Smoothness properties of principal angles between subspaces with applications to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyze their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in [W.-J. Beyn, G. Froyland, and T. Hüls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245–1286], [W.-J. Beyn, and T. Hüls, SIAM J. Appl. Dyn. Syst., 22 (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.