Multiplicative Isometries on Some F-Algebras of Holomorphic Functions

z = (z1, . . . , zn) is denoted by C . The unit polydisk {z ∈ C : |zj| < 1, 1 ≤ j ≤ n} is denoted by U , and the distinguished boundary T is {z ∈ C : |zj|= 1, 1 ≤ j ≤ n}. The unit ball {z ∈ C : ∑ n j=1 |zj| 2 < 1} is denoted byBn and Sn its boundary. In this paper, X denotes the unit polydisk or the unit ball for n ≥ 1, and ∂X denotes T for X = U or Sn for X = Bn. The normalized (in the sense that σ(∂X) = 1) Lebesgue measure on ∂X is denoted by dσ. For each 0 < q ≤ ∞, the Hardy space onX is denoted by

For each 0 <  ≤ ∞, the Hardy space on  is denoted by   () with the norm ‖ ⋅ ‖  .
The Nevanlinna class () on  is defined as the set of all holomorphic functions  on  such that sup holds.It is known that  ∈ () has a finite nontangential limit, also denoted by , almost everywhere on .
It is well-known that the following inclusion relations hold: As shown in [4], for any  > 1, the class   () coincides with the class   (), and the metrics    () and    () are equivalent.Therefore, the topologies induced by these metrics are identical on the set   () =   ().But we note that [4, Theorems 1 and 4] implies that the sets of linear isometries on   () and   () are different.It is known that  ∞ () is a dense subalgebra of   ().The convergence in the metric is stronger than uniform convergence on compact subsets of .
In this paper, we consider surjective multiplicative (but not necessarily linear) isometries from the class   () ( ∈ N) on the open unit disk, the ball, or the polydisk onto itself.
If |  ∞ () is complex-linear, then  is complex-linear on   (), since  ∞ () is dense in   () and the convergence in the original metric is stronger than uniform convergence on compact subsets of .Therefore, the first formula of the conclusion holds by Corollary 2.3 in [1].
are the Taylor coefficients of the function (log(1 + )/)  .Using (15), we can prove that  is isometric in  +  for  ∈ N and all  ∈ N by induction.