Relative Positions of Half-sided Modular Inclusions

Let $K_1\subset H$ and $K_2\subset H$ be half-sided modular inclusions in a common standard subspace $H$. We prove that the inclusion $K_1\subset K_2$ holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions $K_1\subset H$ and $K_2\subset H$. From this we give a characterization of this situation in terms of (operator valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.