Pinchings and norms of scaled triangular matrices

Abstract

Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in non- descending order of magnitude down the diagonal. It is proved that ||D^-1UD||≥||U|| for any matrix norm that is reduced by a pinching. In addition to known examples -weakly unitarily invariant norms - we show that any matrix norm defined by ||A||=def/max RE(x* Ay)/x≠o, y≠oΦ(x)φ(y) where Φ(·) and ψ(·) are two absolute vector norms, has this property. This includes ellp operator norms as a special case.