HOLOMORPHIC MAPPINGS INTO SOME DOMAIN IN A COMPLEX NORMED SPACE. Tatsuhiro Honda. 1. Introduction

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1 J. Korean Math. Soc. 41 (2004), No. 1, pp HOLOMORPHIC MAPPINGS INTO SOME DOMAIN IN A COMPLEX NORMED SPACE Tatsuhiro Honda Abstract. Let D 1, D 2 be convex domains in complex normed spaces E 1, E 2 respectively. When a mapping f : D 1 D 2 is holomorphic with f(0) = 0, we obtain some results like the Schwarz lemma. Furthermore, we discuss a condition whereby f is linear or injective or isometry. 1. Introduction Let = {z C; z < 1} be the unit disc in C. The classical Schwarz lemma in one complex variable is as follows: The Classical Schwarz Lemma. Let f : be a holomorphic mapping with f(0) = 0. Then the following statements hold: (i) f(z) z for any z, (ii) if there exists z 0 \{0} such that f(z 0 ) = z 0, or if f (0) = 1, then there exists a complex number λ of modulus 1 such that f(z) = λz and f is an automorphism of. It is natural to consider an extension of the above results to more general domains or higher dimensional spaces. However, condition (ii) in above no longer holds even for the bidisc. In fact, one can easily construct a holomorphic mapping f : such that f(0) = 0 and f(z) = z for z in an open subset of, but f is not an isometry (cf. J. P. Vigué [18]). Nevertheless, E. Vesentini [15], [16] showed that if f(w) = w holds on B 1 and if every boundary point of the unit ball B 2 is a complex extreme point, then f : B 1 B 2 is a linear isometry, where B 1, B 2 are the open unit balls in normed spaces E 1, E 2 over C respectively. J. P. Vigué [18], [19] proved that if every boundary point of the unit ball B for some norm in C n is a complex extreme Received November 29, Mathematics Subject Classification: 32F45, 32H02. Key words and phrases: Schwarz lemma, complex geodesic.

2 146 Tatsuhiro Honda point of B and if f(w) = w holds on an open subset U of B, then f : B B is a linear automorphism of C n. H. Hamada [6] generalized the above classical Schwarz lemma to the case where f(w) = w holds on some local complex submanifold of codimension 1. The author [10], [11] generalized to the case where f(w) = w holds on a nonpluripolar subset. H. Hamada and the author [8] generalized to the case where f(w) = w holds on a totally real submanifold. In this paper, we consider some condition whereby a holomorphic mapping is linear or injective or isometric. 2. Notation and preliminaries All topologies considered throughout this paper shall be Hausdorff. A vector space E over C is said to be locally convex if E is equipped with the Hausdorff topology defined by some family Π of seminorms such that sup α Π α(x) > 0 for all x E \ {0}, that is, a fundamental system of neighborhoods of x in this topology is made up of finite intersections of sets x + α 1 ([0, a]), α Π, 0 < a <. Then all seminorms in Π are continuous, but the family cs(e) of all continuous semi-norm on E is generally larger than Π. A sequence {z n } n N on a locally convex space E is a Cauchy sequence in E if for each α Π and each ε > 0, there exists n 0 N such that α(z n z m ) < ε for all m, n n 0. A locally convex space E is said to be sequentially complete if any Cauchy sequence converges. Let F be a locally convex space, let E be a sequentially complete locally convex space. Let U be an open subset in F, and let f : U E be a holomorphic mapping. For a U, there uniquely exists a sequence of n-homogeneous polynomials P n : F E such that the expansion f(a + z) = f(a) + P n (z) holds for all z in the largest balanced subset of U a. This series is called the Taylor expansion of f by n-homogeneous polynomials P n at a. Let = {z C; z < 1} be the unit disc in the complex plane C, and let γ(λ) = 1/(1 λ 2 ). The Poincaré distance ρ on is defined for z, w as follows: ρ(z, w) = 1 2 n=1 log 1 + (z w)/(1 z w) 1 (z w)/(1 z w).

