normal matrix. This is a direct energy result, not an unqualified inequality by a scalar rational function evaluated at a logarithmic norm.

A maximal dissipative operator is dissipative and has no proper dissipative extension. Equivalently, for a densely defined dissipative operator, $\lambda I-A$ is onto for some positive $\lambda$.

The Lumer-Phillips theorem says that a densely defined operator generates a contraction C0-semigroup exactly when it is maximal dissipative.

A strongly continuous unitary group is a family $U(t)$ for all real $t$ that is strongly continuous, obeys the group law, and preserves the Hilbert norm. Its generator is skew-adjoint; conversely, every skew-adjoint operator generates such a group.

Every strongly continuous unitary group has a unique possibly unbounded self-adjoint operator $A$ as its generator. Its domain consists of the vectors for which the norm limit of $(U_t\xi -\xi )/(it)$ exists as $t\to 0$. Conversely, the Borel functional calculus for a normal operator constructs a strongly continuous unitary group from every self-adjoint $A$. Smooth averaging supplies a dense generator domain; Laplace integrals of $U_t$ and $U_{-t}$ prove that $A\pm i$ are surjective, which establishes self-adjointness.

An operator $A$ is skew-adjoint when $A^{\ast }=-A$. Both $A$ and $-A$ are then maximal dissipative, and the generated group is unitary.

On a finite-dimensional real inner product space, a skew-adjoint operator has even rank. In an orthonormal basis its matrix is skew-symmetric, so an invertible such matrix must have even dimension by $\det A=\det (-A)=(-1{)}^d\det A$. Moreover $\ker A=(\mathrm{im}A{)}^{\perp }$, so the image and kernel intersect trivially. The restriction of $A$ to its image is an invertible skew-adjoint map, and its dimension, the rank, is therefore even.

On a finite-dimensional positive inner-product space, a skew-adjoint nilpotent operator is zero. In an orthonormal basis, $\Vert A{\Vert }_F^2=\mathrm{tr}(A^{\ast }A)=-\mathrm{tr}(A^2)=0$, since a nilpotent matrix has zero trace and so does its square. The Frobenius norm therefore vanishes. In particular, a nonzero nilpotent adjoint map rules out a compact Lie algebra.

An abstract Cauchy problem has the form $Z^{\prime }(t)=AZ(t)+F(t)$ in a Banach space, where $A$ may be unbounded and generates the homogeneous evolution.

For a closed operator $L$ on a Banach space $H$, a positive-time classical solution belongs to $C([0,T],H)\cap C^1((0,T],H)$, takes values in $D(L)$ for positive time, and has $Lu\in C((0,T],H)$. It satisfies the differential equation and the initial condition in $H$. Classical regularity also at time zero requires the corresponding regularity of the initial datum, in particular $u_0\in D(L)$. For an analytic semigroup and Hölder continuous forcing, the variation-of-constants formula gives such a solution: subtracting the forcing at the current time turns the singular estimate $\Vert LS(r)\Vert \le C/r$ into the integrable bound $C{r}^{\theta -1}$.

A nonautonomous abstract Cauchy problem has the form $u^{\prime }(t)=A(t)u(t)$, with a time-dependent generally closed linear operator. Under stability, common-domain, and regularity hypotheses, it is propagated by an evolution family.

An evolution family consists of bounded operators $U(t,s)$ for $s\le t$ satisfying $U(s,s)=I$ and $U(t,r)U(r,s)=U(t,s)$. For sufficiently regular data it solves ${\partial }_{t}U(t,s)x=A(t)U(t,s)x$ and ${\partial }_{s}U(t,s)x=-U(t,s)A(s)x$.

For a partition $s=t_0<\cdots <t_N=t$, the frozen-generator approximation isUnder the hypotheses of the nonautonomous generation theorem, these products converge strongly and uniformly on compact time triangles to the evolution family.

A mild solution need only be continuous and satisfy the integrated semigroup formula. It need not lie in the generator domain or be differentiable.

If $A$ generates $U(t)$, the variation-of-constants formula isIt is also called Duhamel's formula.

A semilinear abstract Cauchy problem has $Z^{\prime }=AZ+N(Z)$, where the linear part generates a C0-semigroup and the nonlinear map is typically locally Lipschitz on the phase space.

A local mild solution is a fixed point of the nonlinear variation-of-constants map in $C([0,T];X)$. Local Lipschitz continuity of the nonlinearity gives existence and uniqueness for sufficiently small $T$.

For a locally Lipschitz semilinear evolution, a maximal mild solution either exists for all positive time or its phase-space norm becomes unbounded as the finite maximal time is approached.