I will consider the space $ M_{n}(\mathbb{C})$ of $ n\times n$ complex matrices for simplicity, although many of the results below are true in more general Banach spaces.

I have encountered an interesting representation of the famous Grothendieck inequality recently. Basically, the usual representation of the inequality states that there are two (not relevant here) norms $ \|X\|_{(1)}, \|X\|_{(2)}$ such that the relation $ \|X\|_{(2)} \leq \|X\|_{(1)} \leq k \|X\|_{(2)}$ holds for any $ X \in M_{n}(\mathbb{C})$ and the (dimension-independent) constant $ k$ is called Grothendieck’s constant. Now, in the book *Summing and Nuclear Norms in Banach Space Theory* by G. J. O. Jameson (Cambridge, 1987), the author essentially takes the dual norms of the norms involved in Grothendieck’s inequality, giving $ \frac{1}{k} \|X\|^*_{(2)} \leq \|X\|^*_{(1)} \leq \|X\|^*_{(2)}$ . The dual norms can be given as

\begin{align} \|X\|^*_{(1)} &= \min_{U,V \in M_{n}(\mathbb{C})} \left \{ \|U\|_{2\to\infty} \|V\|_{2\to\infty} : X = U V^H \right\}\ \|X\|^*_{(2)} &= \min_{\{x_i\}, \{y_i\} \in \mathbb{C}^n} \left \{ \sum_{i,j} \|x_i \|_\infty \|y_j\|_\infty :X = \sum_{i,j} x_i y^H_j \right\} \end{align} with $ \|U\|_{2\to\infty}$ being an operator norm that in this case is the maximum $ \ell_2$ norm of the rows of a matrix $ U$ , and $ \|x\|_\infty$ denoting the $ \ell_\infty$ norm in $ \mathbb{C}^n$ . Interestingly, for a positive semidefinite matrix $ P$ the computation of $ \|P\|^*_{(1)}$ reduces to computing the largest magnitude of the entries of $ P$ , that is, $ \|P\|^*_{(1)} = \max_{i,j} |P_{ij}| = \max_{i} |P_{ii}|$ .

This motivates my question: is it true that for positive semidefinite matrices $ \|X\|^*_{(1)} = \|X\|^*_{(2)}$ ? If not, is there a non-trivial class of matrices for which it does hold true (e.g. positive semidefinite with further restrictions)?

Put in another way, when is it true that a positive semidefinite matrix $ P$ with largest entry magniutde $ |P_{\max}|$ can be expressed as a convex combination $ P = \sum_{i,j} c_{ij} x_i y^H_j$ with each $ \|x_i\|_{\infty} \leq \sqrt{|P_{\max}|}$ , $ \|y_j\|_{\infty} \leq \sqrt{|P_{\max}|}$ and $ \sum_{i,j} c_{ij} = 1$ ?

I have been experimenting with different sets of positive semidefinite matrices (with fixed trace, fixed operator norm…) but ultimately I was not able to prove the result for any non-trivial $ P$ . Any advice about this problem would be appreciated.