Locally Minimal Projections in B(h)

Given an n-tuple {a1, ..., an} of self-adjoint operators on an infinite dimensional Hilbert space H, we say that a projection p in B(H) is locally minimal for {a1, ..., an} if each pajp (for j = 1, ..., n) is a scalar multiple of p. In Theorem 1.8 we show that for any such {a1, ..., an} and any positive integer k there exists a projection p of rank k that is locally minimal for {a1, ..., an}. If we further assume that {a1, . . . , an, 1} is a linearly independent set in the Calkin algebra, then in Theorem 2.8 we prove that p can be chosen of infinite rank.