In thus talk I will discuss noncommutative (nc) polynomials and more generally rational functions that define bounded left multipliers on the Fock space. Alternatively, these are bounded nc rational function on the row ball. We will show that an nc rational function is an element of the Fock space if and only if it is an element of the nc disc algebra. Moreover, the domain of such an nc rational function contains a row ball of radius strictly greater than 1.

These result allow us to show that the inner parts of rational multipliers are Blaschke and that the spectrum of a rational multipliers is the closure of the spectra of its values at points on finite levels of the row ball. In particular, the spectrum map is continuous at every rational multiplier.

This talk is based on joint work with Michael Jury and Robert Martin.