Multiplicative isometries on the Smirnov class

作者：Hatori Osamu^*; Iida Yasuo

来源：Central European Journal of Mathematics, 2011, 9(5): 1051-1056.

DOI：10.2478/s11533-011-0068-1

• Isometries
• Smirnov class
• Multiplicative maps

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 phi such that T(f) = f circle phi for every class element f or T( f) = f f circle phi for every class element f, where the automorphism phi is a unitary transformation in the case of the ball and phi(z(1,) ... , lambda nz(in)) = (lambda(1),z(1,) ... , lambda nz(in)) for |lambda(j)| = 1, 1 <= j <= n, and (i(n), .., in) is some permutation of the integers from 1 through n in the case of the n - dimensional polydisk.

• 出版日期2011-10