Let $A$ and $B$ be hermitian matrices (a special case that would already help would be $A^{-1} = B^T$). I'm looking for a closed form of the series $$X := \sum_{n=0}^\infty A^n \circ B^n$$ where $\circ$ denotes the Hadamard (element-wise) product. The series is assumed to converge, it might suffice to obtain a formal (non-converging) expression.
What I've tried so far is checking for the individual series $$\alpha := \sum_{n=0}^\infty A^n$$ and $$\beta := \sum_{n=0}^\infty B^n$$ and their Hadamard product, one idea was to introduce an external parameter $\varphi$ that leaves the original series invariant but changes both $\alpha$ and $\beta$: $$ A \rightarrow Ae^{i\varphi}, \quad B\rightarrow Be^{-i\varphi}$$ It should then hold that $$ \alpha_\varphi \circ \beta_\varphi = X + f(\varphi)$$ and it is possible to write down a formula for $\mathrm{d}f/\mathrm{d}\varphi$ that depends on $\alpha_\varphi$ and $\beta_\varphi$. One could then obtain a formula for $f(\varphi)-f(0)$ as a function of $\alpha_\varphi$ and $\beta_\varphi$. I can't see how that could help, though.
Thanks for any kind of help in advance.
