A Counterexample Concerning C₀-Semigroups of Holomorphic Carathéodory Isometries

Abstract

We give an example for a $C_0$-semigroup of non-linear 0-preserving holomorphic Carathéodory isometries of the unit ball.

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 ${\mathbb{C}}^N$, any holomorphic mapping $F:D\to D$ with $F\left(0\right)=0$ 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found various examples, even with holomorphic families, showing that the infinite dimensional version of this fact is no longer valid in general Banach space setting.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 $C_0$-semigroup $[F^t:t\ge 0]$ of non-linear Carathéodory isometries $F^t\in \mathrm{Hol}(\mathbf{D},\mathbf{D})$ on a bounded circular domain $\mathbf{D}$ contained in some complex Banach space $\mathbf{E}$. Our aim in this short note is a $C_0$-semigroup construction (Lemma 2) carried out with slight modifications to familiar methods used in the theory of $C_0$-semigroups of linear operators [4] with respect to delay equations [5] in the fading memory space $C_0({\mathbb{R}}_{+},\mathbf{E})$. Our examples involve bounded convex circular domains $\mathbf{D}$ but rely upon some auxiliary remarks with independent interest in holomorphic invariant distances associated with domains for the type $\mathcal{D}=\left\{x\in \mathcal{X}:\mathrm{range}\left(f\right)\subset \mathbf{D}\right\}$ in a function space $\mathcal{X}=C_0(\mathsf{\Omega },\mathbf{E})$ with some bounded convex domains $\mathbf{D}$ containing $0\in \mathbf{E}$.

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 $C_0$-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 $C_0$-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 $C_0$-semigroup on a topological space X we mean an indexed family $\left[F^t:t\in {\mathbb{R}}_{+}\right]$ of mappings $F^t:X\to X$ with the semigroup properties $F^0={\mathrm{Id}}_X=\left[X\ni x\mapsto x\right]$, $F^t\circ F^s\left(x\right)=F^t\left(F^s\left(x\right)\right)=F^{t+s}\left(x\right)$$(s,t\in {\mathbb{R}}_{+})$ and the continuity of all orbits $t\mapsto F^t\left(x\right)$ for any $x\in X$. Given two metric spaces $(X_j,d_j)$$(j=1,2)$, a mapping $f:X_1\to X_2$ is a $d_1\to d_2$ contraction if $d_2\left(f\left(x\right),f\left(y\right)\right)\le d_1(x,y)$$(x,y\in X_1)$.

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 $D=e^{it}D=\{e^{it}x:x\in D\}$$(t\in \mathbb{R})$.

Throughout this work, let $\mathbf{E}$ denote an arbitrarily fixed complex Banach space with norm $\parallel ·\parallel$ and open unit ball $B_1\left(\mathbf{E}\right)$. As standard notation, we write $\mathbb{C}$ for the complex plane regarded as a 1-dimensional space normed with the absolute value and unit disc $\Delta =B_1\left(\mathbb{C}\right)=\{\zeta :|\zeta |<1\}$ equipped with the Poincaré metric $d_{\Delta }(\alpha ,\beta )={tanh}^{-1}|(\beta -\alpha )/(1-\overline{\alpha })|$ $\left(\alpha ,\beta \in \Delta \right)$. Given any domain (connected open set) $\mathbf{D}\subset \mathbf{E}$,

$$d_{\mathbf{D}}\left(p,q\right)=sup\left\{d_{\Delta }\left(f\left(p\right),f\left(q\right)\right):f\in \mathrm{Hol}(\mathbf{D},\Delta )\right)\}\left(p,q\in \mathbf{D}\right)$$

