I am trying to show that the operator:
$$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$.
So I can re-write this operator as:
$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ so this is in the form of an integral operator.
To show that this is self adjoint i just need to show that:
$\int_0^1 K(s,t)f(t)dt=\int_0^1\overline{K(s,t)}f(t)dt$
But then as $\overline{(5s^2t^2+2)}=5s^2t^2+2$ we are done?
Is the above correct, it seems to simple?
Thanks for any help
Edit:
I am using the following:
Integral operators are of the form:
$Tf(s)=\int_0^1 K(s,t)f(t)dt$
So in order to calculate the adjoint we have that:
$\langle Tf,g\rangle=\int_0^1 Tf(s)\overline{g(s)}ds=\int_0^1\int_0^1 K(s,t)f(t)\overline{g(s)}dtds=\int_0^1f(t)\overline{\int_0^1\overline{K(s,t)}g(s)dsdt}=\langle f,T^*g\rangle$
This gives that the adjoint is of the form:
$\int_0^1 \overline{K(s,t)}g(t)ds$
So once I have written my operator in the form of an integral operator to show it is self adjoint I only have to show what I have above (I think).