We establish a Szegő-type first limit theorem for Toeplitz-like operators acting on the Drury–Arveson space \(H^2_d\) over the unit ball \(\mathbb {B}_d\) . After identifying an appropriate notion of Toeplitz-like operators associated with continuous symbols on \(\partial \mathbb {B}_d\) , we analyze the asymptotic behavior of their finite-dimensional compressions and prove that the normalized traces of these compressions converge to the integral of the symbol against a canonical measure, in direct analogy with the classical one-variable setting. This yields a genuine multivariable analog of Szegő’s first limit theorem within the framework of the Drury–Arveson space. As a consequence of our main result, we obtain corresponding Szegő-type limit theorems for a broad class of regular unitarily invariant spaces on the unit ball, including the weighted Bergman spaces and the Dirichlet space. These corollaries follow from the structural relationship between the Toeplitz algebras of such spaces and that of the Drury–Arveson space. Our results extend the classical Szegő limit theorem to a multivariable setting.
