#### Abstract

We study multiplicative isometries on the following -algebras of holomorphic functions: Smirnov class , Privalov class , Bergman-Privalov class and Zygmund -algebra where is the open unit ball or the open unit polydisk in .

#### 1. Introduction

Complex-linear isometries on function spaces of holomorphic functions have been studied for almost five decades by many mathematicians. In this paper we study multiplicative isometries on certain -algebras of holomorphic functions. Recall that an -algebra is a topological algebra in which the topology arises from a complete metric. For a positive integer let denote the open unit ball in the -dimensional complex vector space and the unit polydisk in . We characterize multiplicative isometries on the Smirnov class, the Privalov class, the Bergman-Privalov class and the Zygmund -algebras on or . Surjective multiplicative maps on the Smirnov class, and the Bergman-Privalov class have already been correspondingly characterized in [1, 2].

#### 2. Preliminaries

In studying surjective isometries in [1, 2] we applied the Mazur-Ulam theorem for surjective maps on certain subspaces, which themselves are Banach spaces, of the given -algebras. Generally we do not assume surjectivity of the isometries in this paper, so instead of the Mazur-Ulam theorem we use Lemma 2.1. Recall that a normed real-linear space is *uniformly convex* if for any there exists a such that the inequality holds for every pair of with , , and . It is well known that Hilbert spaces and -spaces for are uniformly convex.

Lemma 2.1. *Let and be normed real-linear spaces with uniformly convex. Let be an isometry from into such that . Then is real-linear.*

The lemma might be well known, but we give a sketch of the proof for the completeness and the benefit of the reader.

*Proof of Lemma 2.1. *Let , be arbitrary elements of . Put . Then since is an isometry, and . We also have .

Suppose that . Set
Since is uniformly convex and is positive there exists a such that
Then by the triangle inequality
holds, which contradicts to . Thus we get , from which for we obtain . Substituting by in the last equality we get
so that . A routine argument yields , .

For , we denote by its distinguished boundary. For , this is the topological boundary , and for the polydisk , it is the torus . Denote the normalized Lebesgue measure on by . A holomorphic map is inner if exists and lies in for almost all with respect to . We say that is the boundary map of and denote it by . We say that is measure preserving if for every Borel set .

Now we recall definitions and some properties of the Smirnov class, the Privalov class, the Bergman-Privalov class, and the Zygmund -algebra on or . The space of all holomorphic functions on or is denoted by . For each , the Hardy space is denoted by with the norm .

##### 2.1. Smirnov Class

Let . The Nevanlinna class on is defined as the set of all holomorphic functions on such that holds. It is known that every has a finite nontangential limit, denoted by , almost everywhere on .

The Smirnov class is defined as Define a metric for . With the metric the Smirnov class becomes an -algebra and in particular, is a dense subalgebra of . The convergence in the metric is stronger than uniform convergence on compact subsets of .

Complex-linear isometries on the Smirnov class were characterized by Stephenson in [3].

##### 2.2. Privalov Class

Let . The Privalov class , , is defined as (for the original source see [4, 5])

It is well known that is a subalgebra of , hence every has a finite nontangential limit almost everywhere on . Define a metric for . With this metric is an -algebra (cf. [6, 7]) and The Hardy algebra is dense in . The convergence on the metric is stronger than uniform convergence on compacts of .

Complex-linear isometries on are investigated by Iida and Mochizuki [8] for one-dimensional case, and by Subbotin [7] for a general case.

##### 2.3. Bergman-Privalov Class

Let and . The Bergman-Privalov class on the unit ball and the polydisk are defined, respectively, as where for the normalized Lebesgue volume measure on and is a normalization constant, that is . Let . In what follows denotes for and for , respectively. The Bergman-Privalov class is an -algebra with respect to the metric for . For some results in the case see [9].

The weighted Bergman space for and on the unit ball and the polydisk are defined, respectively, as It is known that

Complex-linear isometries on the Bergman-Privalov class on the unit ball were characterized by Matsugu and Ueki in [10] and on the polydisk by Stević in [2].

##### 2.4. Zygmund -Algebra

Let and , where . Let . The Zygmund -algebra on is defined as It is known that This implies that the finite nontangential limit exists almost everywhere on , for any . For defines a complete metric on and is an -algebra with this metric (cf. [11]).

Ueki [12] characterized the complex-linear isometries on the Zygmund -algebra on the balls.

#### 3. Main Results

In this section we formulate and prove the main results in this paper.

##### 3.1. Multiplicative Isometries on

Our first result concerns the Smirnov class.

Theorem 3.1. *Let . Suppose that is a (not necessarily linear) multiplicative isometry. Then there is an inner map on whose boundary map is measure preserving and such that either of the following formulas holds:
*

*Proof. *First we claim that . Since and is a holomorphic function on the connected open set we get or . But is impossible because if it were , then , for each , which contradicts with the assumption that is an isometry. As and is injective, we obtain . Similarly is also observed by making use of . Then assert that or . If , then the first formula of the conclusion will follow and the second one will follow from .

