# 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

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