Vol. 14, No. 6, 2021

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Multipliers and operator space structure of weak product spaces

Raphaël Clouâtre and Michael Hartz

Vol. 14 (2021), No. 6, 1905–1924
Abstract

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space H1 . For complete Nevanlinna–Pick spaces , we characterize all multipliers of the weak product space . In particular, we show that if has the so-called column-row property, then the multipliers of and of coincide. This result applies in particular to the classical Dirichlet space and to the Drury–Arveson space on a finite-dimensional ball. As a key device, we exhibit a natural operator space structure on , which enables the use of dilations of completely bounded maps.

Keywords
complete Nevanlinna–Pick space, weak product, multiplier, Hankel operator, completely bounded map, dilation
Mathematical Subject Classification 2010
Primary: 46E22
Secondary: 46L07, 47A20
Milestones
Received: 27 September 2019
Revised: 10 January 2020
Accepted: 3 March 2020
Published: 7 September 2021
Authors
Raphaël Clouâtre
Department of Mathematics
University of Manitoba
Winnipeg, MB
Canada
Michael Hartz
Fachrichtung Mathematik
Universität des Saarlandes
Saarbrücken
Germany
Fakultät für Mathematik und Informatik
FernUniversität in Hagen
Hagen
Germany