  We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to
abstract evolution problems that fit into the Gelfand-triple framework. We
introduce novel stability notions that are well-suited to this setting and
provide simple, necessary and sufficient, conditions to verify that a DIRK
scheme is stable in our sense and in Bochner-type norms. We use several popular
DIRK schemes in order to illustrate cases that satisfy the required structural
stability properties and cases that do not. In addition, under some mild
structural conditions on the problem we can guarantee compactness of families
of discrete solutions with respect to time discretization.
