Non-commutative Clarkson inequalities for unitarily invariant norms

Abstract

It is shown that if A and B are operators on a separable complex Hilbert space and if ||| · ||| is any unitarily invariant norm, then

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

“…which yields p-uniform smoothness of B (see [2,11]). Indeed, let X = 1 and Y = τ Z with Z = 1 and τ > 0.…”

“…which yields p-uniform convexity of B (see [2,11]). Indeed, (we prove it for the convenience of the reader) let X = Y = 1 and…”

“…Bhatia and Holbrook [4] and Hirzallah and Kittaneh [11] generalized these inequalities for general symmetric norms. Bhatia and Kittaneh [7] obtained Clarkson-McCarthy inequalities for certain N -tuples of operators from S p and the second author [15] extended them to all N -tuples of operators from S p .…”

Commutator Inequalities for the Hilbert–Schmidt Norm

“…This inequality has been extended to L p -spaces and Schatten p-classes to establish the uniform convexity of these spaces (see, e.g., [5,9,16] and the references therein). Our first result is a non-commutative commutator version of the inequality (2) for the case when p is an integer.…”

Non-commutative Clarkson Inequalities for n-Tuples of Operators

“…By a suitable change of variables, one can infer that the first and the second inequalities in each of the Clarkson inequalities (1.4)-(1.11) are equivalent (see [7] and [11]). Related Clarkson inequalities for n-tuples of operators have been recently given by Kissin [13].…”