Assume the Hilbert space $H_w$ is defined as $$H_w = \{ h\in L_{loc}^1(\mathbb{R}_+)|\exists h'\in L_{loc}^1(\mathbb{R}_+)\text{ and }\|h\|_w<\infty \},$$ where $$ \|h\|_w^2 = |h(0)|^2 + \int_{\mathbb{R}_+}|h'(x)|^2w(x)\mathrm{d}x, $$ and $w:\mathbb{R}_+\to[1,\infty]$ is a non-decreasing $C^1$-function such that $w^{-1/3}\in L^1(\mathbb{R}_+).$
Let $A=\frac{\partial}{\partial x}$ be the differential operator on $H_w$. I want to prove that for $\lambda>\omega\in\mathbb{R}$ and for all $n \in \mathbb{N}$ we have that $$B:=\|(\lambda I - A)^{-n}\|\leq \frac{M}{(\lambda - \omega)^n}.$$
However, any help in proving that $B$ is bounded would be of great help.