Hankel Operators on the Drury-Arveson Space

The Drury-Arveson space, initially introduced in the proof of a generalization of von Neumann's inequality, has seen a lot of research due to its intrigue as a Hilbert space of analytic functions. This space has been studied in the context of Besov-Sobolev spaces, Hilbert spaces with complete Nevanlinna Pick kernels, and Hilbert modules. More recently, McCarthy and Shalit have studied the connections between the Drury-Arveson space and Hilbert spaces of Dirichlet series, and Davidson and Cloutare have established analogues of classic results of the ball algebra to the multiplier algebra for the Drury-Arveson Space.

The goal of this dissertation is to contribute to this growing body of research by studying the Hankel operators on the Drury-Arveson Space. We begin by establishing basic results regarding the function theoretic properties of the Drury-Arveson space and general properties of Hankel operators. It is then shown that every invariant subspace of the d-shift on the Drury-Arveson space is an at most countable intersection of kernels of Hankel operators. We then prove that if a function and its reciprocal lie in the Drury-Arveson space, then that function must be a cyclic vector. In addition, we prove that each multiplier invariant subspace on the vector-valued Drury-Arveson space is an intersection of kernels of vectorial Hankel operators, and we characterize a special class of symbols which induce a bounded Hankel operator in terms of a Carleson measure condition on the symbol.