Multiplicative Isometries on Some 𝐹-Algebras of Holomorphic Functions

Abstract

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

1. Introduction

Let 𝑛 be a positive integer. The space of 𝑛-complex variables 𝑧=(𝑧₁,…,𝑧_𝑛) is denoted by ℂ^𝑛. The unit polydisk {𝑧∈ℂ^𝑛∶|𝑧_𝑗|<1,1≤𝑗≤𝑛} is denoted by 𝑈^𝑛, and the distinguished boundary 𝕋^𝑛 is {𝑧∈ℂ^𝑛∶|𝑧_𝑗|= 1, 1≤𝑗≤𝑛}. The unit ball {𝑧∈ℂ^𝑛∶∑^𝑛_𝑗=1|𝑧_𝑗|²<1} is denoted by 𝐵_𝑛 and 𝑆_𝑛 its boundary. In this paper, 𝑋 denotes the unit polydisk or the unit ball for 𝑛≥1, and 𝜕𝑋 denotes 𝕋^𝑛 for 𝑋=𝑈^𝑛 or 𝑆_𝑛 for 𝑋=𝐵_𝑛. The normalized (in the sense that 𝜎(𝜕𝑋)=1) Lebesgue measure on 𝜕𝑋 is denoted by 𝑑𝜎.

For each 0<𝑞≤∞, the Hardy space on 𝑋 is denoted by 𝐻^𝑞(𝑋) with the norm ‖⋅‖_𝑞.

The Nevanlinna class 𝑁(𝑋) on 𝑋 is defined as the set of all holomorphic functions 𝑓 on 𝑋 such that sup_{0≤𝑟<1𝜕𝑋}log1+𝑓(𝑟𝑧)𝑑𝜎(𝑧)<∞ (1) holds. It is known that 𝑓∈𝑁(𝑋) has a finite nontangential limit, also denoted by 𝑓, almost everywhere on 𝜕𝑋.

The Smirnov class 𝑁_∗(𝑋) is defined as the set of all 𝑓∈𝑁(𝑋) which satisfy the equality sup_{0≤𝑟<1𝜕𝑋}𝑓((𝑧)=log1+𝑟𝑧)𝑑𝜎_𝜕𝑋log1+𝑓(𝑧)𝑑𝜎(𝑧). (2) Define a metric 𝑑_{𝑁∗(𝑋)}𝑓,𝑔=_𝜕𝑋log1+𝑓(𝑧)−𝑔(𝑧)𝑑𝜎(𝑧) (3) for 𝑓,𝑔∈𝑁_∗(𝑋). With the metric 𝑑_{𝑁∗(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 class 𝑁^𝑝(𝑋), 1<𝑝<∞, is defined as the set of all holomorphic functions 𝑓 on 𝑋 such that sup_{0≤𝑟<1𝜕𝑋}log1+𝑓(𝑟𝑧)^𝑝𝑑𝜎(𝑧)<∞ (4) holds. It is well-known that 𝑁^𝑝(𝑋) is a subalgebra of 𝑁_∗(𝑋); hence, every 𝑓∈𝑁^𝑝(𝑋) has a finite nontangential limit almost everywhere on 𝜕𝑋. Under the metric defined by 𝑑_𝑁^𝑝_(𝑋)𝑓,𝑔=_𝜕𝑋log(1+|𝑓(𝑧)−𝑔(𝑧)|)^𝑝𝑑𝜎(𝑧)^1/𝑝 (5) for 𝑓,𝑔∈𝑁^𝑝(𝑋), 𝑁^𝑝(𝑋) becomes an 𝐹-algebra (cf. [2]). Complex-linear isometries on 𝑁^𝑝(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 class 𝑀^𝑝(𝑋). For 1≤𝑝<∞, the class 𝑀^𝑝(𝑋) is defined as the set of all holomorphic functions 𝑓 on 𝑋 such that _𝜕𝑋log1+sup_0≤𝑟<1𝑓(𝑟𝑧)^𝑝𝑑𝜎(𝑧)<∞. (6) Define a metric 𝑑_𝑀^𝑝_(𝑋)=𝑓,𝑔_𝜕𝑋log1+sup_0≤𝑟<1|𝑓(𝑟𝑧)−𝑔(𝑟𝑧)|^𝑝𝑑𝜎(𝑧)^1/𝑝 (7) for 𝑓,𝑔∈𝑀^𝑝(𝑋). With this metric, 𝑀^𝑝(𝑋) is also an 𝐹-algebra (see [2]). Complex-linear surjective isometries on 𝑀^𝑝(𝑋) are investigated by Subbotin [2, 4].

