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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multipliers, linear functionals and the Fréchet envelope of the Smirnov class $ N\sb *({\bf U}\sp n)$


Author: Marek Nawrocki
Journal: Trans. Amer. Math. Soc. 322 (1990), 493-506
MSC: Primary 46E10; Secondary 32A35
MathSciNet review: 974523
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Abstract: Linear topological properties of the Smirnov class $ {N_{\ast}}({\mathbb{U}^n})$ of the unit polydisk $ {\mathbb{U}^n}$ in $ {\mathbb{C}^n}$ are studied. All multipliers of $ {N_{\ast}}({\mathbb{U}^n})$ into the Hardy spaces $ {H_p}({\mathbb{U}^n}),\;0 < p \leqslant \infty $, are described. A representation of the continuous linear functionals on $ {N_{\ast}}({\mathbb{U}^n})$ is obtained. The Fréchet envelope of $ {N_{\ast}}({\mathbb{U}^n})$ is constructed. It is proved that if $ n > 1$, then $ {N_{\ast}}({\mathbb{U}^n})$ is not isomorphic to $ {N_{\ast}}(\mathbb{U}{^1})$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0974523-9
PII: S 0002-9947(1990)0974523-9
Keywords: Smirnov class, multipliers, Fréchet envelopes, linear functionals, nuclear spaces
Article copyright: © Copyright 1990 American Mathematical Society