  Given a smooth manifold $M$ (with or without boundary), in this paper we
establish a global functional calculus (without the standard assumption that
the operators are classical pseudo-differential operators) and the G\r{a}rding
inequality for global pseudo-differential operators associated with boundary
value problems. The analysis that we follow is free of local coordinate
systems. Applications of the G\r{a}rding inequality to the global solvability
for a class of evolution problems are also considered.
