Column bounded matrices and Grothendieck’s inequalities - Christensen, Erik (MATHEMATICA SCANDINAVICA, 2025)
It follows from Grothendieck’s little inequality that to any complex m × n matrix X of column norm at most 1, and an 0 <ε <1, there exist a natural number l, a matrix C in M_(m,l)(C) with (1-ε)^2I_m ≤ CC^* ≤ (4/π ) (1 + ε)^2 I_m and an l × n matrix Z with entries in the torus T, such that X= l^-(1/2)CZ. Grothendieck’s little inequality with the constant k_G^C = 4/π follows from this factorization. Grothendieck’s big inequality may be reproduced with the estimate K_G^C ≤ k_G^C/(2- k_G^C).