Given Banach spaces \(X\) and \(Y\) with unconditional Schauder bases \((e_n)_n\) and \((f_m)_m\), respectively, the authors say that a continuous bilinear form \(A\) on \(X\times Y\) is unconditional if the series \[ A(x,y)=\sum_{n=1}^\infty\sum_{m=1}^\infty x_ny_mA(e_n,f_m) \] is unconditionally convergent. The authors use \({\mathcal B}_\nu(X\times Y)\) to denote the space of all unconditional bilinear forms on \(X\times Y\) which becomes a Banach space under the norm \[ \nu(A)=\sup\left\{\sum_{n=1}^\infty\sum_{m=1}^\infty| x_n| | y_m| | A(e_n,f_m)| : \| x\| \leq 1, \| y\| \leq 1\right\}. \] In fact, \({\mathcal B}_\nu(X\times Y)\) is a dual Banach space; as the authors show, there is a crossnorm \(\mu\) on \(X\otimes Y\) under which the dual of \(X\otimes_\mu Y\) is isometrically isomorphic to \({\mathcal B}_\nu(X\times Y)\). Furthermore, \((e_n\otimes f_m)_{n,m}\) is an unconditional Schauder basis for \(X\hat\otimes_\mu Y\).
A function \(P: X\to{\mathbb R}\) is said to be an \(n\)-homogeneous polynomial if there is a (necessarily unique) \(n\)-linear mapping \(\check P: X\times \ldots \times X\to {\mathbb R}\) such that \(P(x)=\check P(x,\ldots,x)\) for all \(x\) in \(E\). When \(X\) has an unconditional basis \((e_n)_n\), the authors say that the \(n\)-homogeneous polynomial \(P\) is unconditional if the series \[ P(x)= \check P(x,\dots,x) \sum_{j_1,\ldots,j_n} \check P(e_{j_1},\dots,e_{j_n})x_{j_1}\ldots x_{j_n} \] is unconditionally convergent. They denote the space of all such polynomials by \({\mathcal P}_\nu(^nX)\) and prove that \({\mathcal P}_\nu(^nX)\) is a Banach space when endowed with the norm \[ \nu(P)=\sup\left\{\sum_{j_1,\dots,j_n}| x_{j_1}| \dots | x_{j_n}| | A(e_{j_1},\dots, e_{j_n})| : \| x\| \leq 1\right\}. \] Every integral homogeneous polynomial is unconditional.
When \(E\) is a real Banach lattice, the positive polynomials are defined as those \(n\)-homogeneous polynomials \(P\) for which \(\check P(x_1,\dots,x_n)\) is positive whenever \(x_1,\dots, x_n\) are positive, while the regular polynomials are those \(n\)-homogeneous polynomials which can be written as the difference of two positive polynomials. The regular norm of a regular polynomial, \(\| P\| _r\), is defined by \(\| P\| _r=||| P|||\), where \(| P| \) is the absolute value of \(P\) with respect to the vector lattice structure.
The main result of the present paper is that when \(X\) is a Banach space with a \(1\)-unconditional Schauder basis, then the Banach spaces of unconditional and regular \(n\)-homogeneous polynomials on \(X\) are isometrically isomorphic.