It is shown that if A and B are
operators on a separable complex Hilbert space and if ||| ⋅ ||| is any unitarily
invariant norm, then
2||||A|p+ |B|p|||
≤||||A + B|p+ |A − B|p|||
≤ 2p−1||||A|p+ |B|p|||
for 2 ≤ p < ∞, and
2p−1||||A|p+ |B|p|||
≤||||A + B|p+ |A − B|p|||
≤ 2||||A|p+ |B|p|||
for 0 < p ≤ 2. These inequalities are natural generalizations of some of the classical
Clarkson inequalities for the Schatten p-norms. Generalizations of these
inequalities to larger classes of functions including the power functions are also
obtained.
