Complexification of multilinear mappings and polynomials (Q2770425)

Let \((X,\|\cdot\|_X)\) be a real normed space. The pair \((X_C,\gamma_C)\) is a complexification of \(X\) if \(X_C\) is the algebraic complexification of \(X\) and \(\gamma_C\) is a complex norm on \(X_C\) such that \(\gamma_C(x)=\|x\|_X\) for all \(x\in X\). If, in addition, \(\gamma_C(\operatorname {Re}(z))\leq\gamma_C(z)\) for every \(z\in X_C\), then \(\gamma_C\) is a desirable norm. A number of desirable norms and their properties are discussed in the paper under review. Also, several estimates for the norms of complexified multilinear mappings and polynomials are obtained using desirable norms.