The Cauchy-Schwarz Norm Inequality for Elementary Operators in Schatten Ideals

Abstract

For bounded Hilbert space operators X, An and Bn, n = 1, 2, …, ‖ ∑ n = 1 ∞ A n X B n ‖ p ⩽ ‖ ( ∑ n = 1 ∞ A n * ( ∑ n = 1 ∞ A n A n * ) p - 1 A n ) 1 / 2 p X ( ∑ n = 1 ∞ B n ( ∑ n = 1 ∞ B n * B n ) p - 1 B n * ) 1 / 2 p ‖ p , for all ⩽ p < ∞ and ‖ ∑ n = 1 ∞ A n X B n ‖ ⩽ ‖ ∑ n = 1 ∞ A n A n * ‖ ‖ ∑ n = 1 ∞ B n * B n ‖ ¯ ‖ X ‖ . These inequalities involve some estimates for the norm of elementary operators with the range contained in the Schatten p-ideals.