Denote by $\Delta$ the Laplacian on $SL_2(\mathbb{R})$ with the standard Haar measure. Denote by $A_{\lambda}$ the operator $$\dfrac{1}{\Delta^2 - \frac{1}{4}\frac{d^2}{d\theta^2} + \lambda}.$$ The operator $A_{\lambda}$ is positive and self adjoint for $\lambda > 0$. It is represented by an integral operator $$A_{\lambda}(f) = \int_{SL_2(\mathbb{R})}k(g,h)f(h)dh,$$ In general, I don't see that there should be any reason why $k(g,h)\in\mathbb{R}$ off the diagonal. However, I would like to prove that there exists a choice of $\lambda > 0$ for which $k(g,h) > 0$ for all $(g,h)\in SL_2(\mathbb{R})\times SL_2(\mathbb{R})$.
I would be quite fascinated with a closed form for $k(g,h)$.
My choice of $A_{\lambda}$ was somewhat arbitrary, and I am willing to consider other bounded resolvents such as $$B_{\lambda_1,\lambda_2} = \dfrac{1}{\Delta - \dfrac{1}{4}\dfrac{d^2}{d\theta^2} + \lambda_1} - \dfrac{1}{\Delta - \dfrac{1}{4}\dfrac{d^2}{d\theta^2} + \lambda_2}.$$
I would appreciate any reference/suggestion.