Is the Dirichlet space a quotient of \(DA_{n}\)? (English) Zbl 1394.46019
Cwikel, Michael (ed.) et al., Functional analysis, harmonic analysis, and image processing: a collection of papers in honor of Björn Jawerth. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2836-5/pbk; 978-1-4704-4166-1/ebook). Contemporary Mathematics 693, 301-307 (2017).
Summary: We show that the Dirichlet space is not a quotient of the Drury-Arveson space on the \(n\)-ball for any finite \(n.\) The proof is based a quantitative comparison of the metrics induced by the Hilbert spaces.
For the entire collection see [Zbl 1378.46003].
For the entire collection see [Zbl 1378.46003].
MSC:
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
Biographic References:
Jawerth, BörnReferences:
| [1] | Agler, Jim; McCarthy, John E., Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44, xx+308 pp. (2002), American Mathematical Society, Providence, RI · Zbl 1010.47001 · doi:10.1090/gsm/044 |
| [2] | Arcozzi, N.; Mozolyako, P.; Perfekt, K-M.; Richter, S.; Sarfattisome, G.; Hilbert Spaces Related with the Dirichlet Space arXiv:1512.07532v2 |
| [3] | Arcozzi, N.; Rochberg, R.; Sawyer, E.; Wick, B. D., Distance functions for reproducing kernel Hilbert spaces. Function spaces in modern analysis, Contemp. Math. 547, 25-53 (2011), Amer. Math. Soc., Providence, RI · Zbl 1236.46023 · doi:10.1090/conm/547/10805 |
| [4] | Coburn, L. A., Sharp Berezin Lipschitz estimates, Proc. Amer. Math. Soc., 135, 4, 1163-1168 (electronic) (2007) · Zbl 1137.47018 · doi:10.1090/S0002-9939-06-08569-8 |
| [5] | Duren, Peter; Weir, Rachel, The pseudohyperbolic metric and Bergman spaces in the ball, Trans. Amer. Math. Soc., 359, 1, 63-76 (2007) · Zbl 1109.32003 · doi:10.1090/S0002-9947-06-04064-5 |
| [6] | Goldman, William M., Complex hyperbolic geometry, Oxford Mathematical Monographs, xx+316 pp. (1999), The Clarendon Press, Oxford University Press, New York · Zbl 0939.32024 |
| [7] | Hartz, M \(.;\) On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, arXiv:1505.05108. (to appear, Canad. J. Math.) · Zbl 1481.47111 |
| [8] | Hartz, M \(.;\) Embedding Dimension, manuscript, 2015 · Zbl 1493.46042 |
| [9] | McCarthy, J., Shalit, O.; preliminary manuscript, 2015 |
| [10] | Rochberg, Richard, Structure in the spectra of some multiplier algebras. The corona problem, Fields Inst. Commun. 72, 177-200 (2014), Springer, New York · Zbl 1320.30092 · doi:10.1007/978-1-4939-1255-1\_9 |
| [11] | Shalit, O.; Operator theory and function theory in Drury-Arveson space and its quotients, arXiv:1308.1081 (to appear in Handbook of Operator Theory). · Zbl 1344.47002 |
| [12] | Shalit, O.; Operator theory and function theory in Drury-Arveson space and its quotients, arXiv:1308.1081 (to appear in Handbook of Operator Theory). · Zbl 1344.47002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
