# Multiplicative Isometries on SomeF-Algebras of Holomorphic Functions

Authors: Yasuo Iida; Kazuhiro Kasuga
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Multiplicative (but not necessarily linear) isometries
of Mp(X) onto Mp(X) will be described, where Mp(X) (p≥1) are F-algebras included in the Smirnov class N∗(X).


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## Body

## 1. Introduction

Letn be a positive integer. The space of n-complex variables z=(z1,…,zn) is denoted by ℂn. The unit polydisk {z∈ℂn:|zj|<1,1≤j≤n} is denoted by Un, and the distinguished boundary 𝕋n is {z∈ℂn:|zj|= 1, 1≤j≤n}. The unit ball {z∈ℂn:∑j=1n‍|zj|2<1} is denoted by Bn and Sn its boundary. In this paper, X denotes the unit polydisk or the unit ball for n≥1, and ∂X denotes 𝕋n for X=Un or Sn for X=Bn. The normalized (in the sense that σ(∂X)=1) Lebesgue measure on ∂X is denoted by dσ.For each0<q≤∞, the Hardy space on X is denoted by Hq(X) with the norm ∥·∥q.The Nevanlinna classN(X) on X is defined as the set of all holomorphic functions f on X such that
(1)sup0≤r<1∫∂X‍log(1+|f(rz)|)dσ(z)<∞
holds. It is known that f∈N(X) has a finite nontangential limit, also denoted by f, almost everywhere on ∂X.The Smirnov classN*(X) is defined as the set of all f∈N(X) which satisfy the equality
(2)sup0≤r<1∫∂X‍log(1+|f(rz)|)dσ(z)=∫∂X‍log(1+|f(z)|)dσ(z).
Define a metric
(3)dN*(X)(f,g)=∫∂X‍log(1+|f(z)-g(z)|)dσ(z)
for f,g∈N*(X). With the metric dN*(X)(·,·), the Smirnov class N*(X) is an F-algebra. Recall that an F-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 [1].The Privalov classNp(X), 1<p<∞, is defined as the set of all holomorphic functions f on X such that
(4)sup0≤r<1∫∂X‍(log(1+|f(rz)|))pdσ(z)<∞
holds. It is well-known that Np(X) is a subalgebra of N*(X); hence, every f∈Np(X) has a finite nontangential limit almost everywhere on ∂X. Under the metric defined by
(5)dNp(X)(f,g)=(∫∂X‍(log(1+|f(z)-g(z)|))pdσ(z))1/p
for f,g∈Np(X), Np(X) becomes an F-algebra (cf. [2]). Complex-linear isometries on Np(X) are investigated by Iida and Mochizuki [3] for one-dimensional case and by Subbotin [2, 4] for a general case.Now, we define the classMp(X). For 1≤p<∞, the class Mp(X) is defined as the set of all holomorphic functions f on X such that
(6)∫∂X‍(log(1+sup0≤r<1|f(rz)|))pdσ(z)<∞.
Define a metric
(7)dMp(X)(f,g)={∫∂X‍(log(1+sup0≤r<1|f(rz)-g(rz)|))pdσ(z)}1/p
for f,g∈Mp(X). With this metric, Mp(X) is also an F-algebra (see [2]). Complex-linear surjective isometries on Mp(X) are investigated by Subbotin [2, 4].It is well-known that the following inclusion relations hold:(8)Hq(X)⊊Np(X)⊊M1(X)⊊N*(X)--------(0<q≤+∞,p>1).
As shown in [4], for any p>1, the class Mp(X) coincides with the class Np(X), and the metrics dMp(X) and dNp(X) are equivalent. Therefore, the topologies induced by these metrics are identical on the set Mp(X)=Np(X). But we note that [4, Theorems 1 and 4] implies that the sets of linear isometries on Mp(X) and Np(X) are different. It is known that H∞(X) is a dense subalgebra of Mp(X). The convergence in the metric is stronger than uniform convergence on compact subsets of X.In this paper, we consider surjective multiplicative (but not necessarily linear) isometries from the classMp(X)(p∈ℕ) on the open unit disk, the ball, or the polydisk onto itself.