the associated Carathéodory distance where $\mathrm{Hol}({\mathbf{D}}_1,{\mathbf{D}}_2)$ stands for the family of all holomorphic maps between two Banach space domains ${\mathbf{D}}_1\subset {\mathbf{E}}_1$ with respect to ${\mathbf{D}}_2\subset {\mathbf{E}}_2$. In the cases of our interest, a function $f:{\mathbf{D}}_2\to {\mathbf{E}}_2$ with bounded range is holomorphic if and only if for any point $p\in \mathbf{D}$ and any unit vector $v\in \mathbf{E}$, it admits a uniformly convergent directional Taylor expansion $\zeta \mapsto f(p+\zeta v)={\displaystyle \sum _{n=0}^{\infty }}{\zeta }^n{a}_n\left(a_n\in {\mathbf{E}}_1,{\displaystyle \sum _{n=0}^{\infty }}\parallel a_n\parallel {\rho }^n<\infty \right)$ whenever the closed ball $p+\rho \overline{B_1\left(\mathbf{E}\right)}$ is contained in $\mathbf{D}$. A fundamental feature of Carathéodory metrics [1] is that all holomorphic maps ${\mathbf{D}}_2\to {\mathbf{D}}_2$ are $d_{{\mathbf{D}}_1}\to d_{{\mathbf{D}}_2}$ contractions; furthermore, if the domain $\mathbf{D}\subset \mathbf{E}$ is bounded then $\left(\mathbf{D},d_{\mathbf{D}}\right)$ is a complete metric space giving rise to the same topology as the distance by the norm on $\mathbf{D}$.

For a locally compact Hausdorff space $\mathsf{\Omega }$, $C_0(\mathsf{\Omega },\mathbf{E})$ will denote the Banach space of all continuous functions $f:\mathsf{\Omega }\to \mathbf{E}$ vanishing at infinity (i.e., $f^{-1}\{p\in \mathbf{E}:\parallel p\parallel \ge \epsilon \}$ is a compact subset for any $\epsilon >0$) equipped with the norm $\parallel f\parallel =\underset{\omega \in \mathsf{\Omega }}{max}\parallel f\parallel$. In particular, $C_0({\mathbb{R}}_{+},\mathbf{E})$ consists of functions with limit 0 at infinity. Given any domain ${\mathbf{D}}_0$ in some Banach space ${\mathbf{E}}_0$, a mapping $f:{\mathbf{D}}_0\to C_0(\mathsf{\Omega },\mathbf{E})$ with bounded range is immediately holomorphic if and only if all pointwise evaluations ${\delta }_{\omega }f:{\mathbf{D}}_0\ni z\mapsto f\left(z\right)\left(\omega \right)$$(\omega \in \mathsf{\Omega })$ are holomorphic.

Given a bounded convex domain $\mathbf{D}\subset \mathbf{E}$ with $0\in \mathbf{D}$, we also introduce the figure $C_0(\mathsf{\Omega },\mathbf{D})=\{f\in C_0(\mathsf{\Omega },\mathbf{E}):\mathrm{range}\left(f\right)\subset \mathbf{D}\}$, which can easily be seen as a bounded convex domain in $C_0(\mathsf{\Omega },\mathbf{E})$. In the course of the verification of Carathéodory isometry properties of holomorphic self-maps of domains $\mathcal{D}$ of the type $C_0(\mathsf{\Omega },\mathbf{D})$, we shall use the following plausible but highly non-trivial relation.

Lemma 1.

For the Carathéodory distance of the domain $\mathcal{D}=C_0(\mathsf{\Omega },\mathbf{D})$ with $0\in \mathbf{D}\subset \mathbf{E}$, we have

$$d_{\mathcal{D}}\left(x,y\right)=\underset{\omega \in \mathsf{\Omega }}{max}{d}_{\mathbf{D}}\left(x\left(\omega \right),y\left(\omega \right)\right)\left(x,y\in \mathcal{D}\right)$$

(1)

provided the underlying topological space Ω has a countable base and the target space $\mathbf{E}$ is separable.

Remark 1.

The special case of (1) with $\mathcal{D}=C_0({\mathbb{R}}_{+},\Delta )$ appears in [1] with a proof relying upon Möbius transformations. Similar arguments can be applied in the case when $\mathbf{D}$ 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, $\mathbf{D}$ denotes an arbitrarily fixed bounded convex domain in $\mathbf{E}$ containing the origin. For short, we can write

$$\mathcal{X}=C_0({\mathbb{R}}_{+},\mathbf{E})\mathrm{and}\mathcal{D}=C_0({\mathbb{R}}_{+},\mathbf{D})=\left\{x\in \mathcal{X}:\mathrm{range}\left(x\right)\subset \mathbf{D}\right\}.$$

Lemma 2.

