In two recent joint papers (Math. Scand. 2007 and preprint, math arxiv, 2006) Hanne Schultz and I define and study unbounded R-diagonal operators and use them as a technical tool to construct spectral subspaces for general operators T in II1 factor. Our key example is the quotient z = x/y of two *-free circular operators (x, y). The operator z is in L p (M, tr) (M = W ∗ (x, y)) for 0 < p < 1, but not for p ≥ 1. Moreover, if T is an operator in a larger II1 factor N, such that T is *-free from (x, y), then the perturbation T ′ = T + az (a > 0) of T has very nice properties: Its Brown measure can be explicitly computed and it converges to the Brown measure of T for a− > 0. Moreover the resolvent of T’, R(s) = (s1 − T ′ ) −1 is in L p (N, tr) for 0 < p < 1, and it is a Lipshitz map for 0 < p < 2/3. The latter property is crucial for our construction of spectral (T-invariant) subspaces K(T, B) for every operator T in a II1 factor and every Borel set B in the complex plane.