Inequalities for Trace Norms of 2 × 2 Block Matrices

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Quantum logarithmic Sobolev inequalities and rapid mixing (2012, background): “where the first inequality follows from a result shown in [42] , and the second inequality follows from hypercontractivity ofT (N−1) t (inductive hypothesis).”
Norm equalities and inequalities for operator matrices (2008, unknown): “, [1,6,10–12], and references therein).”
Norm Inequalities for Sums of Positive Operators. II (2006, methodology): “The second lemma contains recent inequalities of King [7], relating the Schatten p-norm of a positive 2 × 2 operator matrix to the Schatten p-norm of the 2×2 matrix obtained by replacing each block entry by its Schatten p-norm.”