Global and local versions for a PhÓng VŨ theorem for periodic evolution families in Hilbert spaces

A theorem of Gearhart concerning strongly continuous semigroups in Hilbert spaces is extremely useful for stability analysis of concrete equations; see e.g. [20]), and for control theory [27] or [13, page 475]. Phóng Vũ introduced an equivalent condition in [23]. The aim of this article is to extend these results from the autonomous case to time dependent 1-periodic evolution equations in Hilbert spaces. Both cases (continuous and discrete) are analyzed and global and local versions of the Phóng Vũ theorem are provided.