On bounds for the operator norm of the modulus of an \(n\times n\) matrix (Q1239218)

Let \(A\) be a complex \(n\times n\)-matrix. lt is shown that for any permutation-invariant lattice-norm \(p\) on \(\mathbb{C}^n\) the norm of the modulus \(| A|\), considered as an operator on \((\mathbb{C}^n,p)\), is bounded by \(\sqrt{n}\| A\|\). If \(p\) is an arbitrary lattice norm on \(\mathbb{C}^n\), the inequality \(\|| A|\|\leq \frac{n+1}{2}\| A\|\) holds. We discuss the question whether these estimates are the best possible, and describe matrices for which \(\|| A|\|\) is actually equal to \(\sqrt{n}\| A\|\).