DECOMPOSITION OF BILINEAR FORMS AS SUMS OF BOUNDED FORMS

摘要

The problem of decomposition of bilinear forms which satisfy a certain condition has been studied by many authors, by example in [4]: Let H and K be Hilbert spaces and let A, C is an element of B (H), B, D is an element of B (K). Assume that u : H x K -> C a bilinear form satisfies vertical bar u (x, y) vertical bar <= parallel to Ax parallel to parallel to By parallel to + parallel to Cx parallel to parallel to Dy parallel to for all x is an element of H and y is an element of K. Then u can be decomposed as a sum of two bilinear forms u = u(1) + u(2) where vertical bar u(1) (x, y)vertical bar <= parallel to Ax parallel to parallel to By parallel to, vertical bar u(2) (x, y)vertical bar <= parallel to Cx parallel to parallel to Dy parallel to for all x is an element of H, y is an element of K. U. Haagerup conjectured that an analogous decomposition as a sum of bounded bilinear forms is not always possible for more than two terms. In this paper we give a necessary and sufficient condition for such a decomposition to exist and use this criterion to show that indeed it is not always possible for more than two terms.