The column-row property fails for the multipliers of the Fock space
Let S be a set of operators. We say S satisfies the column-row property if there exists a constant C>0 such that for any sequence from S (finite or infinite,) ‖ If such a C can be chosen to equal 1, we say \mathcal{S} satisfies the true column-row property. The column-row property when \mathcal{S} is taken to be a space of multipliers is important in the theory of interpolating sequences. As far as the speaker knows, there is no known commutative complete Nevanlinna-Pick space for which the multipliers do not satisfy the (true) column-row property. We showed, in joint work with Augat and Jury, that the column-row property fails for the Fock space in two or more variables. Creating further discord, under a suitable model, a randomly chosen infinite sequence of multipliers satisfies \|\sum S_iS_i^*\| \leq \| \sum S_i^*S_i\| for any space such that the monomials satisfy \|z^\alpha\|\|z^\beta\| \geq \|z^{\alpha+\beta}\| in the Hilbert space norm. A deep question is whether or not the Drury-Arveson space satisfies the true column-row property. Our results suggest that naive random search may be unlikely to produce a counter-example, even if one exists.