Adjoint of $T$, infinite dimensional space

Question

$T: L_2 (0,1) \rightarrow L_2(0,1), T(g) =\int_{0} ^{x} g(t) dt $ Could anyone tell me how to compute adjoint of of $T$?

Thank you for your help.

Answer 1

The scalar product on $L_2(0,1)$ is defined by (in case of real valued functions) :

$$\langle f,g \rangle= \int_0^1 f(x)g(x)dx.$$

By definition $T^*$, the adjoint of $T$ is such that :

$$\langle Tf,g \rangle=\langle f,T^*g \rangle.$$

We have :

$$\langle Tf,g \rangle=\int_0^1 \left( \int_{0} ^{x} f(t)dt\right)g(x)dx=\int_0^1 \left( \int_{0} ^{1} f(t)g(x) \mathbf{1}_{0\le t \le x}dt\right)dx.$$

Where $\mathbf{1}_{0\le t \le x}$ is the indicator function of $[0,x]$ for $t$.

Can you take it from here ?

Edit : Now try to reverse the order of integration using Fubini's theorem.