TY - JOUR

T1 - Multiplier Tests and Subhomogeneity of Multiplier Algebras

AU - Aleman, Alexandru

AU - Hartz, Michael

AU - McCarthy, John E.

AU - Richter, Stefan

N1 - Funding Information:
The second named author is grateful for valuable discussions with Ken David-son regarding Proposition 7.2. He also thanks Jörg Eschmeier for asking two questions that led to the results in Section 8. M. H. was partially supported by a GIF grant. J. M. was partially supported by National Science Foundation Grant DMS 2054199.
Publisher Copyright:
© 2022

PY - 2022

Y1 - 2022

N2 - Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of n 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.

AB - Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of n 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.

KW - Reproducing kernel Hilbert space

KW - multiplier

KW - subhomogeneous operator algebras

UR - http://www.scopus.com/inward/record.url?scp=85134695251&partnerID=8YFLogxK

U2 - 10.25537/dm.2022v27.719-764

DO - 10.25537/dm.2022v27.719-764

M3 - Article

AN - SCOPUS:85134695251

SN - 1431-0635

VL - 27

SP - 719

EP - 764

JO - Documenta Mathematica

JF - Documenta Mathematica

ER -