simplify the notation, we write $A(i\sigma)=A_{\pm, V}(i\sigma)=I\pm \vhalf R_0(i\sigma)\vhalf$, and note that for $\sigma>0$ sufficiently large, $$\label{eq:Abd}
\|A^{-1}(i\sigma)-I\|=O(1/\sigma^2),\; \|A^{-1/2}(i\sigma)-I\|=O(1/\sigma^2).$$ Now with $B_1$ the operator defined in (\[eq:B1\]), $$\begin{aligned}
\label{eq:diffB1}
B_1-\vhalf  T \vhalf & = (A^{-1/2}-I)\vhalf T  
\vhalf A^{-1/2}  
+ \vhalf T \vhalf (A^{-1/2}-I).\end{aligned}$$ Because $V$ is radial, multiplication by either $V$ or $\vhalf$ commutes with $P_l$. Since $R_0$ commutes with $P_l$, so do $A$, $A^{-1}$, and $A^{-1/2}$. Thus $$\begin{gathered}
\| (B_1-\vhalf  T \vhalf)P_l\|  \\ \leq 
\|(A^{-1/2}-I)\| \| \vhalf T  
\vhalf P_l\| \| A^{-1/2}\|  
+ \|\vhalf T  \vhalf P_l\| \| (A^{-1/2}-I) \|.\end{gathered}$$ Thus using (\[eq:Abd\]) we are done.

Using the notation of [@ab-st], let $J_\nu$ and $Y_\nu$ denote the Bessel functions of the first and second kinds, respectively, and recall that $H^{(1)}_\nu(z)= J_\nu(z)+iY_{\nu(z)}$. For $l\in \Natural_0$, set $\nu_l= l+(d-2)/2$ and notice that $\nu_l$ is an integer since $d$ is even. We can now expand $R_0(\lambda)$ using spherical harmonics. When $0<\arg \lambda<\pi$ and $g\in L^2(\Real^d)$, $$\label{eq:R0exp}
(R_0(\lambda)g)(r\theta)= 
\sum_{l=0}^{\infty} \sum_{\mu=1}^{\mu(l)}
\int_0^\infty  \int_{\Sphere^{d-1}} G_{\nu_l}(r,r';\lambda) Y^{\mu}_l( \theta)
\overline{Y}^\mu_l(\omega) g(r'\omega) (r')^{d-1}dS_\omega dr'$$ with $$\label{eq:Gnu}G_{\nu_l}(r,r';\lambda)=
\left\{ \begin{array}{ll}\frac{\pi}{2i}(r r')^{-(d-2)/2}
J_{\nu_l}(\lambda r)H_{\nu_l}^{(1)}(\lambda r'),\; & \text{if } 
r<r'\\\frac{\pi}{2i}(rr')^{-(d-2)/2}
H_{\nu_l}^{(1)}(\lambda r)J_{\nu_l}(\lambda r'), & \text{if}\; r\geq r'
\end{array}
\right.$$ As noted earlier, for compactly supported, bounded $\chi$, $\chi R_0(\lambda)\chi$ has an analytic 