3 Holomorphic mappings into some domain in a complex normed space 147 Let D be a domain in a sequentially complete locally convex space E. The gauge N D of D is defined for z E as follows: N D (z) = inf{α > 0; z αd}. The Carathéodory pseudodistance C D on D is defined for p, q D as follows: C D (p, q) = sup{ρ(f(p), f(q)); f H(D, )}. The Lempert function δ D of D is defined for p, q D as follows: δ D (p, q) = inf{ρ(ξ, η); ξ, η, ϕ H(, D) such that ϕ(ξ) = p, ϕ(η) = q}. The Kobayashi pseudodistance K D on D is defined for p, q D as follows: { m } K D (p, q) = inf δ D (x k, x k+1 ); m N, {p = x 1, x 2,..., x m+1 = q} D. k=1 Then we have established between the various pseudodistances on a domain D: C D K D δ D on D D. The infinitesimal Carathéodory pseudometric c D for D is defined for z D, v E as follows: c D (z, v) = sup{ dψ(z)(v) ; ψ H(D, )}. The infinitesimal Kobayashi pseudometric κ D for D is defined for z D, v E as follows: (2.1) κ D (z, v) = inf{γ(λ) α ; ϕ H(, D), λ such that ϕ(λ) = z, αϕ (λ) = v}. Then holomorphic mappings ϕ H(, D) as in (2.1) certainly exist. In fact, if R is the radius of the open disc {λ C; λv U(z)}, where U(z) is a neighborhood of z, we may take the mapping ϕ(λ) = z + λ ζ v for ζ 1/R. Hence κ D (z, v) 1/R. Moreover, for any ψ H(D, ) with ψ(z) = 0, we have (ψ ϕ) (0) = dψ(z)(ϕ (0)). It follows from this that c D κ D on D E. We use convexity to obtain the relationship among the pseudodistances or pseudometrics (S. Dineen [3], T. Franzoni and E. Vesentini [5], M. Hervé [9] etc).

4 148 Tatsuhiro Honda Proposition 2.1. If D is a balanced convex domain in a sequentially complete locally convex space E, then (i) C D (0, x) = K D (0, x) = δ D (0, x) = ρ(0, N D (x)) for any x D, (ii) c D (0, v) = κ D (0, v) = N D (v) for any v E. Let D be a balanced pseudoconvex domain in a sequentially complete locally convex space E. Then we have the following proposition as the gauge N D is plurisubharmonic on E. Proposition 2.2. If D is a balanced pseudoconvex domain in a sequentially complete locally convex space E, then (i) K D (0, x) = δ D (0, x) = ρ(0, N D (x)) for any x D, (ii) κ D (0, v) = N D (v) for any v E. Using the above proposition, we obtain the following generalization of part (i) of the Schwarz lemma to balanced pseudoconvex domains in sequentially complete locally convex spaces. Proposition 2.3. Let E j be a sequentially complete locally convex space and let D j be a balanced pseudoconvex domain in E j for j = 1, 2. Let f : D 1 D 2 be a holomorphic mapping with f(0) = 0. Then N D2 f(z) N D1 (z). Proof. By Proposition 2.2 (i), we have ρ(0, N D1 (z)) = δ D1 (0, z) δ D2 (0, f(z)) = ρ(0, N D2 f(z)). Since ρ(0, r) is strictly increasing for 0 r < 1, we obtain this proposition. The following definition of a complex geodesic due to E. Vesentini [15, 16, 17]. Definition 2.4. Let D be a domain in a sequentially complete locally convex space E endowed with a pseudodistance d D. A holomorphic mapping ϕ : D is said to be a complex d D -geodesic for (x, y) if d D (x, y) = ρ(ξ, η) for any points ξ, η such that ϕ(ξ) = x and ϕ(η) = y. A holomorphic mapping ϕ : D is said to be a complex c D - geodesic for (z, v) if c D (z, v) = γ(λ) α holds for any λ and any α C such that ϕ(λ) = z and αϕ (λ) = v.