## 2.  The Results

Proposition 1.
Letn be a positive integer, and let X be either Bn or Un. Let p∈ℕ, and suppose that T:Mp(X)→Mp(X) is a surjective isometry. If T is 2-homogeneous in the sense that T(2f)=2T(f) holds for every f∈Mp(X), then either
(9)T(f)=αf∘Φforeveryf∈Mp(X),
or
(10)T(f)=αf∘Φ¯¯foreveryf∈Mp(X),
where α is a complex number with the unit modulus and, for X=Bn, Φ is a unitary transformation; for X=Un, Φ(z1,…,zn)=(λ1zi1,…,λnzin), where |λj|=1, 1≤j≤n and (i1,…,in) is some permutation of the integers from 1 through n.Proof.
We follow [5, Proposition 2.1] and [4, Theorem 3]. Let f,g∈Hp(X). By 2-homogenuity of isometry T, the equation
(11)∫∂X‍(log(1+sup0≤r<1|f(rz)2n-g(rz)2n|))pdσ(z)=∫∂X‍(log(1+sup0≤r<1|(Tf)(rz)2n-(Tg)(rz)2n|))pdσ(z)
holds. In a way similar to the proof of Theorem 2.1 in [1], we see that
(12)∫∂X‍sup0≤r<1|f(rz)-g(rz)|pdσ(z)=∫∂X‍sup0≤r<1|(Tf)(rz)-(Tg)(rz)|pdσ(z).
This equality implies that T is isometric in the norm
(13)∥f∥Hmp:={∫∂X‍sup0≤r<1|f(rz)|pdσ(z)}1/p
of the space Hp(X), which is equivalent to the standard norm in Hp(X). From (12) with g=0, we obtain T(Hp(X))⊆Hp(X) since T(0)=0, which is observed by just letting f=0 in the equation T(2f)=2T(f). Furthermore, the restricted map T|Hp(X) is an isometry with respect to the metric induced by the Hp-norm ∥·∥p. The same argument for T-1 shows that T-1(Hp(X))⊆Hp(X). Thus, we see that T(Hp(X))=Hp(X). By the Mazur-Ulam theorem [6], T|Hp(X) is a real-linear isometry since T(0)=0.
Using the limit(14)limε→0+1εp+1{(εt)p-(log(1+εt))p}=p2tp+1,t≥0,
we show that T is also isometric in the norm ∥f∥Hmp+1. If p>0, then for any k∈ℕ, there exists the following limit:
(15)limε→0+1εp+k{(log(1+εt))p-∑n=0k-1‍cn(εt)n+p}=cktp+k,t≥0,
where cn are the Taylor coefficients of the function (log(1+t)/t)p. Using (15), we can prove that T is isometric in Hmp+k for p∈ℕ and all k∈ℕ by induction.
Sincedσ is a finite measure, we verify that
(16)limp→∞∥f∥Hmp=∥f∥Hm∞
holds for every f∈H∞(X), and it is clear that ∥f∥Hm∞=∥f∥∞. Moreover, ∥f∥p=∥T(f)∥p for every f∈H∞(X) and limp→∞∥T(f)∥p=∥T(f)∥∞, so we have T(f)∈H∞(X), and ∥f∥∞=∥T(f)∥∞ for every f∈H∞(X). Similarly, we see that f∈H∞(X) if T(f) belongs to H∞(X). Therefore T|H∞(X) is a surjective  isometry with respect to ∥·∥∞ from H∞(X) onto itself. We may suppose that H∞(X) 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 T|H∞(X) is complex-linear or conjugate linear by [7, Theorem].
IfT|H∞(X) is complex-linear, then T is complex-linear on Mp(X), since H∞(X) is dense in Mp(X) and the convergence in the original metric is stronger than uniform convergence on compact subsets of X. Therefore, the first formula of the conclusion holds by Corollary 2.3 in [1].
IfT|H∞(X) is conjugate linear, then T is conjugate linear on Mp(X) as before. Let T~:Mp(X)→Mp(X) be defined as T~(f)=T(f~) for every f∈Mp(X), where
(17)f~(z1,…,zn)=f(z1¯,…,zn¯)¯
for f∈Mp(X). Then, T~ is complex-linear isometry from Mp(X) onto itself. Applying Corollary 2.3 in [1] to T~, the second formula of the conclusion holds.Let1≤p<∞. We say a map T:Mp(X)→Mp(X) is multiplicative if T(fg)=T(f)T(g) for every f,g∈Mp(X). Next, we characterize multiplicative isometries from Mp(X)(p∈ℕ) onto itself. Let Φ be a transformation described in Proposition 1. Then, T(f)=f∘Φ defines a complex-linear multiplicative isometry from Mp(X) onto itself, and T(f)=f∘Φ¯¯ defines a conjugate linear multiplicative isometry from Mp(X) onto itself. We show that they are the only multiplicative isometries from Mp(X) onto itself.Theorem 2.
Letp∈ℕ, and let T be a multiplicative (not necessarily linear) isometry from Mp(X) onto itself. Then, there exists a holomorphic automorphism Φ on X such that either of the following holds:
(18)T(f)=f∘Φforeveryf∈Mp(X)
or
(19)T(f)=f∘Φ¯¯foreveryf∈Mp(X),
where Φ is a unitary transformation for X=Bn; Φ(z1,…,zn)=(λ1zi1,…,λnzin)  for X=Un, where |λj|=1 for every 1≤j≤n and (i1,…,in) is some permutation of the integers from 1 through n.Proof.
SinceT is multiplicative, we see by the same way as in the proof of Theorem 2.2 in [5] that T(1)=1, T(2)=2, and T(1/2)=1/2. Therefore, T is a surjective isometry which satisfies T(2f)=2T(f) as T is multiplicative. It follows by Proposition 1 that
(20)T(f)=αf∘Φ,f∈Mp(X),
or
(21)T(f)=αf∘Φ¯¯,f∈Mp(X),
holds for a complex number α and the holomorphic automorphism Φ as described in Proposition 1. The constant α=1 is observed as T(1)=1; hence, the conclusion holds.Remark 3.
We note that surjective multiplicative isometries of the classMp(X)(p∈ℕ) 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 p.



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*Source: 681637-2013-06-06.xml*