Multipliers, linear functionals and the Fréchet envelope of the Smirnov class 𝑁_{*}(𝑈ⁿ)

Abstract

Linear topological properties of the Smirnov class N ∗ ( U n ) {N_{\ast }}({\mathbb {U}^n}) of the unit polydisk U n {\mathbb {U}^n} in C n {\mathbb {C}^n} are studied. All multipliers of N ∗ ( U n ) {N_{\ast }}({\mathbb {U}^n}) into the Hardy spaces H p ( U n ) , 0 > p ⩽ ∞ {H_p}({\mathbb {U}^n}),\;0 > p \leqslant \infty , are described. A representation of the continuous linear functionals on N ∗ ( U n ) {N_{\ast }}({\mathbb {U}^n}) is obtained. The Fréchet envelope of N ∗ ( U n ) {N_{\ast }}({\mathbb {U}^n}) is constructed. It is proved that if n > 1 n > 1 , then N ∗ ( U n ) {N_{\ast }}({\mathbb {U}^n}) is not isomorphic to N ∗ ( U 1 ) {N_{\ast }}(\mathbb {U}{^1}) .