Definition 1: Let $X$ be a Banach space. A semigroup is a family $\{T(t)\}_{t\geq 0}$ of continuous linear operators $T(t):X\to X$ such that
$(i)\;\;T(0)=I$, where $I$ is the identity operator;
$(ii)\;\;T(s)\circ T(t)=T(t+s)$ for all $t,s\geq 0$.Definition 2: the infinitesimal generator of a semigroup $\{T(t)\}_{t\geq 0}$ is the operator $A:D(A)\to X$ where: $$D(A)=\left\{x\in X;\;\lim_{h\to 0^+}\frac{T(h)x-x}{h}\text{ exists in } X \right\}$$ and $$A(x)=\lim_{h\to 0^+}\frac{T(h)x-x}{h}$$ for all $x\in D(A)$.
Definition 3: the translation of the function $f:\mathbb{R}\to\mathbb{R}$ is the function $f_t:\mathbb{R}\to\mathbb{R}$given by $f_t(x)=f(x+t)$ for all $x\in\mathbb{R}$.
Take $X=L^2(\mathbb{R})$ in definition 1 and consider the semigroup $T:=\{T(t)\}_{t\geq 0}$ where $T(t)f=f_t$ for all $f\in L^2(\mathbb{R})$.
My problem is to find the infinitesimal generator of $T$. First of all I need to find $D(A)$, that is, I need to find all $f\in L^2(\mathbb{R})$ such that $$\lim_{h\to 0^+}\frac{f_h-f}{h}=\lim_{h\to 0^+}\frac{T(h)f-f}{h}=g\tag{1}$$ for some $g\in L^2(\mathbb{R})$.
Could someone explain me how can we conclude? Any help is appreciated.
Thanks.
Best Answer
The limit $$ \lim_{h\to 0^+}\frac{f_h-f}{h}, $$ exists if the derivative of $f$ lies in $L^2(\mathbb R)$. More precisely, if there exists a $g\in L^2(\mathbb R)$, such that $$ \lim_{h\to 0^+} h^{-1}\|f_h-f-hg\|_{L^2(\mathbb R)}=0. $$ Clearly, the functions $f$ with the property above are dense in $L^2(\mathbb R)$, as every continuously differentiable function with compact support has this property, and such functions are indeed dense in $L^2(\mathbb R)$. So ${\mathcal D}(A)$ is dense in $L^2(\mathbb R)$, and $A=\frac{d}{dx}$.