PACIFIC JOURNAL OF MATHEMATICS Vol. 202, No. 2, 2002 NON-COMMUTATIVE CLARKSON INEQUALITIES FOR UNITARILY INVARIANT NORMS Omar Hirzallah and Fuad Kittaneh 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 ||| ≤ 2 p-1 ||| |A| p + |B| p ||| for 2 ≤ p< ∞, and 2 p-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. 1. Introduction. The classical Clarkson inequalities for the Schatten p-norms of Hilbert space operators assert that 2 ( ‖A‖ p p + ‖B‖ p p ) ≤‖A + B‖ p p + ‖A - B‖ p p ≤ 2 p-1 ( ‖A‖ p p + ‖B‖ p p ) (1) for 2 ≤ p< ∞, 2 p-1 ( ‖A‖ p p + ‖B‖ p p ) ≤‖A + B‖ p p + ‖A - B‖ p p ≤ 2 ( ‖A‖ p p + ‖B‖ p p ) (2) for 0 <p ≤ 2, 2 ( ‖A‖ p p + ‖B‖ p p ) q/p ≤‖A + B‖ q p + ‖A - B‖ q p (3) for 2 ≤ p< ∞; 1 p + 1 q = 1, and ‖A + B‖ q p + ‖A - B‖ q p ≤ 2 ( ‖A‖ p p + ‖B‖ p p ) q/p (4) for 1 <p ≤ 2; 1 p + 1 q = 1. These inequalities, which can be found in [11], are non-commutative ver- sions of the celebrated Clarkson inequalities for the classical sequence spaces. These inequalities have useful applications in operator theory and in math- ematical physics (see, e.g., [2], [5], [7], [10], [12], and references therein). In 363