Asao Arai, Yasumichi Matsuzawa Time Operators of a Hamiltonian with Purely Discrete Spectrum (62K, Latex2e) ABSTRACT. We develop a mathematical theory of time operators of a Hamiltonian with purely discrete spectrum. The main results include boundedness, unboundedness and spectral properties of them. In addition, possible connections of a time operator of $H$ with regular perturbation theory are discussed.