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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.

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.

Consider the operators on all of $C_0(X)$ or $L^2(\mu )$, extending the displayed positive-function formula linearly. Here $C_0(X)$ is the space of continuous functions vanishing at infinity. The multiplier has modulus at most one, so $\Vert P_{t}f\Vert \le \Vert f\Vert$, and multiplication of exponentials gives $P_{s+t}=P_s{P}_t$ and $P_0=I$. For strong continuity on $C_0(X)$, choose a compact $K$ outside which $|f|$ is small. On $K$ the continuous $k$ is bounded, making $e^{-tk}\to 1$ uniform; outside $K$, $|(e^{-tk}-1)f|\le |f|$. For $L^2(\mu )$, use pointwise convergence and the dominated convergence theorem with bound $|(e^{-tk}-1)f{|}^2\le |f{|}^2$. Thus both are contraction semigroups, even when $k$ is unbounded.

The generator of a multiplication semigroup isIndeed pointwise difference quotients converge to $-kf$. A norm limit therefore forces the displayed domain condition, using an almost-everywhere convergent subsequence for the $L^2$ case. Conversely $|(e^{-tk}-1)/t|\le k$, so the dominated convergence theorem proves the $L^2$ limit when $kf\in L^2$. In $C_0(X)$ write the error as $kf$ times $1-(1-e^{-tk})/(tk)$, with the value at $k=0$ supplied by continuity. That second factor tends uniformly to zero where $k$ is bounded and has modulus at most one; the compact-set and small-tail argument applied to $kf$ proves uniform convergence. To prove directly that $L$ is a closed operator, take $f_n\to f$ and $L{f}_n\to g$. Pointwise limits in $C_0$, or a common almost-everywhere convergent subsequence in $L^2$, give $g=-kf$. Thus $f$ is in the appropriate generator domain and $Lf=g$. In $L^2$ the domain is dense because $f{1}_{\{k\le n\}}\to f$; in $C_0$ compactly supported continuous functions are a dense subspace contained in the domain.

A real multiplication operator on its maximal $L^2$ domain is an unbounded self-adjoint operator. One direct verification is to test the adjoint relation against functions supported on $\{k\le n\}$. If $h$ is in the adjoint operator domain with representing vector $g$, those tests force $g=-kh$ on each such set, hence $kh\in L^2$. The converse follows by integration. On the sigma-finite measure spaces used below, the spectrum of a real multiplication operator is its essential range; spectral values are detected by unit vectors supported where the multiplier is arbitrarily close to that value, while outside the essential range its reciprocal is a bounded resolvent multiplier.

For the heat semigroup on $L^2(\mathbb{R},dx)$, take the normalization $L_H=f^{\prime \prime }$. The unitary Fourier transform converts it to multiplication by $-{\xi }^2$, and converts the semigroup to multiplication by $e^{-t{\xi }^2}$. ConsequentlyThe Sobolev space domain is exactly $\{f:{\xi }^2\hat{f}\in L^2\}$. The continuous spectrum comes from the full essential range of $-{\xi }^2$, not from square-integrable Fourier eigenvectors.

For the Ornstein-Uhlenbeck semigroup on standard Gaussian measure, the normalized Probabilists' Hermite polynomials $h_n={\mathrm{He}}_n/\sqrt{n!}$ form an orthonormal basis. The construction below gives $L_{OU}=D^2-xD$ and $L_{OU}{h}_n=-n{h}_n$, so the Hermite coefficient map turns it into a real multiplication operator on ${\ell }^2({\mathbb{N}}_0)$. ThereforeFinally, the heat semigroup on the whole real line has no positive spectral gap, whereas the standard Ornstein-Uhlenbeck semigroup has gap one. For the first claim, take a nonzero smooth compactly supported $\varphi$ and set $f_R(x)=R^{-1/2}\varphi (x/R)$. Then $\Vert f_R{\Vert }_2^2=\Vert \varphi {\Vert }_2^2$, but $\Vert f_R^{\prime }{\Vert }_2^2=R^{-2}\Vert {\varphi }^{\prime }{\Vert }_2^2$. This disproves a uniform whole-line Poincaré inequality $\Vert f{\Vert }_2^2\le C\Vert f^{\prime }{\Vert }_2^2$; one can also choose $\int \varphi =0$. Lebesgue measure here is infinite, so it has no normalized probability mean to subtract. For Gaussian measure, the Hermite expansion instead gives the sharp Gaussian Poincaré inequalityEquality holds for affine functions. The identity on the right is interpreted on the energy-form domain, which is larger than the full operator domain.