Let $\left[{\phi }^t:t\in {\mathbb{R}}_{+}\right]$ be a $C_0$-semigroup of (norm)-contractions $\mathbf{D}\to \mathbf{D}$. Then, the maps ${\Phi }^t:\mathcal{D}\to \mathcal{X}$$(t\in {\mathbb{R}}_{+})$ defined by

$${\Phi }^t\left(x\right):{\mathbb{R}}_{+}\ni \tau \mapsto \left[{\phi }^{t-\tau }\left(x\left(0\right)\right)if0\le \tau \le t,x(\tau -t)if\tau \ge t\right]$$

form a $C_0$-semigroup of isometries $\mathcal{D}\to \mathcal{D}$.

Proof. 

Consider any function $x\in \mathcal{D}$. Since, by definition, the function $\tau \mapsto {\phi }^t\left(x\left(0\right)\right)$ is continuous and ranges in $\mathbf{D}$, we have ${\Phi }^t\left(x\right)\in \mathcal{D}$. Given another function $y\in \mathcal{D}$,

$$\begin{array}{cc}\hfill \parallel {\Phi }^t\left(x\right)-{\Phi }^t\left(y\right)\parallel & =max\left\{\underset{0\le \tau \le t}{max}\parallel {\phi }^{t-\tau }\left(x\left(0\right)\right)-{\phi }^{t-\tau }\left(y\left(0\right)\right)\parallel ,\underset{\sigma \ge t}{max}\parallel x(\sigma -t)-y(\sigma -t)\parallel \right\}\hfill \\ & \le \parallel x\left(0\right)-y\left(0\right)\parallel ,\underset{\sigma \ge t}{max}\parallel x(\sigma -t)-y(\sigma -t)\parallel \}\hfill \\ & =\underset{\tau \ge 0}{max}\parallel x\left(\tau \right)-y\left(\tau \right))\parallel =\parallel x-y\parallel .\hfill \end{array}$$

Since, trivially,

$$\parallel {\Phi }^t\left(x\right)-{\Phi }^t\left(y\right)\parallel \ge \underset{\sigma \ge t}{max}\parallel x(\sigma -t)-y(\sigma -t)\parallel \}=\underset{\tau \ge 0}{max}\parallel x\left(\tau \right)-y\left(\tau \right)\parallel \}=\parallel x-y\parallel ,$$

we conclude that each map ${\Phi }^t$ is a $\mathcal{D}$-isometry.

Next, we check the semigroup property of $[{\Phi }^t:t\in {\mathbb{R}}_{+}]$. Let $s,t\ge 0$. Then, we have

$$\begin{array}{cc}\hfill {\Phi }^s\circ {\Phi }^t\left(x\right)& :\tau \mapsto \left[{\phi }^{s-\tau }\left({\Phi }^t\left(x\right)\left(0\right)\right)\mathrm{if}\tau \le s,{\phi }^t\left(x\right)(\tau -s)\mathrm{if}\tau \ge s\right],\hfill \\ \hfill {\Phi }^{s+t}\left(x\right)& :\tau \mapsto \left[{\phi }^{(s+t)-\tau }\left(x\left(0\right)\right)\mathrm{if}\tau \le s+t,x\left(\tau -(s+t)\right)\mathrm{if}\tau \ge s+t\right].\hfill \end{array}$$

Thus, if $0\le \tau \le s$, then

$$\begin{array}{cc}\hfill {\Phi }^s\circ {\Phi }^t\left(x\right)\left(\tau \right)& ={\phi }^{s-\tau }\left({\Phi }^t\left(x\right)\left(0\right)\right)={\phi }^{s-\tau }\left({\phi }^t\left(x\left(0\right)\right)\right)\hfill \\ & ={\phi }^{s-\tau }\circ {\phi }^t\left(x\left(0\right)\right)={\phi }^{(s+t)-\tau }\left(x\left(0\right)\right)={\Phi }^{s+t}\left(x\right)\left(\tau \right).\hfill \end{array}$$

