Decomposition of bilinear forms as sums of bounded forms
Let \(H,K\) be a Hilbert spaces and let \(A,C\in B(H)\) and \(B,D \in B(K)\). A~result noted by \textit{Q.-H. Xu} [Duke Math. J. 131, No. 3, 525--574 (2006; Zbl 1129.46048)] states that, if \(u\) is a bilinear form on \(H \times K\) such that \(| u(x,y)| \leq \| A(x)\| \| B(y)\| +\| C(x)\| +\| D(y)\| \) for all \(x \in H\),\(y\in K\), then \(u\) can be decomposed as sum \(u = u_1+u_2\) of bilinear forms such that \(| u_1(x,y)| \leq \| A(x)\| \| B(y)\| \) and \(| u_2(x,y)| \leq \| C(x)\| \| D(y)\| \). In this interesting paper, the author gives complete proof of this result and goes on to give a necessary and sufficient condition in the case of finite dimensional Hilbert spaces, for the decomposition theorem to hold when there are \(n\) terms in the bound for \(u\). This is then used to given an example of a \(u\) where there are three terms in the bound but \(u\) is not decomposable as a sum of \(3\)-forms, answering a question raised by Haagerup.
scientific article; zbMATH DE number 6658782