in terms of W(A) and W(B). In [l] we obtained results on this problem by restricting B to be nonnegative. Here and throughout A and B are “operators”, which means that they are bounded linear operators on the complex Hilbert space H. For any collection of sets (8, : 01 E A}, by cl. conv.{& : OL E A} we mean the smallest closed convex set containing S, for each 01 E A. It is well known that for any operator A we have (4) I( A (1 < w(A) < 11 A II and if A is normal then w(A) equals II A II . In the second section we prove a theorem which implies that (4) I( A I] = w(A) when AH is orthogonal to A*H. An application of this fact uses the result of Sz-Nagy, Foias, and Berger that w(A) < 1 if and only if A has a 2-dilation, i.e. there exists a unitary operator U on a complex Hilbert space R 3 H such that An = 2QU” for 71 = 1, 2,... with Q the orthogonal projection of R onto H. Thus if II A /I < 2 and if AH is orthogonal to A*H then A has a 2-dilation. The above theorem of Sz-Nagy, Foias, and Berger also permits applications of our third section. In particular if A has such a 2-dilation as above and B is an isometry commuting with A then AB has such a 2-dilation. Another corollary of our third section is the following: let A have polar factorization UR; if (AH)= (A*H)and if U*A = AU* then r(A) = w(A) = 11 A I] where r(A) is the spectral radius of A. More generally the conclusions of our third section give evidence relevant to the conjecture that w(AB) < I/ A 11 w(B) when A and B commute. This conjecture is the analog of the Banach algebra theorem that r(AB) < r(A) r(B) when A and B commute. The final section uses the machinery of [I] along with some perturbation