If $s\le \tau \le s+t$, then

$$\begin{array}{cc}\hfill {\Phi }^s\circ {\Phi }^t\left(x\right)\left(\tau \right)& ={\Phi }^t\left(x\right)(\tau -s){=}^{\tau -s\le t}={\phi }^{t-(\tau -s)}\left(x\left(0\right)\right)\hfill \\ & ={\phi }^{(s+t)-\tau }\left(x\left(0\right)\right)={\Phi }^{s+t}\left(x\right)\left(\tau \right).\hfill \end{array}$$

If $s+t\le \tau$, then

$$\begin{array}{cc}\hfill {\Phi }^s\circ {\Phi }^t\left(x\right)\left(\tau \right)& ={\Phi }^t\left(x\right)(\tau -s){=}^{\tau -s\ge t}=x\left((\tau -s)-t\right)={\Phi }^{s+t}\left(x\right)\left(\tau \right).\hfill \end{array}$$

We complete the proof by checking strong continuity, i.e., that $\parallel {\Phi }^t\left(x\right)-{\Phi }^s\left(x\right)\parallel \to 0$ whenever $s\to t$ in ${\mathbb{R}}_{+}$. Recall that the moduli of continuity

$$M(z,\delta ):=\underset{|t_1-t_2|\le \delta }{max}\parallel z\left(t_1\right)-z\left(t_2\right)\parallel ,m(e,\delta ):=\underset{|t_1-t_2|\le \delta }{max}\parallel {\phi }^{t_1}\left(e\right)-{\phi }^{t_2}\left(e\right)\parallel$$

associated with any function $z\in \mathcal{X}$ with respect to any vector $e\in \mathbf{E}$ are well-defined and converge to 0 as $\delta \searrow 0$. Let $0\le t_1\le t_2$. Since we have

$${\Phi }^{t_2}\left(x\right)\left(\tau \right)-{\Phi }^{t_1}\left(x\right)\left(\tau \right)=\left\{\begin{array}{cc}{\phi }^{t_2-\tau }\left(x\left(0\right)\right)-{\phi }^{t_1-\tau }\left(x\left(0\right)\right)\hfill & \mathrm{if}\tau \le t_1,\hfill \\ {\phi }^{t_2-\tau }\left(x\left(0\right)\right)-x(\tau -t_1)\hfill & \mathrm{if}{t}_1\le \tau \le t_2,\hfill \\ x(\tau -t_2)-x(\tau -t_1)\hfill & \mathrm{if}{t}_2\le \tau ,\hfill \end{array}\right.$$

it follows that

$$\parallel {\Phi }^{t_2}\left(x\right)-{\Phi }^{t_1}\left(x\right)\parallel \le \left\{\begin{array}{cc}m\left(x\left(0\right),t_2-t_1\right)\hfill & \mathrm{if}\tau \le t_1,\hfill \\ \parallel {\phi }^{t_2-\tau }\left(x\left(0\right)\right)-x\left(0\right)\parallel +\parallel x(\tau -t_1)-x\left(0\right)\parallel \le \hfill & \\ \le m\left(x\left(0\right),t_2-t_1\right)+M(x,t_2-t_1)\hfill & \mathrm{if}{t}_1\le \tau \le t_2,\hfill \\ M(x,t_2-t_1)\hfill & \mathrm{if}{t}_2\le \tau .\hfill \end{array}\right.$$

Hence, we see the uniform continuity of the function $t\mapsto {\Phi }^t\left(x\right)$ with the modulus of continuity $\delta \mapsto m\left(x\left(0\right),\delta \right)+M(x,\delta )$. □

Remark 2.

The conclusion of Lemma 2 holds even if $\mathbf{E}$ is only assumed to be a real Banach space.

