Multiplier tests and subhomogeneity of multiplier algebras

TL;DR

This paper investigates the conditions under which multiplier algebras of reproducing kernel Hilbert spaces can be characterized by positivity of finite-sized matrices, linking the problem to subhomogeneity of operator algebras, with key results showing many such algebras are not subhomogeneous.

Contribution

It establishes that for many Hilbert spaces of analytic functions, multiplier algebras are not subhomogeneous, requiring testing of arbitrarily large matrices, and connects this to the embedding of certain weighted Dirichlet space multipliers.

Findings

Multiplier algebras of Dirichlet and Drury-Arveson spaces are not subhomogeneous.

Finite matrix size testing is insufficient for these spaces, necessitating arbitrarily large matrices.

Weighted Dirichlet space multipliers embed into the Drury-Arveson space multiplier algebra.

Abstract

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