5 Holomorphic mappings into some domain in a complex normed space 149 A holomorphic mapping ϕ : D is said to be a complex κ D - geodesic for (z, v) if κ D (z, v) = γ(λ) α holds for any λ and α C such that ϕ(λ) = z and αϕ (λ) = v. The following results about a complex geodesic are well-known (cf. S. Dineen [3], T. Franzoni and E. Vesentini [5], M. Hervé [9] etc). Proposition 2.5. Let D be a domain in a sequentially complete locally convex space E endowed with a pseudodistance d D or a pseudometric µ D. Then the following statements hold: (i) a holomorphic mapping ϕ : D is a complex d D -geodesic for (x, y) if and only if there exists only one pair (ξ, η) 2 with (ξ η) such that ϕ(ξ) = x, ϕ(η) = y and d D (x, y) = ρ(ξ, η), (ii) a holomorphic mapping ϕ : D is a complex µ D -geodesic for (z, v) if and only if there exists only one point λ such that ϕ(λ) = z, αϕ (λ) = v and µ D (ϕ(λ), ϕ (λ)) = α γ(λ). A point x of the closure D of D is said to be a complex extreme point of D if y = 0 is the only vector in E such that the function : ζ x + ζy maps into D. For example, C 2 -smooth strictly convex boundary points are complex extreme points. For a bounded balanced pseudoconvex domain D, the holomorphic mapping ϕ(ζ) = ζa/n D (a) is a complex δ D -geodesic and κ D -geodesic for (0, a) for any a D with N D (a) > 0. In fact, M. Hervé [9] has given the following characterization of the uniqueness of complex geodesics (see e.g. E. Vesentini [15], [16], [17]). Proposition 2.6. Let D be a balanced convex domain in a sequentially complete locally convex space E. Let a D be such that N D (a) > 0, and let ϕ : D be the holomorphic mapping defined by ϕ(ζ) = ζa/n D (a). Then the following conditions are equivalent : (i) the point b = a/n D (a) is a complex extreme point of D; (ii) ϕ is the unique (modulo Aut( )) complex C D -geodesic for (0, a); (iii) ϕ is the unique (modulo Aut( )) complex K D geodesic for (0, a); (iv) ϕ is the unique (modulo Aut( )) complex δ D geodesic for (0, a); (v) ϕ is the unique (modulo Aut( )) complex c D -geodesic for (0, a); (vi) ϕ is the unique (modulo Aut( )) complex κ D -geodesic for (0, a).

6 150 Tatsuhiro Honda Using the uniqueness of complex geodesics, we obtain the linearity of complex geodesics as in the following proposition. Proposition 2.7. Let D j be a bounded balanced convex domain in complex normed spaces E j for j = 1, 2, and let f : D 1 D 2 be a holomorphic mapping with f(0) = 0. Let x D 1 \ {0} and let ϕ(ζ) = ζx/n D1 (x). We assume that f(x)/n D2 f(x) is a complex extreme point of D 2. If one of the following conditions is satisfied, then f ϕ is a linear complex δ D2 -geodesic. (i) N D2 f(x) = N D1 (x). (ii) δ D2 (f(0), f(x)) = δ D1 (0, x). (iii) K D2 (f(0), f(x)) = K D1 (0, x). (iv) C D2 (f(0), f(x)) = C D1 (0, x). Proof. By Proposition 2.1 (i), the conditions (i), (ii), (iii) and (iv) are equivalent. Suppose that (i) is satisfied. By Proposition 2.1 (i), δ D2 (f ϕ(0), f ϕ N D1 (x)) = δ D2 (0, f(x)) = δ D1 (0, x) = ρ(0, N D1 (x)). By Proposition 2.5 (i), f ϕ is a complex δ D2 -geodesic for (0, f(x)). By Proposition 2.6, we have for some θ R. f ϕ(ζ) = ζe iθ f(x) N D2 f(x) 3. Special versions of the Schwarz Lemma Now we introduce the projective space P(E). Let E be a locally convex space. Let z and z be points in E \ {0}. z and z are said to be equivalent if there exists λ C such that z = λz. We denote by P(E) the quotient space of E \ {0} by this equivalence relation. Then P(E) is a Hausdorff space. The Hausdorff space P(E) is called the complex projective space induced by E. We denote by Q the quotient map from E \ {0} to P(E) (see M. Nishihara [14]). Theorem 3.8. Let E j be a complex normed space, let D j be a bounded balanced convex domain in E j for j = 1, 2 and let f : D 1 D 2 be a holomorphic mapping with f(0) = 0. Let X be a non-empty subset

