Geodesic
Multiplier tests and subhomogeneity of multiplier algebras
Documenta mathematica, Tome 27 (2022), pp. 719-764.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 46E22, 47B32, 47L55
Keywords: reproducing kernel Hilbert space, multiplier, subhomogeneous operator algebras, Drury-Arveson space 
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     author = {Aleman, Alexandru and Hartz, Michael and McCarthy, John E. and Richter, Stefan},
     title = {Multiplier tests and subhomogeneity of multiplier algebras},
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Aleman, Alexandru; Hartz, Michael; McCarthy, John E.; Richter, Stefan. Multiplier tests and subhomogeneity of multiplier algebras. Documenta mathematica, Tome 27 (2022), pp. 719-764. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a48/