The joint spectrum of a tuple of projections

Abstract

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 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.