The joint spectrum of a tuple of projections

We study the joint spectrum of projections in a Hilbert space, and calculate the joint spectrum of a pair of projections. In particular, by calculating the joint spectrum for a tuple $[I,P,Q]$ in which $I$ is the identity and $P$, $Q$ is a regular pair of projections,we get specific characterizations of the spectrums of $P+Q$ and $P-Q$, respectively. Conversely, we prove that two closed subsets of $\\mathbb~C$ in particular forms are respectively the spectrums of the sum and the difference of a regular pair of projections. We also calculate the joint spectrum of a $3$-tuple of projections $[P,Q,R]$ for $P$, $Q$ and $R$ satisfying some specific conditions.