7 Holomorphic mappings into some domain in a complex normed space 151 of D 1 such that X is mapped homeomorphically onto an open subset Ω in the complex projective space P(E 1 ) by the quotient map Q from E 1 \ {0} onto P(E 1 ). We assume that f(x)/n D2 (f(x)) is a complex extreme point of D 2 for any x X and that there exists w 0 X such that w 0 /N D1 (w 0 ) is a complex extreme point of D 1. If one of the following conditions is satisfied, then f is linear and injective. (i) N D2 (f(x)) = N D1 (x) for any x X. (ii) C D2 (f(0), f(x)) = C D1 (0, x) for any x X. (iii) K D2 (f(0), f(x)) = K D1 (0, x) for any x X. (iv) δ D2 (f(0), f(x)) = δ D1 (0, x) for any x X. Proof. By Proposition 2.1 (i), the conditions (i), (ii), (iii) and (iv) are equivalent. Suppose that (i) is satisfied. We take a point w X\{0} and set ϕ(ζ) = ζw/n D1 (w) for ζ. Then ϕ is a complex δ D1 -geodesic. We have δ D2 (f ϕ(0), f ϕ(n D1 (w))) = ρ(0, N D1 (w)). By Proposition 2.7, f ϕ is a complex δ D2 -geodesic. It follows from this that there exists a point y D 2 \ {0} such that (3.1) y f ϕ(ζ) = ζ N D2 (y). On the other hand, let f(x) = P n (x) be the Taylor expansion of n=1 f by n-homogeneous polynomials P n in a neighborhood V of 0 in E 1. Then we have w f ϕ(ζ) = P n (ζ N D1 (w) ) = ( ) ζ n (3.2) P n (w) N D1 (w) n=1 n=1 in a neighborhood of 0 in. By (3.1) and (3.2), we obtain P n (w) = 0 for w X, n 2. We take any point y C X = {tx; t C, x X}. Then there exist t C and x X such that y = tx. Hence P n (y) = P n (tx) = t n P n (x) = 0, that is, P n 0 on C X E 1 for every n 2. Since Q is continuous, the set C X = Q 1 (Ω) contains an open subset U of E. By the identity theorem, P n 0 on E 1 for every n 2.

8 152 Tatsuhiro Honda Therefore f = P 1, that is, f is linear. Next we show that f is injective. Let z be a point of E 1 with f(z) = 0. Since f is linear, we have N D2 f(tx) = N D2 (tf(x)) = t N D2 f(x) = t N D1 (x) = N D1 (tx) for every t C, x X. It follows from this that N D2 f(y) = N D1 (y) for all y C X. Since C X is open, there exists a positive number r > 0 such that w 0 + ζz C X for ζ C, ζ < r. Then we have (3.3) On the other hand, (3.4) By (3.3) and (3.4), we have Hence N D2 f(w 0 + ζz) = N D1 (w 0 + ζz). N D2 f(w 0 + ζz) = N D2 (f(w 0 ) + ζf(z)) N D1 ( w0 = N D2 f(w 0 ) = N D1 (w 0 ). N D1 (w 0 + ζz) = N D1 (w 0 ). N D1 (w 0 ) + ) ζ N D1 (w 0 ) z = 1 for ζ < r. Since w 0 /N D1 (w 0 ) is a complex extreme point of D 1, we have Therefore f is injective. z = 0. Since complex Hilbert spaces are endowed with the norm which is induced from its inner products, we have the following corollary. Corollary 3.9. Let H j be a complex Hilbert space with the inner product <, > j, let B j be the open unit ball of H j for the norm j =<, > 1 2 j for j = 1, 2. Let f : B 1 B 2 be a holomorphic map with f(0) = 0. Let X be a non-empty subset of B 1 such that X is mapped onto an open subset Ω in the projective space P(H 1 ) by the quotient map Q. If w 1 = f(w) 2 holds for every w X, then f is a linear isometry.

9 Holomorphic mappings into some domain in a complex normed space 153 If H 1 = H 2 = C n with the Euclidean unit ball B, then f is a linear automorphism of B. Proof. Since every point of the boundary B j = {z H j ; z j 1 = 0} of B j is a complex extreme point of the closure B j of B j for j = 1, 2, by Theorem 3.8, f is linear and injective. We consider a function g(z) = z 2 1 f(z) 2 2 for z H 1. By Proposition 2.3, we have g 0 on H 1. Since g 0, the non-negative valued function g is plurisubharmonic on H 1. Hence log g is plurisubharmonic on H 1. Since w 1 = f(w) 2 for every w X, log g on an open subset C X = Q 1 (Ω). Therefore f is a linear isometry. 4. Infinitesimal pseudometrics Proposition Let D j be a bounded balanced convex domain in a complex normed space E j for j = 1, 2, and let f : D 1 D 2 be a holomorphic mapping with f(0) = 0. Let x D \ {0} and let ϕ(ζ) = ζx/n D1 (x). We assume that df(0)x/n D2 (df(0)x) is a complex extreme point of D 2. If one of the following conditions is satisfied, then f ϕ is a linear complex κ D2 -geodesic. (i) N D2 f(x) = N D1 (x). (ii) c D2 (f(0), f(x)) = c D1 (0, x). (iii) κ D2 (f(0), f(x)) = κ D1 (0, x). Proof. By Proposition 2.1 (ii), κ D2 (0, df(0)x) = κ D1 (0, x) = N D1 (x). Since N D1 (x)(f ϕ) (0) = df(0)x, f ϕ is a complex κ D2 -geodesic for (0, df(0)x). By Proposition 2.6, we have for some θ R. f ϕ(ζ) = ζe iθ df(0)x N D2 (df(0)x)