Next we show . Put . Suppose that on a set of positive measure on . Then there exists a subset of positive measure and with on . Since
there is a positive integer such that
From this and since is a multiplicative isometry on we have that
which is a contradiction proving almost everywhere on . Hence holds almost everywhere on as almost everywhere on . Since
we have that and almost everywhere on .

Since and
it is easy to check that almost everywhere on . Hence holds. As is multiplicative, is -homogeneous in the sense that holds for every .

Let . It requires only elementary calculation applying the -homogeneity of to check that
holds. Multiplying (3.7) by and then letting we get
by the monotone convergence theorem, since nondecreases monotonically to as for any , which can be easily proved by considering the function . From (3.8) for , we obtain and the restricted map is an isometry with respect to the metric induced by the -norm .

Let the function on the interval be defined as
It is easy to check that is positive and continuous on and . Hence is bounded on , so that

We claim that the inclusion and is isometric with respect to the metric induced by the -norm. For this purpose let . Now note that since , equality (3.7) holds and as well as the next equality
By subtracting (3.7) from (3.11) and then multiplying such obtained equation by we obtain
As is bounded the function is an integrable function dominating the integrand in the left-hand side integral in (3.12). Letting and applying the Lebesgue theorem on dominated convergence to the left-hand side and Fatou’s lemma to the right-hand side (as is positive on ) we obtain
From this and since we get that the function is integrable. Letting again in (3.12) we have that
by the Lebesgue theorem on dominated convergence now applied to both integrals in (3.12). Hence for every pair of . For , we get and consequently , as claimed.

Since is a Hilbert space, it is uniformly convex. Hence by Lemma 2.1 the restriction is real-linear. Since the operations of scalar multiplication and addition on are continuous and is dense in we see that is real-linear on .

First assume . As is real-linear and multiplicative, is complex-linear in this case. Then by [3, Theorem 2.2] and since , there is an inner map such that for every .

Now assume . Let be defined as for every , where
for . Then is well defined and a complex-linear isometry from into itself. Again by [3, Theorem 2.2] we have that there is an inner map on whose boundary map is measure preserving such that for every . This implies that for every .

Corollary 3.2 (see [1]). *Let . Suppose that is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism on such that either of the following formulas holds:
**
where is a unitary transformation for , while for , for some real numbers for and a permutation of the integers from 1 to .*

*Proof. *By Theorem 3.1, is complex-linear or conjugate linear. If is complex-linear, then the result holds by [3, Corollary 2.3]. If is conjugate linear, then put for , where is defined as in (3.15). Then , for every , and for an inner map on whose boundary map is measure preserving. Since is a surjective isometry, the desired property of again follows from [3, Corollary 2.3].

##### 3.2. Multiplicative Isometries on

The next result concerns the Privalov class.

Theorem 3.3. *Let and . Suppose that is a (not necessarily linear) multiplicative isometry. Then there is an inner map on whose boundary map is measure preserving and such that either of the following formulas holds:
*

*Proof. *Since is multiplicative we see by the same way as in the proof of Theorem 3.1 that , and or . Also we see that . It follows by the proof of Theorem 3.1 that for every pair and in ,
holds. Multiplying (3.18) by and then letting we get
Thus . The Hardy space can be seen as a subspace of . Since is uniformly convex, so is for . Then by Lemma 2.1 the operator is real-linear on . Since is a dense subspace of we see that is real-linear on . As we have already learnt that or , we obtain that is complex-linear or conjugate linear on . The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [7, Theorem 1] instead of [3, Theorem 2.2]. We omit the details.

Corollary 3.4. *Let and . Suppose that is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism on such that either of the following formulas holds:
**
where is a unitary transformation for , while for , for some real numbers for and a permutation of the integers from 1 to .*

*Proof. *By Theorem 3.3, is complex-linear or conjugate linear. If is complex-linear, then the result follows directly from [7, Corollary and Remark 3]. If is conjugate linear, then put for , where is defined as in (3.15). Then is a complex-linear isometric surjection from onto itself. Hence by [7, Corollary and Remark 3] there is a desired automorphism on such that for every .

##### 3.3. Multiplicative Isometries on

The next result concerns the Bergman-Privalov class.

Theorem 3.5. *Let , and . Suppose that is a (not necessarily linear) multiplicative isometry. Then there is a holomorphic self-map on with the property that
**
for every bounded or positive Borel function on such that either of the following formulas holds:
*

*Proof. *We can prove the theorem in a way similar to that in the proofs of Theorem 3.1 for and Theorem 3.3 for . For the case of , instead of using the Hardy spaces and we make use of the weighted Bergman spaces and . For the case of , instead of using the Hardy space we make use of the weighted Bergman space . We also apply [10, Theorem 1] for and [2, Theorem 2] for to represent complex-linear isometries instead of [3, Theorem 2.2].