It is well-known that the following inclusion relations hold: 𝐻^𝑞(𝑋)⊊𝑁^𝑝(𝑋)⊊𝑀¹(𝑋)⊊𝑁_∗(𝑋)0<𝑞≤+∞,𝑝>1. (8) As shown in [4], for any 𝑝>1, the class 𝑀^𝑝(𝑋) coincides with the class 𝑁^𝑝(𝑋), and the metrics 𝑑_𝑀^𝑝_(𝑋) and 𝑑_𝑁^𝑝_(𝑋) are equivalent. Therefore, the topologies induced by these metrics are identical on the set 𝑀^𝑝(𝑋)=𝑁^𝑝(𝑋). But we note that [4, Theorems 1 and 4] implies that the sets of linear isometries on 𝑀^𝑝(𝑋) and 𝑁^𝑝(𝑋) are different. It is known that 𝐻^∞(𝑋) is a dense subalgebra of 𝑀^𝑝(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 class 𝑀^𝑝(𝑋)(𝑝∈ℕ) on the open unit disk, the ball, or the polydisk onto itself.

2. The Results

Proposition 1. Let 𝑛 be a positive integer, and let 𝑋 be either 𝐵_𝑛 or 𝑈^𝑛. Let 𝑝∈ℕ, and suppose that 𝑇∶𝑀^𝑝(𝑋)→𝑀^𝑝(𝑋) is a surjective isometry. If 𝑇 is 2-homogeneous in the sense that 𝑇(2𝑓)=2𝑇(𝑓) holds for every 𝑓∈𝑀^𝑝(𝑋), then either 𝑇𝑓=𝛼𝑓∘Φ𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑓∈𝑀^𝑝(𝑋), (9) or 𝑇𝑓=𝛼𝑓∘Φ𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑓∈𝑀^𝑝(𝑋), (10) where 𝛼 is a complex number with the unit modulus and, for 𝑋=𝐵_𝑛, Φ is a unitary transformation; for 𝑋=𝑈^𝑛, Φ(𝑧₁,…,𝑧_𝑛)=(𝜆₁𝑧_𝑖1,…,𝜆_𝑛𝑧_𝑖𝑛), where |𝜆_𝑗|=1, 1≤𝑗≤𝑛 and (𝑖₁,…,𝑖_𝑛) is some permutation of the integers from 1 through 𝑛.

