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arXiv · 2008.00981

Multiplier tests and subhomogeneity of multiplier algebras

Alexandru Aleman·Michael Hartz·John E. McCarthy·Stefan Richter
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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.

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BibTeXRIS

Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter. 2020-08-03. Multiplier tests and subhomogeneity of multiplier algebras. https://doi.org/10.25537/dm.2022v27.719-764

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