Structural Properties of the K\"othe Dual of the Matricial Bloch Space

We study the Köthe dual $\mathcal{B}(D,\ell_2)^K$ of the matricial Bloch space.

A 2015 conjecture [Publ.

Math.

Debrecen \textbf{87} (2015), 351--370] proposed that this space coincides with the dyadic mixed-norm space determined by the operator norms of the diagonals.

We disprove the conjecture by revealing a structural obstruction: membership in $\mathcal{B}(D,\ell_2)^K$ is sensitive to the placement of the entries within the diagonals and cannot be detected from diagonal data alone; in particular the trace-norm variant fails as well.

However, testing against Toeplitz matrices exactly recovers the trace-norm variant, a matricial analogue of the Anderson--Shields theorem, proved via analytic majorants.

Finally, we establish two-sided estimates: row-wise and column-wise $\ell(2,1)$ conditions are sufficient, while the trace-norm dyadic condition is necessary; the latter inclusion is strict.