Proposition 1.

Under the hypothesis of Lemma 1, if the maps ${\phi }^t$ above are additionally holomorphic and leave the origin of $\mathbf{E}$ fixed, then furthermore the underlying Banach space $\mathbf{E}$ is separable or $\mathbf{D}$ is a circular holomorphically symmetric domain and then each term ${\Phi }^t$ is a holomorphic 0-preserving $d_{\mathcal{D}}\to d_{\mathcal{D}}$ isometry.

Proof. 

Since the domain $\mathbf{D}$ is bounded, the holomorphy of the maps ${\Phi }^t$ with holomorhic terms ${\phi }^t$ is an immediate consequence of the fact that all the pointwise evaluations ${\delta }_{\omega }\mathsf{\Psi }:\mathcal{D}\ni x\mapsto \mathsf{\Psi }\left(x\right)\left(\omega \right)$$(\omega \in \mathsf{\Omega })$ are holomorphic. Indeed, we have ${\delta }_{\tau }{\Phi }^t=\left[x\mapsto x(\tau -t)\right]$ or ${\delta }_{\tau }{\Phi }^t=\left[x\mapsto {\phi }^{\tau -t}\left(x\left(0\right)\right)\right]$ with holomorhic maps by assumption.

Since the maps $\phi \in \mathrm{Hol}(\mathbf{D},\mathbf{D})$ are $d_{\mathcal{D}}\to d_{\mathcal{D}}$ contractions, by the aid of Lemma 1 we can see that each term ${\Phi }^t$ is a $d_{\mathcal{D}}$-isometry, as follows. Given any pair of functions $x,y\in \mathcal{D}$, we have $d_{\mathbf{D}}({\phi }^t\left(x\left(0\right)\right),{\phi }^t\left(y\left(0\right)\right)\le d_{\mathbf{D}}\left(x\left(0\right),y\left(0\right)\right)$$(t\ge 0)$. Hence,

$$\begin{array}{cc}& d_{\mathcal{D}}\left({\Phi }^t\left(x\right),{\Phi }^t\left(y\right)\right)=\underset{\tau \ge 0}{max}{d}_{\mathbf{D}}\left({\delta }_{\tau }{\Phi }^t\left(x\right)\left(\tau \right),{\delta }_{\tau }{\Phi }^t\left(y\right)\left(\tau \right)\right)=\hfill \\ & =max\left\{d_{\mathbf{D}}\left({\phi }^{{[t-\tau ]}_{+}}\left(x\left(0\right)\right),{\phi }^{{[t-\tau ]}_{+}}\left(y\left(0\right)\right)\right),d_{\mathbf{D}}\left(x\left({[\tau -t]}_{+}\right),y\left({[\tau -t]}_{+}\right)\right):t\ge 0\right\}=\hfill \\ & =d_{\mathbf{D}}\left(x(\tau -t),y(\tau -t)\right),max\left\{d_{\mathbf{D}}\left(x\left(0\right),y\left(0\right)\right),d_{\mathbf{D}}\left(x\left(\tau \right),y\left(\tau \right)\right):\tau \ge 0\right\}=\hfill \\ & =\underset{\tau \ge 0}{max}{d}_{\mathbf{D}}\left(x\left(\tau \right),y\left(\tau \right)\right)=d_{\mathcal{D}}(x,y)\hfill \end{array}$$

which completes the proof. □

Remark 3.

It is well known from [11] that, given a continuously differentiable function $f\in \mathcal{X}$, we have

$$\frac{d^{+}}{dt}\parallel f\left(t\right)\parallel :=\underset{h\searrow 0}{lim\; sup}\left[\parallel f(t+h)\parallel -\parallel f\left(t\right)\parallel \right]/h=\underset{L\in \mathcal{S}\left(f\right(t\left)\right)}{sup}\mathrm{Re}〈L,f^{\prime }\left(t\right)〉$$

in terms of the family of supporting bounded linear functionals

$$\mathcal{S}\left(y\right):=\left\{L\in {\mathbf{E}}^{*}:\parallel L\parallel =1,\langle L,y\rangle =\parallel y\parallel \right\}(y\in \mathbf{E}).$$

In particular, f is non-increasing whenever $\mathrm{Re}〈L,f^{\prime }\left(t\right)〉\le 0$ for any $t\in {\mathbb{R}}_{+}$ and for any functional $L\in \mathcal{S}\left(f\right(t\left)\right)$.