10 154 Tatsuhiro Honda Theorem Let E j be a complex normed space, let D j be a bounded balanced convex domain in E j for j = 1, 2, and let f : D 1 D 2 be a holomorphic mapping. Let V be a connected open neighborhood of the origin in D 1. We assume that κ D2 (0, df(0)x) = κ D1 (0, x) for x V. If f(0) = 0 and df(0)x/n D2 (df(0)x) is a complex extreme point of D 2 for any x V \ {0}, and if there exists w V \ {0} such that w/n D1 (w) is a complex extreme point of D 1, then f is linear and injective. Proof. Let f(z) = n=1 P n(z) be the expansion of f by n-homogeneous polynomials P n in a neighborhood of 0 in E 1. Since κ D2 (f(0), df(0)v) = κ D1 (0, v) for any v V, by Proposition 4.10, f(ζx/n D1 (x)) is the restriction of a linear map for any x V. Then we have P n (x) = 0 on V for n 2 as in the proof of Theorem 3.8. By the analytic continuation theorem, we have P n is identically 0 for n 2. Therefore f is the restriction of a linear map. Let ϕ(ζ) = ζw/n D1 (w). By Proposition 2.6, f ϕ is a complex δ D2 -geodesic for (0, f(v)). By Proposition 2.1, ρ(0, N D2 (f(v))) = δ D2 (0, f(v)) = ρ(0, N D1 (v)). This implies that N D2 (f(v)) = N D1 (v). The rest of the proof is same as Theorem 3.8. We note that the map f is not necessarily a linear isometry under the assumption of the above theorem (cf. J. P. Vigué [18]). The following theorem was obtained by H.Cartan for bounded domain in C 2 (see T. Franzoni and E. Vesentini [5] etc). Theorem Let D be a bounded domain in a complex normed space E, and let f : D D be a holomorphic mapping. If there exists x 0 D such that f(x 0 ) = x 0 and df(x 0 ) is an identity, f is the identity map. Using the above theorem of Cartan, we obtain the following theorem. Theorem Let E j be a complex normed space, let D j be a bounded balanced convex domain in E j for j = 1, 2, and let f : D 1 D 2 be a holomorphic mapping. Let V be a connected open neighborhood of the origin in D 1. We assume that κ D2 (0, df(0)x) = κ D1 (0, x) for x V. If the inverse df(0) 1 exists, then f(0) = 0 and f is the restriction of df(0) to D 1.

11 Holomorphic mappings into some domain in a complex normed space 155 Proof. First we will show f(0) = 0. We assume that a = f(0) 0. Since a D, there exists a point v E 1 such that N D1 (v) = 1 and df(0)v = a/n D2 (a). Then we have Therefore κ D2 (a, a/n D2 (a)) = κ D2 (f(0), df(0)v) κ D1 (0, v) = N D1 (v) = 1. (4.1) κ D2 (a, a) N D2 (a). On the other hand, we set ϕ(ζ) = ζa/n D2 (a) for ζ. Then ϕ is a complex κ D2 -geodesic for (0, a). So we have κ D2 (a, a) = κ D2 (ϕ(n D2 (a)), N D2 (a)ϕ (N D2 (a))) = κ (N D2 (a), N D2 (a)) N D2 (a) = 1 {N D2 (a)} 2. Therefore κ D2 (a, a) > N D2 (a). This contradicts with (4.1). We obtain f(0) = 0. By the assumptions, we have N D1 (df(0) 1 (w)) < 1 for w D 2. Now we consider a holomorphic mapping g = df(0) 1 f. Then g is a holomorphic mapping from D 1 to D 1 such that g(0) = 0 and dg(0) is identity. By Theorem 4.12, g is identity. In the Hilbert space case, since every boundary point of the unit ball is a complex extreme point, by the proofs of Corollary 3.9, Theorem 4.11 and Theorem 4.13, we obtain the following corollary. Corollary Let H j be a complex Hilbert space with the inner product <, > j, let B j be the open unit ball of H j for the norm j =<, > 1 2 j for j = 1, 2. Let f : B 1 B 2 be a holomorphic map. Let V be a connected open neighborhood of the origin in B 1. We assume that κ B2 (0, df(0)x) = κ B1 (0, x) for x V. Then f(0) = 0 and f is a linear isometry. If H 1 = H 2 = C n with the Euclidean unit ball B, then f is a linear automorphism of B.

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