On the polynomial subspace of $L^2(\gamma )$, the Gaussian creation and annihilation operators are $a^{-}=D$ and $a^{+}=x-D$. Gaussian integration by parts makes them adjoints there and gives $[a^{-},a^{+}]=I$. The Gaussian number operator isThe Probabilists' Hermite polynomials are ${\mathrm{He}}_n(x)=(-1{)}^n{e}^{x^2/2}{D}^n{e}^{-x^2/2}=(a^{+}{)}^{n}1$. Their generating function is $e^{sx-s^2/2}={\sum }_n{\mathrm{He}}_n(x)s^n/n!$, giving $D{\mathrm{He}}_n=n{\mathrm{He}}_{n-1}$. The commutator also yieldsThe orthogonality follows by repeated integration by parts; the leading coefficient is one, so these polynomials span every polynomial. By the allowed density assumption $h_n={\mathrm{He}}_n/\sqrt{n!}$ is an orthonormal basis. Define the closed number operator by $N\sum c_n{h}_n=\sum n{c}_n{h}_n$ on $\sum n^2|c_n{|}^2<\infty$. This diagonal multiplication operator is self-adjoint and positive semidefinite; polynomial truncations show it is the closure of the polynomial operator.

The Ornstein-Uhlenbeck semigroup is $P_t=e^{-tN}$, or $P_{t}f={\sum }_n{e}^{-nt}{c}_n{h}_n$. It also has the Mehler formula for the Ornstein-Uhlenbeck semigroupTo verify the formula, apply its right side to $e^{sx-s^2/2}$: the Gaussian moment-generating function makes the result $e^{s{e}^{-t}x-s^2{e}^{-2t}/2}$, proving the eigenvalue identity on every Hermite polynomial. Positivity and invariance of Gaussian measure give $L^2$ contraction by Jensen inequality, so polynomial density extends the equality to all $L^2(\gamma )$. The coefficient expansion givesFor continuously differentiable $f$ with bounded derivative, differentiate the Mehler expectation by the dominated convergence theorem to get the Ornstein-Uhlenbeck gradient commutation identityThe same identity extends to the Gaussian Sobolev form domain. Mehler's formula also gives pointwise convergence to the Gaussian mean for such $f$, since bounded derivative permits at most linear growth.

The Gaussian Dirichlet energy is the energy form of $N$:For polynomials this follows from $N=(a^{-}{)}^{\ast }{a}^{-}$, and closure extends it to the form domain. For the given continuously differentiable $f$, Gaussian integration by parts yields $⟨f^{\prime },h_{n-1}⟩=\sqrt{n}{c}_n$, so Parseval identity proves the derivative-energy equality directly. Bounded derivative puts $f$ in the Gaussian Sobolev space, hence in that form domain; it need not be in the full operator domain, so $⟨f,Nf⟩$ is not always a legitimate initial definition.

Use the entropy functional ${\mathrm{Ent}}_{\gamma }(h)=\int h\log hd\gamma -(\int hd\gamma )\log \int hd\gamma$. Begin with bounded smooth $h>0$ bounded away from zero, and put $h_t=P_{t}h$. Invariance and Gaussian integration by parts give the Ornstein-Uhlenbeck entropy dissipation identityThe gradient identity and the permitted weighted Cauchy-Schwarz inequality, applied with weight $h$, implyThe entropy tends to zero as $t\to \infty$ by Mehler's formula and bounded convergence. Integrating in time and using invariance yields ${\mathrm{Ent}}_{\gamma }(h)\le \frac{1}{2}\int |h^{\prime }{|}^2/hd\gamma$. Set $h=f^2$ to obtain the sharp Gaussian logarithmic Sobolev inequalityFor the stated possibly unbounded $f$, apply the argument to smooth positive clipped approximations with a common positive lower bound and uniformly bounded derivatives. They can converge pointwise together with their derivatives and have a common linear-growth bound. Since Gaussian measure integrates every polynomial moment, dominated convergence theorem passes both the entropy and the derivative energy to the limit. This also justifies use of the supplied inequality when $2f{f}^{\prime }$ itself is unbounded. The normalization $\Vert f{\Vert }_1=1$ concerns $f$, and does not imply $\int f^{2}d\gamma =1$; the second term in the entropy definition must be retained.