Essential norms of some singular integral operators

Let a and a be bounded measurable functions on the unit circle T. The singular integral operator S-alpha,S-beta is defined by S(alpha,beta)f = alpha Pf + beta Qf(f epsilon L-2(T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of S-alpha,S-beta was calculated in general, using alpha,beta and alpha (B) over bar + H-infinity where H-infinity is a Hardy space in L-infinity (T). In this paper, the essential norm parallel to S(alpha,beta)parallel to(e) of S-alpha,S-beta is calculated in general, using alpha (B) over bar + H-infinity + C where C is a set of all continuous functions on T. Hence if alpha (B) over bar is in H-infinity + C then parallel to S(alpha,beta)parallel to(e) = max (parallel to alpha parallel to(infinity), parallel to beta parallel to(infinity)). This gives a known result when alpha,beta are in C.