Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Multiplier Tests and Subhomogeneity of Multiplier Algebras

Aleman, Alexandru LU ; Hartz, Michael ; McCarthy, John E. and Richter, Stefan (2022) In Documenta Mathematica 27. p.719-764
Abstract

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... (More)

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.

(Less)
Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
multiplier, Reproducing kernel Hilbert space, subhomogeneous operator algebras
in
Documenta Mathematica
volume
27
pages
46 pages
publisher
Deutsche Mathematiker-Vereinigung (DMV) & Fakultät für Mathematik, Universität Bielefeld.
external identifiers
  • scopus:85134695251
ISSN
1431-0635
DOI
10.25537/dm.2022v27.719-764
language
English
LU publication?
yes
id
fc5e1345-5999-4b83-a705-282fe199cce0
date added to LUP
2022-09-06 13:15:47
date last changed
2022-09-06 13:15:47
@article{fc5e1345-5999-4b83-a705-282fe199cce0,
  abstract     = {{<p>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.</p>}},
  author       = {{Aleman, Alexandru and Hartz, Michael and McCarthy, John E. and Richter, Stefan}},
  issn         = {{1431-0635}},
  keywords     = {{multiplier; Reproducing kernel Hilbert space; subhomogeneous operator algebras}},
  language     = {{eng}},
  pages        = {{719--764}},
  publisher    = {{Deutsche Mathematiker-Vereinigung (DMV) & Fakultät für Mathematik, Universität Bielefeld.}},
  series       = {{Documenta Mathematica}},
  title        = {{Multiplier Tests and Subhomogeneity of Multiplier Algebras}},
  url          = {{http://dx.doi.org/10.25537/dm.2022v27.719-764}},
  doi          = {{10.25537/dm.2022v27.719-764}},
  volume       = {{27}},
  year         = {{2022}},
}