Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms positivity $n \times n$ matrices analogous to the classical Pick matrix. We study for which it suffices consider bounded size $n$. connect this problem notion subhomogeneity non-selfadjoint operator algebras. Our main results show that multiplier algebras many analytic functions, such as Dirichlet space and Drury-Arveson space, are not subhomogeneous, hence one has test arbitrarily large matrix To treat we certain weighted on disc embed completely isometrically into algebra space.