Corollary 3.6 (see [2]). *Let , and . Suppose that is a (not necessarily linear) surjective multiplicative isometry. Then there is a holomorphic automorphism on such that either of the following formulas holds:
**
where is a unitary transformation for , while for , for some real numbers for and a permutation of the integers from 1 to .*

*Proof. *By Theorem 3.5, is complex-linear or conjugate linear. Suppose that is complex-linear. If , then the conclusion follows by [10, Theorem 2], while for the conclusion follows similar to the corresponding part of the proof of [2, Theorem 3]. If is conjugate linear, then the conclusion follows from the similar argument in the proof of Corollary 3.2.

##### 3.4. Isometries on

In [12] Ueki characterized complex-linear isometries on the Zygmund -algebra on . For the following result is proved similar to [12, Theorem 1]. Hence it is omitted.

Theorem 3.7. *Let . If is a complex-linear isometry of into itself, then there exist an inner function and an inner map on whose boundary map is measure preserving on such that
**
Conversely, for given such and , the weighted composition operator is an injective linear isometry of .*

For the surjective isometries the result is as follows.

Corollary 3.8. *An isometry of is surjective if and only if where with and for some real numbers , and a permutation of the integers from 1 to .*

To prove Corollary 3.8 we need the next auxiliary result.

Lemma 3.9. *For any function , if and only if and
**
where denotes the Poisson kernel for ;
**
for , and
**
is the Poisson kernel for the unit disk .*

*Proof. *If , then Fatou’s lemma shows that . The inclusion (2.18) implies , and so we see that has the least -harmonic majorant. Since the least -harmonic majorant of is the Poisson integral , we obtain the following inequality:
Note that is strictly increasing and convex on , and the measures are normalized on , which follows from the well-known equality
Applying Jensen’s inequality to (3.28), we obtain the desired inequality (3.25).

Conversely we put in (3.25). By integrating with respect to and applying Fubini’s theorem, we have that
By the symmetric property and the normalization property of the Poisson kernel, we obtain that
Hence the condition implies that .

Now we give a proof of Corollary 3.8.

*Proof of Corollary 3.8. *Suppose that is surjective. Then Theorem 3.7 gives that . A standard argument shows that is an automorphism of . So there are conformal maps () of onto and there is a permutation of the integers from 1 to such that
The mean value theorem shows that
for each . Here denotes the one-dimensional normalized Lebesgue measure on the unit circle .

On the other hand, the measure-preserving property of gives that
By (3.33) and (3.34) we see that fixes the origin, and so each is the rotation transform.

Next we prove that is a unimodular constant. If is such that , then . Inequality (3.25) in Lemma 3.9 gives that
and so we have on . Since is inner, is a unimodular constant.

Now we show results on multiplicative isometries on the Zygmund -algebras on and .

Theorem 3.10. *Let . Suppose that is a (not necessarily linear) multiplicative isometry. Then there exists an inner map on whose boundary map is measure preserving on , such that either of the following formulas holds:
*

Note that multiplicative isometries of the Privalov class and the Zygmund -algebra have the same form as multiplicative isometries of the Smirnov class.

*Proof of Theorem 3.10. *As is multiplicative we obtain , , and or . Since
holds for every , the equation is proved similarly as in Theorem 3.1.

Let . Then we can prove that
following the lines of the corresponding part of the proof in Theorem 3.1. By some calculation we see that
holds for every . Hence we get
almost everywhere on and is an integrable function dominating . We get
by the Lebesgue dominated convergence theorem since
On the other hand, applying Fatou’s lemma we get
from which for we get . Since
follows from (3.40), the function is an integrable function dominating . Hence
holds by the Lebesgue dominated convergence theorem. Consequently
holds. As and are arbitrary elements of we obtain that is isometric on with respect to the metric induced by the -norm.

We also obtain that there exists a bounded positive continuous function on such that and
Applying this equality we obtain that and is a real-linear isometry on , hence is a complex-linear (if ) or conjugate linear isometry (if ) on , similar as in the proof of Theorem 3.1. The rest of the proof is similar to the last part of the proof of Theorem 3.1 applying [12, Theorem 1] for and Theorem 3.7 for instead of [3, Theorem 2.2]. We omit the details.

Corollary 3.11. *Let . Suppose that is a (not necessarily linear) surjective multiplicative isometry. Then there exists a holomorphic automorphism on such that either of the following formulas holds:
**
where is a unitary transformation for , while for , for some real numbers , and a permutation of the integers from 1 to .*

Note that surjective multiplicative isometries of the Privalov class, the Bergman-Privalov class, and the Zygmund -algebra have the same form as surjective multiplicative isometries of the Smirnov class.

*Proof of Corollary 3.11. *By Theorem 3.10, is complex-linear or conjugate linear. Suppose that is complex-linear. Applying [12, Corollary 1] for and Corollary 3.8 for the result follows in this case. If is conjugate linear, then the result follows by similar arguments as in the proof of Corollary 3.2.

#### Acknowledgments

The first and fourth authors are partly supported by the Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science. The second author is partly supported by the Grant from Keiryokai Research Foundation no. 97. The third author is partially supported by the Serbian Ministry of Science (Projects III41025 and III44006).