Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X γ I, for the positive real number γ , and p, q > 1 with 1/p + 1/q = 1, then |A − B|2 p|A|2 + q|B|2 with equality if and only if (1 − p)A = B and γ ||||A − B|2||| |||p|A|2X + qX|B|2||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert–Schmidt operator, then δ2 A,B(X) 2 p|A|2X + qX|B|2 2 with equality if and only if (1 − p)AX = XB.  2003 Elsevier Inc. All rights reserved