Conditional expectattions and interpolation of linear operators

We study in details interpolation of linear operatorson order ideals of L^1(0,1). As a part of our results we obtain a considerable improvement of the famous Calderon - Ryff interpolation theorem. The book contains the results obtained by the author during his lifetime research and complements the results obtained by numerous mathematicians, including C. Bennet, D.W. Boyd, S.G. Krein, B.S. Mityagin, V.I. Ovchinnikov, J. Peetre, Y.I. Petunin, R. Sharply, A.A. Sedaev, E.M. Semenov, V.A. Schestakov.