I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing a self adjoint operator $L$ over smooth functions: \[ Lf(x)=xf'(x) - \left(\frac{1}{m(x)}\int_{x}^a s m(s) ds \right) f''(x) \] where $m(x)$ is a symmetric probability density function over $(-a, a)$. It can be verified by definition of $L$ that which it has 0 and 1 eigenvalues corresponding to $f(x) \equiv 1$ and $f(x) \equiv x$ respectively.

The paper then goes on to claim that by Theorem XIII.7.40 of Linear Operators Part II: Spectral Theory, the remaining spectrum of L is contained in $(1, \infty)$. I was not able to prove this part or get a copy of this book to see the above mentioned theorem. Can someone point me to an easily available reference to understand for this spectral result?

I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self adjoint operator): \[ Lf(x)=xf'(x) - \left(\frac{1}{m(x)}\int_{x}^a s m(s) ds \right) f''(x) \] where $m(x)$ is a symmetric probability density function over $(-a, a)$. It can be verified by definition of $L$ that which it has 0 and 1 eigenvalues corresponding to $f(x) \equiv 1$ and $f(x) \equiv x$ respectively.

The paper then goes on to claim that by Theorem XIII.7.40 of Linear Operators Part II: Spectral Theory, the remaining spectrum of L is contained in $(1, \infty)$. I was not able to prove this part or get a copy of this book to see the above mentioned theorem. Can someone point me to an easily available reference to understand for this spectral result?