I am trying to understand how one uses the theory of quadratic forms to prove self adjointness of various operators.
Here are some example questions on the SE proving self adjointness in this way.
(1) quantum harmonic oscillator operator
(2) associated Legendre operator
They have provided sources (for example Kato's book) which develop the theory of quadratic forms and lead to the theorems that are claimed to be used to show self adjointness of the above linked operators.
What I do not understand however, is that the theory presented in Kato's book is done over a complex Hilbert space.
In examples (1) and (2) the Hilbert space is $L^2(-1,1)$ which is real. How is it they can use the theory developed in Kato to show self adjointness?
Is there a way to show that the above real scenario can be done as a special case of the complex scenario?
$\newcommand{\IC}{\mathbb C}$
As discussed in the comments, $L^2(\Omega)$ can mean either the space of real- or of complex-valued square-integrable functions, and in quantum mechanics, you really need complex Hilbert spaces for results like Stone's theorem.
For quadratic forms however, real Hilbert spaces work just as well. If you have a quadratic form $q$ on a real Hilbert space $H$, define a quadratic form $q^{\IC}$ on the complexification $H^\IC$ by $$ q^\IC(f+ig)=q(f)+q(g) $$ for $f,g\in H$. It is not hard to see that the relevant properties of $q$ (dense domain, closability, closedness) carry over to $q^\IC$. Moreover, if $q$ is densely defined and closed and $A$ the positive self-adjoint operator associated with $q^\IC$, then the property $q^\IC(\bar u)=q^\IC(u)$ translates to $u\in D(A)$ iff $\bar u\in D(A)$ and $A\bar u=\overline {Au}$. In particular, $A$ maps $D(A)\cap H$ into $H$, and one can show that $A|_{D(A)\cap H}$ is a positive self-adjoint (real-linear) operator on $H$ associated with $q$.