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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 681637, 3 pages
Multiplicative Isometries on Some -Algebras of Holomorphic Functions
1Department of Mathematics, Iwate Medical University, Yahaba, Iwate 028-3694, Japan
2Academic Support Center, Kogakuin University, Tokyo 192-0015, Japan
Received 30 November 2012; Accepted 24 May 2013
Academic Editor: Gestur Ólafsson
Copyright © 2013 Yasuo Iida and Kazuhiro Kasuga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Multiplicative (but not necessarily linear) isometries of onto will be described, where () are -algebras included in the Smirnov class .
Let be a positive integer. The space of -complex variables is denoted by . The unit polydisk is denoted by , and the distinguished boundary is = 1, . The unit ball is denoted by and its boundary. In this paper, denotes the unit polydisk or the unit ball for , and denotes for or for . The normalized (in the sense that ) Lebesgue measure on is denoted by .
For each , 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 holds. It is known that has a finite nontangential limit, also denoted by , almost everywhere on .
The Smirnov class is defined as the set of all which satisfy the equality Define a metric for . With the metric , the Smirnov class is an -algebra. Recall that an -algebra is a topological algebra in which the topology arises from a complete metric. Complex-linear isometries on the Smirnov class are characterized by Stephenson in .
The Privalov class , , is defined as the set of all holomorphic functions on such that holds. It is well-known that is a subalgebra of ; hence, every has a finite nontangential limit almost everywhere on . Under the metric defined by for , becomes an -algebra (cf. ). Complex-linear isometries on are investigated by Iida and Mochizuki  for one-dimensional case and by Subbotin [2, 4] for a general case.
Now, we define the class . For , the class is defined as the set of all holomorphic functions on such that Define a metric for . With this metric, is also an -algebra (see ). Complex-linear surjective isometries on are investigated by Subbotin [2, 4].
It is well-known that the following inclusion relations hold: As shown in , for any , 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 on the open unit disk, the ball, or the polydisk onto itself.
2. The Results
Proposition 1. Let be a positive integer, and let be either or . Let , and suppose that is a surjective isometry. If is 2-homogeneous in the sense that holds for every , then either or where is a complex number with the unit modulus and, for , is a unitary transformation; for , , where , and is some permutation of the integers from through .
Proof. We follow [5, Proposition 2.1] and [4, Theorem 3]. Let . By 2-homogenuity of isometry , the equation
holds. In a way similar to the proof of Theorem 2.1 in , we see that
This equality implies that is isometric in the norm
of the space , which is equivalent to the standard norm in . From (12) with , we obtain since , which is observed by just letting in the equation . Furthermore, the restricted map is an isometry with respect to the metric induced by the -norm . The same argument for shows that . Thus, we see that . By the Mazur-Ulam theorem , is a real-linear isometry since .
Using the limit we show that is also isometric in the norm . If , then for any , there exists the following limit: where are the Taylor coefficients of the function . Using (15), we can prove that is isometric in for and all by induction.
Since is a finite measure, we verify that holds for every , and it is clear that . Moreover, for every and , so we have , and for every . Similarly, we see that if belongs to . Therefore is a surjective isometry with respect to from onto itself. We may suppose that is a uniform algebra on the maximal ideal space and the maximal ideal space is connected by the Šilov idempotent theorem; hence, we see that is complex-linear or conjugate linear by [7, Theorem].
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 .
If is conjugate linear, then is conjugate linear on as before. Let be defined as for every , where for . Then, is complex-linear isometry from onto itself. Applying Corollary 2.3 in  to , the second formula of the conclusion holds.
Let . We say a map is multiplicative if for every . Next, we characterize multiplicative isometries from onto itself. Let be a transformation described in Proposition 1. Then, defines a complex-linear multiplicative isometry from onto itself, and defines a conjugate linear multiplicative isometry from onto itself. We show that they are the only multiplicative isometries from onto itself.
Theorem 2. Let , and let be a multiplicative (not necessarily linear) isometry from onto itself. Then, there exists a holomorphic automorphism on such that either of the following holds: or where is a unitary transformation for ; for , where for every and is some permutation of the integers from through .
Proof. Since is multiplicative, we see by the same way as in the proof of Theorem 2.2 in  that , , and . Therefore, is a surjective isometry which satisfies as is multiplicative. It follows by Proposition 1 that or holds for a complex number and the holomorphic automorphism as described in Proposition 1. The constant is observed as ; hence, the conclusion holds.
Remark 3. We note that surjective multiplicative isometries of the class have the same form as surjective mutiplicative isometries of the Smirnov class [5, Theorem 2.2] and the Privalov class [8, Corollary 3.4]. The authors do not know whether this result holds for noninteger .
The authors wish to express their sincere gratitude to Professor O. Hatori, who introduced this subject and kindly directed them. The authors also would like to thank the referee for the detailed comments and valuable suggestions. The first author was partly supported by the Grant from Keiryokai Research Foundation no. 97.
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