# Journal of Operator Theory

Volume 75, Issue 2, Spring 2016  pp. 319-336.

Decomposition of bilinear forms as sums of bounded forms

Authors:  Mohamed Elmursi
Author institution: Mathematics Department, Sohag University, Sohag, Egypt

Summary: The problem of decomposition of bilinear forms which satisfy a certain condition has been studied by many authors, by example in \cite{H08}: Let $H$ and $K$ be Hilbert spaces and let $A,C \in B(H),B,D\in B(K)$. Assume that $u:H\times K\rightarrow \C$ a bilinear form satisfies $|u(x,y)|\leqslant \|Ax\| \|By\|+ \|Cx\| \|Dy\|$ for all $x\in H$ and $y\in K$. Then $u$ can be decomposed as a sum of two bilinear forms $u=u_1+u_2$ where $|u_1(x,y)|\leqslant \|Ax\| \|By\|,\ |u_2(x,y)|\leqslant \|Cx\| \|Dy\|,\ \forall x\in H,y\in K.$ 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.

DOI: http://dx.doi.org/10.7900/jot.2015mar11.2074
Keywords: tensor products, bilinear forms, trace class, finite rank operators