Norms of some singular integral operators and their inverse operators

Let alpha and beta be bounded measurable functions on the unit circle T. Then the singular integral operator S-alpha,(beta) is defined by S-alpha,(beta)f = alpha P(+)f + beta P(-)f, (f is an element of L-2(T) where P+ is an analytic projection and P- is a co-analytic projection. In this paper, the norms of S-alpha,(beta) and its inverse operator on the Hilbert space L-2(T) are calculated in general, using alpha,beta and alpha<(beta)over bar> + H-infinity Moreover, the relations between these and the norms of Hankel operators are established. As an application, in some special case in which alpha and beta are nonconstant functions, the norm of S-alpha,(beta) is calculated in a completely explicit form. If alpha and beta are constant functions, then it is well known that the norm of S-alpha,(beta) on L-2(T) is equal to max{\alpha\, \beta\} If alpha and beta are nonzero constant functions, then it is also known that S-alpha,(beta) On L-2(T) has an inverse operator S-alpha-1,S-beta-1 whose norm is equal to max{\alpha\(-1),\beta\(-1)}.