The Schur multiplier norm and its dual norm

For a complex self-adjoint n × n matrix A we show that its Schur multiplier norm is determined by ‖ A ‖ S = min ⁡ { ‖ diag ( P ) ‖ ∞ : − P ≤ A ≤ P } . For a norm ‖ | . ‖ | on M ( m , n ) ( C ) we define for an X in M ( m , n ) ( C ) the dual norm, ‖ | X ‖ | ⁎ , of X by the formula: ‖ | X ‖ | ⁎ : = sup ⁡ { | Tr n ( Y ⁎ X ) | : Y ∈ M ( m , n ) ( C ) , ‖ | Y ‖ | ≤ 1 } . For a vector λ in C n we let Δ ( λ ) denote the diagonal n × n matrix with entries from λ . The dual norm of the Schur multiplier norm on M n ( C ) is for a self-adjoint matrix A given by the formula: ‖ A ‖ S ⁎ = min ⁡ { Tr n ( Δ ( λ ) ) : λ ∈ R n , − Δ ( λ ) ≤ A ≤ Δ ( λ ) } . We study this minimization problem as a formal linear program.