1. Introduction
It is a well known consequence of Cartan’s classical Uniqueness Theorem [
1] that given a bounded circular domain
D in the
N-dimensional complex space
, any holomorphic mapping
with
and preserving the Carathéodory (or Kobayashi) distance associated with
D is necessarily linear and surjective. In contrast, in 1994, E. Vesentini [
2] (p. 508), [
3] (Sec. 3) found various examples, even with holomorphic families, showing that the infinite dimensional version of this fact is no longer valid in general Banach space setting. Actually Cartan’s result ensured the linearity of the one-parameter semigroups of holomorphic automorphisms, fixing the origin and hence gave rise to a Lie theoretic approach by means of the infinitesimal generators to the precise algebraic description of the group of holomorphic automorphisms of a finite dimensional bounded homogeneous circular domain. However, Vesentini’s techniques seem unsuitable in constructing a
-semigroup
of non-linear Carathéodory isometries
on a bounded circular domain
contained in some complex Banach space
. Our aim in this short note is a
-semigroup construction (Lemma 2) carried out with slight modifications to familiar methods used in the theory of
-semigroups of linear operators [
4] with respect to delay equations [
5] in the fading memory space
. Our examples involve bounded convex circular domains
but rely upon some auxiliary remarks with independent interest in holomorphic invariant distances associated with domains for the type
in a function space
with some bounded convex domains
containing
.
As for the background to our motivation, the approach by von Neumann to classical Quantum Mechanics proposed modeling the evolution of wave functions with one-parameter
-groups of unitary operators in complex Hilbert spaces. Towards the beginning of the 1970s, exigences occurred to extend the related framework beyond the setting of linear operators and to regard this evolution as not necessarily reversible. To achieve this aim, natural candidates are one-parameter
-semigroups of holomorphic self-mappings preserving some automorphism-invariant distance on a bounded Banach space domain. Physical symmetry properties can be provided by the circularity or more generally by the holomorphic symmetry of the underlying domain. According to Kaup’s celebrated Riemann Mapping Theorem [
6], up to holomorphic equivalence, bounded symmetric domains are circular and convex.
At first sight, Theorem 2 seems to provide a negative result. However, the construction may reveal interesting geometric properties and links to delay equations for further investigation. Actually, our arguments require no deep knowledge of symmetric spaces and invariant distances: in
Section 2, we recall the necessary ingredients fom Banach space holomorphy and, after
Section 3, containing our results, we include
Appendix A giving a generalization of a distance formula known thus far only in the setting of symmetric domains.
2. Preliminaries
To establish terminology, by a one-parameter -semigroup on a topological space X we mean an indexed family of mappings with the semigroup properties , and the continuity of all orbits for any . Given two metric spaces , a mapping is a contraction if .
A subset D in a complex topological vector space E is said to be circular if it is connected, contains the origin of E and if .
Throughout this work, let
denote an arbitrarily fixed complex Banach space with norm
and open unit ball
. As standard notation, we write
for the complex plane regarded as a 1-dimensional space normed with the absolute value and unit disc
equipped with the
Poincaré metric . Given any domain (connected open set)
,
the associated
Carathéodory distance where
stands for the family of all
holomorphic maps between two Banach space domains
with respect to
. In the cases of our interest, a function
with bounded range is holomorphic if and only if for any point
and any unit vector
, it admits a uniformly convergent directional Taylor expansion
whenever the closed ball
is contained in
. A fundamental feature of Carathéodory metrics [
1] is that all holomorphic maps
are
contractions; furthermore, if the domain
is bounded then
is a complete metric space giving rise to the same topology as the distance by the norm on
.
For a locally compact Hausdorff space , will denote the Banach space of all continuous functions vanishing at infinity (i.e., is a compact subset for any ) equipped with the norm . In particular, consists of functions with limit 0 at infinity. Given any domain in some Banach space , a mapping with bounded range is immediately holomorphic if and only if all pointwise evaluations are holomorphic.
Given a bounded convex domain with , we also introduce the figure , which can easily be seen as a bounded convex domain in . In the course of the verification of Carathéodory isometry properties of holomorphic self-maps of domains of the type , we shall use the following plausible but highly non-trivial relation.
