Committee Spaces and the Random Column–Row Property

Abstract

A committee space is a Hilbert space of power series, perhaps in several or noncommuting variables, such that $\Vert z^\alpha \Vert \Vert z^\beta \Vert \ge \Vert z^{\alpha +\beta }\Vert .$ Such a space satisfies the true column–row property when ever the map transposing a column multiplier to a row multiplier is contractive. We describe a model for random multipliers and show that such random multipliers satisfy the true column–row property. We also show that the column–row property holds asymptotically for large random Nevanlinna–Pick interpolation problems where the nodes are chosen identically and independently. These results suggest that finding a violation of the true column–row property for the Drury–Arveson space via naïve random search is unlikely.