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Multiplicative isometries on the Smirnov class

Osamu HatoriYasuo Iida — 2011

Open Mathematics

We show 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 Φ such that T(f)=f ○ Φ for every class element f or T(f) = f ϕ ¯ ¯ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = ( λ 1 z i 1 , . . . , λ n z i n ) for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from...

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