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