A problem on spectrum of a self-adjoint operator

Let $A$ be a positive self-adjoint linear operator (not necessarily bounded) in the Hilbert space $mathcal{H}$. Then $a=(1+A)^{-1}inmathcal{B}(mathcal{H})$. For a given $n$, let us consider the projection $e_n:={chi}_n(a)$ in $mathcal{R}({a})$ ($mathcal{R}({a})$ stands for the von Neumann algebra generated by the element $a$). Then both of $Ae_n$ and $ae_n$ are bounded self-adjoint operators in $mathcal{R}({a})$.

Problem: Prove that for $lambdaneq0$, $frac{1-lambda}{lambda}insigma(Ae_n)implieslambdainsigma(ae_n)$.

P.S. I got stuck in this while reading Section $9.10$ of the book ‘Lectures on von Neumann algebras’ by Strătilă and Zsidó. More specifically, where they consider the map $mathcal{B}(sigma(Ae_n))ni fmapsto F_f(ae_n)inmathcal{B}(mathcal{H})$. To make sense of this, we need to have $F_f$ defined on $sigma(ae_n)$ for $f$ defined on $sigma(Ae_n)$. But $F_f(lambda):=begin{cases}
0 &text{ if } lambda=0,\
f((1-lambda)/lambda) &text{ if } lambdain (0,1].
end{cases}$.
Hence I got the above problem. I was trying with the property that $frac{1-lambda}{lambda}insigma(Ae_n)implies f((1-lambda)/lambda)insigma(f(Ae_n))$ for any continuous function $f$ on $sigma(Ae_n)$ but did not get anything. Thanks in advance for any help.