Lemma 1. For the Carathéodory distance of the domain with , we haveprovided the underlying topological space Ω
has a countable base and the target space is separable. Remark 1. The special case of (1) with appears in [1] with a proof relying upon Möbius transformations. Similar arguments can be applied in the case when is a (necessarily convex) holomorphically symmetric bounded circular domain, even without countability restrictions using Kaup’s JB*-triple calculus [6,7,8]. In its full generality, Lemma 1 can be deduced from a far-reaching theorem [9] due to Dineen-Timoney and Vigué (extending Lempert’s result [10] on the coincidence of the Carathéodory- and Kobayashi pseudometrics in finite dimensions) for convex domains in separable locally convex spaces. Since we do not have a reference, we give a detailed proof in Appendix A. 3. Results
Throughout this section,
denotes an arbitrarily fixed bounded convex domain in
containing the origin. For short, we can write
Lemma 2. Let be a -semigroup of (norm)-contractions . Then, the maps defined byform a -semigroup of isometries . Proof. Consider any function
. Since, by definition, the function
is continuous and ranges in
, we have
. Given another function
,
Since, trivially,
we conclude that each map
is a
-isometry.
Next, we check the semigroup property of
. Let
. Then, we have
Thus, if
, then
We complete the proof by checking strong continuity, i.e., that
whenever
in
. Recall that the moduli of continuity
associated with any function
with respect to any vector
are well-defined and converge to 0 as
. Let
. Since we have
it follows that
Hence, we see the uniform continuity of the function with the modulus of continuity . □
Remark 2. The conclusion of Lemma 2 holds even if is only assumed to be a real Banach space.
Proposition 1. Under the hypothesis of Lemma 1, if the maps above are additionally holomorphic and leave the origin of fixed, then furthermore the underlying Banach space is separable or is a circular holomorphically symmetric domain and then each term is a holomorphic 0-preserving isometry.
Proof. Since the domain is bounded, the holomorphy of the maps with holomorhic terms is an immediate consequence of the fact that all the pointwise evaluations are holomorphic. Indeed, we have or with holomorhic maps by assumption.
Since the maps
are
contractions, by the aid of Lemma 1 we can see that each term
is a
-isometry, as follows. Given any pair of functions
, we have
. Hence,
which completes the proof. □
Remark 3. It is well known from [11] that, given a continuously differentiable function , we havein terms of the family of supporting bounded linear functionals In particular, f is non-increasing whenever for any and for any functional .
We are going to complete our constructions with the aid of
flows of vector fields. Conveniently, in our context we simply identify a vector field
W on a domain
with a mapping
. A flow of
W is a one-parameter family of the form
of mappings
with
for
, such that
and for any point
the orbit
is defined on a nondegenerate interval
satisfying the relation
for
. Due to the classical Picard–Lindelöf Theorem, any Lipschitzian vector field
W admits a unique maximal flow [
11].
Lemma 3. Let be a Lipschitzian vector field with . Then, given any constant , the maximal flow of the vector field is a well-defined uniformly continuous one-parameter semigroup consisting of contractive self maps of .
Proof. Let
with
be the maximal flow of
W. Then, for any point
, the orbit
is the solution of the initial value problem
It is well known that by writing
for the domain of maximal solution of (
2), we necessarily have
; furthermore,
whenever
. We are going to exclude such cases due to the contraction properties of the maps
.
Let
and consider the function
defined on the interval
. Observe that, given any functional
, we have
Hence, we conclude that the function is decreasing; in particular, we have the contraction property for . Assumption implies with . Hence, we also see that for all and . This is possible only if . Therefore, the maximal flow of W is defined for all (time) parameters and consists of -contractions .
It is well known that flows parametrized on are automatically strongly continuous semigroups. The uniform continuity in our case is a consequence of the fact that , which shows that . □
Example 1. Let with and let . Since , we can apply Lemma 3 with . For the flow of W, we obtain the holomorphic maps Indeed, the solution of the initial value problemis , as one can check by direct computation. As for heuristics, we obtain a real-valued solution with real calculus for (3) with initial values , and the obtained formula extends holomorphically to Δ.
Theorem 2. Given a complex Banach space with symmetric or separable unit ball, there is a -semigroup of non-linear holomorphic 0-preserving norm and Carathéodory isometries of the open unit ball of the function space .
Proof. We can apply the construction of Proposition 1 with a semigroup obtained with the construction of Lemma 3 with any -polynomial polynomial vector field . □
Example 2. Let and . Then, the mapsform a -semigroup of non-linear holomorphic 0-preserving norms and Carathéodory isometries of the unit ball . Question 1.
Is any holomorphic norm-isometry of the unit ball of a complex Banach space automatically a Carathéodory isometry as well?