Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in mind a decaying exponential weight). All inner products written below are taken to be on $S_1$.
Also consider the space of real valued functions on the real line $S_2=L^2_{\mathbb{R}}(\mathbb{R},a(x)dx)$
Let $R(\lambda)=(M-\lambda I)^{-1}$.
It is standard fact that for any $f$ in $S_1$ (and hence, also for any $f$ in $S_2$), we have the
$A(\lambda)=\langle R(\lambda)f,f\rangle$ maps $\lambda$ from upper half plane to upper half plane.
My question is:
1). SPECIFIC CASE: Suppose $f$ is taken to be in $S_2$. Then under what conditions can we claim that :
$B(\lambda)=\langle R(\lambda)f,f^3\rangle$ also maps $\lambda$ from upper half plane to upper half plane.
The motivation is that I find numerically the spectrum of a perturbed operator to be real. If I do a expansion for eigenvalues in terms of the resolvent $R$, I get the term like B. If one can prove B has the desired properties, it can be used to prove the spectrum is real.
2). GENERAL CASE: Now suppose we have two functions $g$ in $S_2$ and $f$ in $S_2$.
Under what conditions can we claim that
$C(\lambda)=\langle R(\lambda)f,g\rangle$ also maps $\lambda$ from upper half plane to upper half plane.
