# Multiplier tests and subhomogeneity of multiplier algebras

### Alexandru Aleman

Lund University, Mathematics, Faculty of Science, P.O. Box 118, S-221 00 Lund, Sweden### Michael Hartz

Fachrichtung Mathematik, Universität des Saarlandes, 66123 Saarbrücken, Germany### John E. McCarthy

Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130, USA### Stefan Richter

Department of Mathematics, University of Tennessee, 1403 Circle Drive, Knoxville, TN 37996-1320, USA

## 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.

## Cite this article

Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter, Multiplier tests and subhomogeneity of multiplier algebras. Doc. Math. 27 (2022), pp. 719–764

DOI 10.4171/DM/883