# Finding the adjoint of an operator A

1) Consider the operator $A:L_2[0,1]\longrightarrow L_2[0,1]$ defined by $Ax(t)=\int_0^1t x(s)ds$. What is the adjoint of A?

2) Consider the operator $A:L_2[0,1]\longrightarrow L_2[0,1]$ defined by $Ax(t)=\int_0^1t x(t)dt$. What is the adjoint of A?

• $f(x)$ appears nowhere in the right hand side of either of your expressions. – helloworld112358 Jun 15 '17 at 6:19
Hint: Currently, 2) is the only question which kind of makes sense as phrased, but my best interpretation is that $$[Af](x) = \int_0^1 t f(t)\,dt$$ With that, $$\langle Af,g \rangle = \int_0^1 \left(\int_0^1 t f(t)\,dt\right) \overline{g(s)}\,ds \\ = \int_0^1 \int_0^1 t \,f(t)\, \overline{g(s)}\,dt\,ds\\ = \int_0^1 \int_0^1 t \,f(t)\, \overline{g(s)}\,ds\,dt \\ = \int_0^1 f(t)\left(\overline{t\int_0^1 g(s)\,ds}\right)\,dt = \langle f, ??? \rangle$$