Outer Kernel Theorem for Co-orbit Spaces of Localised Frames

We prove a so-called outer kernel theorem for bounded linear operators between co-orbit spaces generated from a localised frame Ψ. In particular, we show that there is a bijective correspondence between the bounded linear operators mapping the co-orbit space of test functions $H_w^1({{\Psi }})$ to the co-orbit space of distributions $H_{1/w}^\infty ({{\Psi }})$ and their kernels in $H_{1/w}^{\infty , \otimes }({{\Psi }})$. The proof of the theorem relies on general properties of localised frames, tensor products and Galerkin's method for matrix represention of operators.