A new kind of Hankel - Toeplitz type operatorconnected with the complementary seriesby

0. Introduction. Let us begin by recalling brieey some salient facts from classical Hankel theory (cf. Ni]). Consider the Hardy class H 2 (T), where T is the unit circle in C , and let H 2 (T) ? be its orthogonal complement in the space L 2 (T). If is a holomorphic function, one deenes the Hankel operator H with symbol by the formula H f = P ? f (f 2 H 2 (T)) where P ? stands for orthogonal projection onto H 2 (T) ?. (If we instead project onto H 2 (T) we get the (classical) Toeplitz operator.) In 1979 Peller (cf. Pel]) proved that the Hankel operator H is in the Schatten-von Neumann class S p if and only if its symbol belongs to the Besov space B p A(T), provided that 1 p < 1; later, this result was extended to cover the case 0 < p < 1 also. A basic property of the Hankel operator is its conformal invariance: the map 7 ! H intertwines with certain natural actions of the group SU(1; 1). This has lead mathematicians to study more general operators with a similar property. This generalization can be done in several steps: replacing the Hardy class H 2 (T) by the family of (generalized) weighted Bergman spaces A = A (D) (> 0) over the unit disc D ; considering higher order analogues of Hankel operators; using, instead of operators, bilinear or even multi-linear forms; passing to the limit ! 1, which leads to the famous Fock space and a degenerate limit of SU(1; 1), the Heisenberg group; replacing T, or rather its \interior" D , by a symmetric domain { this leads to other semi-simple groups than SU(1; 1) { or other domains, or even complex manifolds, with or without a group action; passing to dual (\compact") situations; etc. In particular, much of the classical Hankel theory can be extended to the case of the group SU(n; 1); instead of the disc there appears now the ball B in C n. In this case we have an obvious extension of the previous weighted Bergman spaces; we are going to use the same letter for them, writing thus A = A (B). (Clearly, if n = 1 we have B = D so the notation A is unambiguous.) Generally speaking, Hankel theory forms a link between operator theory and …