This question is motivated by investigation of the operator space structure of Hankel matrices (which is surely well-known to the experts).
Consider a lacunary Hankel matrix, i.e. a matrix $(a_{i+j})$, where the entries are non-zero only when $i+j=2^{n}$ for some natural number $n$. It is known that it is bounded iff there exists a "symbol" $\varphi \in L^{\infty}(\mathbb{T})$ such that $a_{i+j} = \widehat{\varphi}(i+j)$; $\varphi$ can be chosen in such a way that $\|(a_{i+j})\| = \|\varphi\|_{\infty}$. This is general truth but if the matrix is lacunary then its norm turns out to be equivalent to the $L^2$-norm of $\varphi$; this uses the $H^1-BMO$ duality and Paley's inequality, if I understand correctly.
It is still true that if one takes $(a_{i+j})$ to be $n \times n$ matrices themselves then there exists a matrix-valued symbol $\varphi \in L^{\infty}(M_{n})$ such that $a_{i+j} = \widehat{\varphi}(i+j)$; again we can achieve the equality of norms. I would like to show that square of the norm of $\varphi$ is equivalent (with constant independent of $n$, of course) to $max( \|\sum_{i=1}^{\infty} (\widehat{\varphi}(i))^{\ast} \widehat{\varphi}(i)\|_{M_{n}}, \|\sum_{i=1}^{\infty} \widehat{\varphi}(i) (\widehat{\varphi}(i))^{\ast}\|_{M_{n}}.$
Maybe it follows from work of Tao Mei on operator-valued Hardy and BMO spaces but I hope that there is an easier way.