Semigroup sign and generator conventions

Generation and automatic exponential-bound results below are understood under the DC hypotheses of their supplier theorems.

This track consistently writes the abstract evolution equation as $u^{\mathrm{\prime }}=Au+f$ and defines the generator by $Ax={\lim }_{t\downarrow 0}(T(t)x-x)\mathrm{/}t$; the heat flow on a Dirichlet domain is therefore generated by $A={\mathrm{\Delta }}_D$, the Dirichlet Laplacian (the negative of the $L^2$ operator associated with the Dirichlet energy form), not by $-{\mathrm{\Delta }}_D$. Sources writing $u^{\mathrm{\prime }}+Bu=0$ or $u_t+Au=0$ use $B=-A$ or generate $e^{-tA}$: Brezis's maximal-monotone chapter is stated for $u^{\mathrm{\prime }}+Au=0$ with $A$ m-accretive, so its $A$ corresponds to $-A$ here, and Pazy-type statements $\dot{u}+Au=0$ translate the same way. Resolvents are normalised as $R(\lambda ,A)=(\lambda I-A{)}^{-1}$; a source using $(A-\lambda {)}^{-1}$ has the opposite sign, and its resolvent equals $-R(\lambda ,A)$ at the same parameter, so the norms of all powers are unchanged. The contraction case $\mathrm{\parallel }T(t)\mathrm{\parallel }\le 1$ corresponds to $M=1$, $\omega =0$; boundedness ($M>1$) is not the same as contractivity, and the general generation theorem keeps all resolvent powers.

The convention of this track. The abstract evolution equation is written $u^{\mathrm{\prime }}(t)=Au(t)+f(t),u(0)=x,$ and the generator is defined by $Ax={\lim }_{t\downarrow 0}(T(t)x-x)\mathrm{/}t$ on its domain (Infinitesimal generator of a C0-semigroup). The resolvent is normalised as $R(\lambda ,A)=(\lambda I-A{)}^{-1}$. If $\mathrm{\parallel }T(t)\mathrm{\parallel }\le M{e}^{\omega t}$ for all $t\ge 0$, then for every real $\lambda >\omega$ the Laplace representation is $R(\lambda ,A)x={\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}T(t)xdt$ (Resolvent and spectrum of a closed operator on a Banach space, Laplace transform formula for the resolvent); membership in $\rho (A)$ alone does not guarantee convergence of this integral.

Translation dictionary. A source that writes the homogeneous equation as $u^{\mathrm{\prime }}+Bu=0$ or $u_t+Au=0$ is using the opposite sign: its $B$ equals $-A$ in this track, and its solutions are $e^{-tB}$ in its own notation, that is $e^{tA}$ here. In the same way a source whose resolvent is $(A-\lambda {)}^{-1}$ instead of $(\lambda I-A{)}^{-1}$ has the opposite shift convention; $(A-\lambda I{)}^{-1}=-R(\lambda ,A)$ at the same $\lambda$; its $n$th power is $(-1{)}^{n}R(\lambda ,A{)}^n$, so its norm is unchanged. Negating the operator itself is a separate change of generator sign.

Heat flow. With this convention the Dirichlet heat flow ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u$ on a Dirichlet domain is generated by $A={\mathrm{\Delta }}_D$, the Dirichlet Laplacian (the negative of the $L^2$ operator associated with the Dirichlet energy form), and not by $-{\mathrm{\Delta }}_D$; the sign of the generator is the sign of the spatial operator in the equation, not its negative. The dissipativity used by Lumer-Phillips is therefore the inequality $\mathrm{Re}⟨{\mathrm{\Delta }}_{D}u,u⟩\le 0$ on the domain of the Dirichlet Laplacian (Infinitesimal generator of a C0-semigroup names the generator whose behaviour is being discussed), and this is the identification used downstream when the heat semigroup is realised from the Laplacian.

Contraction versus boundedness. The contraction case is the pair $M=1$, $\omega =0$ of the general generation theorem; a bound with $M>1$ alone does not imply that the semigroup is contractive, and the general theorem keeps all resolvent power estimates, the first estimate alone guaranteeing all powers when $M=1$ (with exponential rescaling if $\omega \mathrm{\ne }0$). This dictionary is the one applied when Hille-Yosida generation theorem is specialised to $A={\mathrm{\Delta }}_D$ and when variation of constants is written in the form $u(t)=T(t)x+{\int }_0^{t}T(t-s)f(s)ds$.