# $L^2$-valued integral as parameter integral

## Setting

Let us regard the Hilbert space $L^2(0,1)$ and the $C_0$-semigroup $(T(t))_{t\geq 0}$ defined by $$T(t):\left\{ \begin{array}{rml} L^2(0,1) & \to & L^2(0,1), \\ [f]_{\sim} &\mapsto &\left[x \mapsto \begin{cases} f(x+t), & \text{if}\; x+t<1\\ 0, & \text{else} \end{cases} \right]_{\sim}. \end{array} \right.$$ It is easy to verify that this is indeed a $C_0$-semigroup. Therefore, the mapping $t \mapsto T(t)f$ is a continuous mapping from $L^2(0,1)$ to $L^2(0,1)$. Consequently the $L^2(0,1)$-valued integral $$g := \int_0^1 T(t)f \,\mathrm{d}t$$ exists.

## Question

In order to get some information about the behavior of $g$ it would be nice to regard $g$ as a parameter integral. Hence I am interested in the following equality $$g(x) = \Big(\int_0^1 T(t)f \,\mathrm{d}t\Big) (x)\stackrel{?}{=} \int_0^1 \big(T(t)f\big)(x)\,\mathrm{d}t .$$ Or with a different notation $$g = \int_0^1 \big(x \mapsto \big(T(t)f\big)(x) \big)\,\mathrm{d}t \stackrel{?}{=} \Big(x\mapsto\int_0^1 \big(T(t)f\big)(x)\,\mathrm{d}t\Big) .$$ The evaluation mapping is neither continuous nor well-defined on $L^2$. So I think it is not trivial to justify this step.

It seems quite common to evaluate such $L^2(0,1)$-valued integrals by interpreting it as a parameter integral, so I guess that there is a theorem which justifies that. It would be really great if someone had a reference.

## Solution for this special case

In this particular case I think I have a solution. I know that every convergent sequence in $L^2$ has a subsequence which converges even point-wise a.e.. Since $$g_n := x\mapsto \sum_{i=1}^{n} \frac{1}{n} \Big(T\Big(\frac{i}{n}\Big)f\Big)(x)$$ converges to $g$ and every subsequence of $g_n(x)$ converges in $\mathbb{R}$ to the same limit for a.e. $x\in (0,1)$, the point-wise limit of $g_n$ has to coincide with $g$ a.e..

• Very good question! I've also encountered this issue several times, and found it quite subtle. I have two remarks: (1) I have difficulties to follow your solution of the special case: why does $g_n$ converge to $g$ almost everywhere? More precisely, how do you know that the mapping $t \mapsto T(t)f(x)$ is Riemann integrable for almost every $x$? (2) Do you have any specific situations/applications in mind where you wish to apply this? I often found it quite helpful to drop the almost-everywhere-perspective and work with duality instead. May 18, 2018 at 21:01
• @JochenGlueck (1) You have a good point I don't know if $t\mapsto T(t)f(x)$ is Riemann integrable. I know that it is $L^1$. Maybe someone can find a justification for this. (2) The specific situation is the setting I stated. I want to calculate the domain of the infinitesimal generator of this $C_0$-semigroup. May 19, 2018 at 21:03
• Do you really have difficulties with the particular example? Or does it only exemplify a general problem? Note that already your definition of $T_t$ is formally not correct (because the elements of $L^2$ aren't functions but equivalent classes. You could define $T_t(f)$ by the formula for continuous functions, check continuity and extend by general abstract nonsense to all of $L^2$. May 22, 2018 at 14:08
• @JochenWengenroth As I already mentioned in my question: It seems quite common to interpret $L^2$-valued integrals as parameter integrals eventhough they are a priori different things. I realized that when I stumbled on this example. I edited the definition of $T(t)$. I already have an answer to my question. I will post it soon. May 23, 2018 at 8:24

The trick is to show that both functions $$\Big(\int_0^1 T(t)f \,\mathrm{d}t\Big) (x)$$ and $$\int_0^1 \big(T(t)f\big)(x)\,\mathrm{d}t$$ induce the same element in the dual space. Let $$h \in L^2(0,1)$$ be arbitrary. Since the scalar product is continuous in both arguments, we have $$\Big\langle h, \int_0^1 T(t)f \,\mathrm{d}t \Big\rangle = \int_0^1 \big\langle h, T(t)f \big\rangle \,\mathrm{d}t = \int_0^1 \int_0^1 h(x) \big(T(t)f\big)(x) \,\mathrm{d}x\,\mathrm{d}t$$ It is easy to check that $$(t,x) \mapsto \big(T(t)f\big)(x)$$ is an element of $$L^2\big((0,1)\times (0,1)\big)$$ which allows us to use Fubini $$= \int_0^1 h(x)\int_0^1 \big(T(t)f\big)(x) \,\mathrm{d}t\,\mathrm{d}x = \Big\langle h, x \mapsto \int_0^1 \big(T(t)f\big)(x)\,\mathrm{d}t \Big\rangle .$$
Actually, if $$\Phi: (0,1) \to L^2(0,1)$$ is integrable and $$\Phi(t)(x)$$ is $$L^2\big((0,1)\times (0,1)\big)$$, then we can also regard the $$L^2(0,1)$$-valued integral $$\int_0^1 \Phi(t) \,\mathrm{d}t$$ as the parameter integral $$x\mapsto \int_0^1 \Phi(t)(x) \,\mathrm{d}t$$.