The Drury-Arveson space H^2_d is the reproducing kernel Hilbert space on the open unit ball of C^d with kernel k(z,w) = 1/(1 - <z,w>); equivalently it is the symmetric Fock space over C^d, and its shift is the model for commuting row contractions. Students learn why it, and not the Hardy or Bergman space of the ball, is the correct several-variable analogue of H^2 of the disc: every pure contractive Hilbert module is a quotient of an ampliation of it, and every irreducible complete Pick kernel is a rescaled restriction of its kernel, which makes Nevanlinna-Pick interpolation, its multiplier algebra, and the associated nonselfadjoint operator algebras of subvarieties the natural objects of study. The topic connects function theory on the ball (Carleson measures, interpolating sequences, the corona problem) to operator theory (dilation and model theory, essential normality and the Arveson-Douglas conjecture, the curvature invariant, and the isomorphism problem for multiplier algebras of varieties).
The Drury-Arveson space H^2_d is the Hilbert space of holomorphic functions on the unit ball of C^d in which point evaluation is a bounded functional and the reproducing kernel is k(z,w) = 1/(1 - <z,…