On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces

Hartz, Michael

Canadian journal of mathematics, Tome 69 (2017) no. 1, pp. 54-106

Résumé

We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

Détail

DOI : 10.4153/CJM-2015-050-6

Mots-clés : 47L30, 46E22, 47A13, non-selfadjoint operator algebras, reproducing kernelHilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problem.