We study the dynamics of hydration water in the protein lysozyme in the temperature range 180 K<T<360 K using Fourier-transform-infrared and nuclear magnetic resonance (NMR) spectroscopies. By analyzing the thermal evolution of spectra of the OH-stretching vibration modes and the NMR self-diffusion (DS) and spin-lattice relaxation time (T1), we demonstrate the existence of two dynamical transitions in the protein hydration water. Below the first transition, at about 220 K, the hydration water displays an unambiguous fragile-to-strong dynamic crossover, resulting in the loss of the protein conformational flexibility. Above the second transition, at about 346 K, where the protein unfolds, the dynamics of the hydration water appears to be dominated by the non-hydrogen-bonded fraction of water molecules.