Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators of the form H l/z(k) : F(2)(vertical bar z vertical bar(2)) -> L(2)(vertical bar z vertical bar(2)). Here, H l/z(k)(f):= (Id - P(l))((z) over bar (k)f) and P(l) is the projection onto A(l)(2)(C, vertical bar z vertical bar(2)) := cl (span{(z) over bar (m)z(n)vertical bar m, n is an element of N, m <= l}). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators H l/z(k) are not bounded for l < k - 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L(2) symbols, generalizing arguments from [6], seems possible and is planned for future work. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim