Geometric approach to a dilation theorem

Problem: Given operators A_j ? O on Hilbert space $\text{H}$, with ΣA_j = 1, to find commuting projectors E_j on a Hilbert space $\text{H}$ ? $\text{H}$ such that (for all _j) x¹A_jy = x¹E_jy for, x, y ∈ $\text{H}$. This paper gives an explicit construction, quite different from the familiar solution.