Contraction semigroup ($\Vert S(t)\Vert \le 1$)

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 $n\ge 1$.

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 $1\notin C_0(X)$.

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.