  We extend Hardy's inequality from sequences of non-negative numbers to
sequences of positive semi-definite operators if the parameter p satisfies
1<p<=2, and to operators under a trace for arbitrary p>1. Applications to trace
functions are given. We introduce the tracial geometric mean and generalize
Carleman's inequality.
  Key words and phrases: Hardy's inequality, positive operator, trace function,
geometric mean, Carleman's inequality.
