I am reading this [paper](https://www.sciencedirect.com/science/article/pii/S0022123696900300) 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](https://books.google.com/books/about/Linear_Operators_Part_2.html?id=C9iYugEACAAJ), 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?