Proof. We follow [5, Proposition 2.1] and [4, Theorem 3]. Let 𝑓,𝑔∈𝐻^𝑝(𝑋). By 2-homogenuity of isometry 𝑇, the equation _𝜕𝑋log1+sup_0≤𝑟<1𝑓(𝑟𝑧)2^𝑛−𝑔(𝑟𝑧)2^𝑛𝑝(𝑧)=𝑑𝜎_𝜕𝑋log1+sup_0≤𝑟<1(𝑇𝑓)(𝑟𝑧)2^𝑛−(𝑇𝑔)(𝑟𝑧)2^𝑛𝑝𝑑𝜎(𝑧) (11) holds. In a way similar to the proof of Theorem 2.1 in [1], we see that _𝜕𝑋sup_0≤𝑟<1𝑓(𝑟𝑧)−𝑔(𝑟𝑧)^𝑝(𝑧)=𝑑𝜎_𝜕𝑋sup_0≤𝑟<1𝑇𝑓(𝑟𝑧)−𝑇𝑔(𝑟𝑧)^𝑝𝑑𝜎(𝑧). (12) This equality implies that 𝑇 is isometric in the norm 𝑓_𝐻^𝑝_𝑚∶=_𝜕𝑋sup_0≤𝑟<1𝑓(𝑟𝑧)^𝑝𝑑𝜎(𝑧)^1/𝑝 (13) of the space 𝐻^𝑝(𝑋), which is equivalent to the standard norm in 𝐻^𝑝(𝑋). From (12) with 𝑔=0, we obtain 𝑇(𝐻^𝑝(𝑋))⊆𝐻^𝑝(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|_𝐻^𝑝_(𝑋) is an isometry with respect to the metric induced by the 𝐻^𝑝-norm ‖⋅‖_𝑝. The same argument for 𝑇⁻¹ shows that 𝑇⁻¹(𝐻^𝑝(𝑋))⊆𝐻^𝑝(𝑋). Thus, we see that 𝑇(𝐻^𝑝(𝑋))=𝐻^𝑝(𝑋). By the Mazur-Ulam theorem [6], 𝑇|_𝐻^𝑝_(𝑋) is a real-linear isometry since 𝑇(0)=0.
Using the limit lim_𝜀→0+1𝜀^𝑝+1(𝜀𝑡)^𝑝−log(1+𝜀𝑡)^𝑝=𝑝2𝑡^𝑝+1,𝑡≥0, (14) we show that 𝑇 is also isometric in the norm ‖𝑓‖_𝐻𝑚^𝑝+1. If 𝑝>0, then for any 𝑘∈ℕ, there exists the following limit: lim_𝜀→0+1𝜀^𝑝+𝑘log(1+𝜀𝑡)^𝑝−^𝑘−1_𝑛=0𝑐_𝑛(𝜀𝑡)^𝑛+𝑝=𝑐_𝑘𝑡^𝑝+𝑘,𝑡≥0, (15) where 𝑐_𝑛 are the Taylor coefficients of the function (log(1+𝑡)/𝑡)^𝑝. Using (15), we can prove that 𝑇 is isometric in 𝐻_𝑚^𝑝+𝑘 for 𝑝∈ℕ and all 𝑘∈ℕ by induction.
Since 𝑑𝜎 is a finite measure, we verify that lim_𝑝→∞𝑓_𝐻^𝑝_𝑚=𝑓_𝐻∞𝑚 (16) holds for every 𝑓∈𝐻^∞(𝑋), and it is clear that ‖𝑓‖_𝐻∞𝑚=‖𝑓‖_∞. Moreover, ‖𝑓‖_𝑝=‖𝑇(𝑓)‖_𝑝 for every 𝑓∈𝐻^∞(𝑋) and lim_𝑝→∞‖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].
If 𝑇|_𝐻^∞_(𝑋) is complex-linear, then 𝑇 is complex-linear on 𝑀^𝑝(𝑋), since 𝐻^∞(𝑋) is dense in 𝑀^𝑝(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].
If 𝑇|_𝐻^∞_(𝑋) is conjugate linear, then 𝑇 is conjugate linear on 𝑀^𝑝(𝑋) as before. Let 𝑇∶𝑀^𝑝(𝑋)→𝑀^𝑝(𝑋) be defined as 𝑇(𝑓)=𝑇(𝑓) for every 𝑓∈𝑀^𝑝(𝑋), where 𝑓𝑧₁,…,𝑧_𝑛=𝑓𝑧₁,…,𝑧_𝑛 (17) for 𝑓∈𝑀^𝑝(𝑋). Then, 𝑇 is complex-linear isometry from 𝑀^𝑝(𝑋) onto itself. Applying Corollary 2.3 in [1] to 𝑇, the second formula of the conclusion holds.

Let 1≤𝑝<∞. We say a map 𝑇∶𝑀^𝑝(𝑋)→𝑀^𝑝(𝑋) is multiplicative if 𝑇(𝑓𝑔)=𝑇(𝑓)𝑇(𝑔) for every 𝑓,𝑔∈𝑀^𝑝(𝑋). Next, we characterize multiplicative isometries from 𝑀^𝑝(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 𝑀^𝑝(𝑋) onto itself, and 𝑇(𝑓)=𝑓∘Φ defines a conjugate linear multiplicative isometry from 𝑀^𝑝(𝑋) onto itself. We show that they are the only multiplicative isometries from 𝑀^𝑝(𝑋) onto itself.

Theorem 2. Let 𝑝∈ℕ, and let 𝑇 be a multiplicative (not necessarily linear) isometry from 𝑀^𝑝(𝑋) onto itself. Then, there exists a holomorphic automorphism Φ on 𝑋 such that either of the following holds: 𝑇𝑓=𝑓∘Φ𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑓∈𝑀^𝑝(𝑋) (18) or 𝑇𝑓=𝑓∘Φ𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑓∈𝑀^𝑝(𝑋), (19) where Φ is a unitary transformation for 𝑋=𝐵_𝑛; Φ(𝑧₁,…,𝑧_𝑛)=(𝜆₁𝑧_𝑖1,…,𝜆_𝑛𝑧_𝑖𝑛) for 𝑋=𝑈^𝑛, where |𝜆_𝑗|=1 for every 1≤𝑗≤𝑛 and (𝑖₁,…,𝑖_𝑛) is some permutation of the integers from 1 through 𝑛.

Proof. Since 𝑇 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 𝑇𝑓=𝛼𝑓∘Φ,𝑓∈𝑀^𝑝(𝑋), (20) or 𝑇𝑓=𝛼𝑓∘Φ,𝑓∈𝑀^𝑝(𝑋), (21) 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 class 𝑀^𝑝(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.

Acknowledgments

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.