Decomposition of bilinear forms as sum of bounded forms

TL;DR

This paper investigates the decomposition of bilinear forms into sums of bounded forms, providing a criterion in finite dimensions and demonstrating limitations in decomposing forms into sums of three bounded forms.

Contribution

It establishes a necessary and sufficient criterion for bilinear form decomposition in finite dimensions and presents a counterexample for decomposition into three bounded forms.

Findings

Finite-dimensional criterion for bilinear form decomposition

Counterexample showing limitations for three-term decompositions

Demonstration that some forms cannot be decomposed into three bounded forms

Abstract

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 $u:H\times Karrow \C$ a bilinear form satisfies \[ |u(x,y)|\leq\|Ax\|\ \|By\|+\|Cx\|\|Dy\| \] 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 \[ |u_1(x,y)|\leq \|Ax\|\ \|By\|, |u_2(x,y)|\leq \|Cx\|\|Dy\|, \forall x\in H,y\in K. \] U.Haagerup conjectured that an analogous decomposition as a sum of bounded bilinear forms is not always possible for more than two terms. The aim of current paper is to investigate this problem. In the finite dimensional case, we give a necessary and sufficient criterion for such a decomposition. Finally, we use this criterion to give an example of a sesquilinear form $u$, even on a two-dimensional Hilbert space, which is majorized by the sum of the moduli of three bounded forms $b_1,b_2$ and $b_3$, but can not be decomposed as a sum of three sesquilinear forms $u_i$ where each $u_i$ is majorized by the corresponding $|b_i