Contraction generator from an exponentially weighted difference operator ($(Qf{)}_0=-f_0,(Qf{)}_n=4^n(f_{n-1}/2-f_n)$)

On its maximal domain in ${\ell }^2$, put $g_n=2^n{f}_n$. Then $g_n-g_{n-1}=-2^{-n}(Qf{)}_n$, so $g_n$ has a finite limit ${\ell }_f$. Summation gives $\mathrm{Re}⟨Qf,f⟩=-\frac{1}{2}(|g_0{|}^2+|{\ell }_f{|}^2+{\sum }_{n\ge 1}|g_n-g_{n-1}{|}^2)$. For $\alpha >0$, the resolvent equation is the recursion $g_n=(g_{n-1}+2^{-n}{h}_n)/(1+\alpha 4^{-n})$, which produces a bounded $g_n$ and hence $f\in {\ell }^2$. The Lumer-Phillips theorem proves contraction-semigroup generation.

// Target: analysis.bigb