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Multiplicative isometries on the Smirnov class. (English) Zbl 1264.30042

The authors prove that \(T\) is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if, there exists a holomorphic automorphism \(\Phi\) such that \(T(f) = f \circ \Phi\) for every class element \(f\) or \(T(f) = \overline{f \circ \overline{\Phi}}\) for every class element \(f\), where, in the case of the ball, the automorphism \(\Phi\) is a unitary transformation, and, in the case of the \(n\)-dimensional polydisk, \(\Phi(z_{1}, \dots, z_{n}) = (\lambda_1 z_{i_1},\dots,\lambda_n z_{i_n})\) if \(|\lambda_{j}| = 1\), \(1 \leq j \leq n\), and \((i_{1},\dots, i_{n})\) is a permutation of the integers from 1 through \(n\).

MSC:

30H15 Nevanlinna spaces and Smirnov spaces
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46B04 Isometric theory of Banach spaces
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References:

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