Generators of the Ring of Bounded Operators

J. A. Dieudonne suggested in conversation that some small number of projections might suffice to generate the ring 43 of all bounded operator on separable Hilbert space 3C. There is some analogy between such a result and the theorem that a compact connected metric group can be generated by two elements.1 The analogy is still closer for Theorem 2 below. In this paper, operators A, • ■ ■ will be said to "generate" the smallest ring (i.e., weakly closed self-adjoint algebra) containing A, ■ • ■ and the constants.