| Research Abstract |
Let alpha and beta be bounded measurable functions on the unit circle T, let W be a positive weight function on T, and let P_+ be the Riesz projection on the weighted space L^2 (W,T). The one-demensional linear singular integral operator S_<alpha, beta> is defined by S_<alpha, beta>f=alphaP_+f+betaP_-f, (f*L^2(W,T)) where P_-=I-P_+. This operator is related to Toeplitz operators and Hankel operators. At first, we study the boundedness of S_<alpha, beta> on L^2 (W,T). It follows from the Koosis theorem that there are many alpha, beta, W such that P_+ is not bounded and S_<alpha, beta> is bounded. Next, we study the norm of S_<alpha, beta>. Let h be an outer function such that W=|h|^2, and let phi be an unimodular function such thatphi=h^^_/h. By the Hilbert space argument and the Cotlar-Sadosky's lifting theorem, we get three formulas to compute the norm of S_<alpha, beta> on L^2 (W,T) using alpha, beta and phi. The first formula gives the convex function of a real variable whose fixed point is the norm of S_<alpha, beta> on L^2 (W,T). The second formula is similar to the Feldman-Krupnik-Marcus formula (cf.Gohberg-Krupnik's books). The third formular is very different from their formula. If W is a constant, thenphi becomes a constant, and hence our results become more simple as the following. Let m=(|alpha|^2+|beta|^2)/2, n=||alpha|^2-|beta|^2|/2. Then m+n=max{|alpha|^2, |beta|^2} and m-n=min{|alpha|^2, |beta|^2}. Let ||f|| denote the L^2(T) norm of f. Let H^*(T) denote the Hardy space. Then the following equalities hold. sup{||S_<alpha, beta>f||^2/||f||^2 ; f*L^2(T)}=inf{ess.sup_T(m+(|alphabeta^^_-k|^2+n^2)^<1/2>) ; k*H^*(T)}. inf{||S_<alpha, beta>f||^2/||f||^2 ; f*L^2(T)}=sup{ess.inf_T(m-(|alphabeta^^_-k|^2+n^2)^<1/2>) ; k*H^*(T)}.
|