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 $\mathbf{U}\subset \mathbf{E}$ with a mapping $W:\mathbf{U}\to \mathbf{E}$. A flow of W is a one-parameter family of the form $\left[{\phi }^t:t\in [0,T)\right]$ of mappings ${\phi }^t:{\mathbf{D}}_t\to \mathbf{U}$ with ${\mathbf{D}}_s\supset {\mathbf{D}}_t$ for $0\le s\le t$, such that ${\phi }^0={\mathrm{Id}}_{\mathbf{U}}$ and for any point $x\in \mathbf{U}$ the orbit $\tau \mapsto {\phi }^{\tau }\left(x\right)$ is defined on a nondegenerate interval $[0,T_x)$ satisfying the relation $d{\phi }^{\tau }\left(x\right)/d\tau =W\left({\phi }^{\tau }\left(x\right)\right)$ for $0\le \tau <T_x$. Due to the classical Picard–Lindelöf Theorem, any Lipschitzian vector field W admits a unique maximal flow [11].

Lemma 3.

Let $V:B_1\left(\mathbf{E}\right)\to \mathbf{E}$ be a Lipschitzian vector field with $V\left(0\right)=0$. Then, given any constant $\mu \ge \mathrm{Lip}\left(V\right)={sup}_{f_1,f_2\in B_1\left(\mathbf{E}\right)}\parallel f_1-f_2{\parallel }^{-1}\parallel V\left(f_1\right)-V\left(f_2\right)\parallel$, the maximal flow of the vector field $W:B_1\left(\mathbf{E}\right)\ni e\mapsto V\left(e\right)-\mu e$ is a well-defined uniformly continuous one-parameter semigroup $[{\phi }^t:t\in {\mathbb{R}}_{+}]$ consisting of contractive self maps of $B_1\left(\mathbf{E}\right)$.

Proof. 

Let $[{\phi }^t:t\in I]$ with ${\phi }^t:{\mathbf{D}}_t\to B_1\left(\mathbf{E}\right)$ be the maximal flow of W. Then, for any point $e\in B_1\left(\mathbf{E}\right)$, the orbit $t\mapsto {\phi }^t\left(e\right)$ is the solution of the initial value problem

$$\frac{d}{dt}z\left(t\right)=W\left(z\left(t\right)\right),z\left(0\right)=e,$$

(2)

It is well known that by writing $I_e=[0,T_e)$ for the domain of maximal solution of (2), we necessarily have $T_e=sup{I}_e>0$; furthermore, ${lim}_{t\to T_e}\parallel z\left(t\right)\parallel =1$ whenever $T_e<\infty$. We are going to exclude such cases due to the contraction properties of the maps ${\phi }^t$.

Let $e_1,e_2\in B_1\left(\mathbf{E}\right)$ and consider the function $f\left(t\right):={\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)$ defined on the interval $I_{e_1}\cap I_{e_2}$. Observe that, given any functional $L\in \mathcal{S}\left({\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)\right)$, we have

$$\begin{array}{cc}& \mathrm{Re}〈L,f^{\prime }\left(t\right)〉=\mathrm{Re}〈L,W\left({\phi }^t\left(e_1\right)\right)-W\left({\phi }^t\left(e_2\right)\right)〉=\hfill \\ & =\mathrm{Re}〈L,V\left({\phi }^t\left(e_1\right)\right)-V\left({\phi }^t\left(e_2\right)\right)〉-\mu \mathrm{Re}〈L,{\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)〉=\hfill \\ & =\mathrm{Re}〈L,V\left({\phi }^t\left(e_1\right)\right)-V\left({\phi }^t\left(e_2\right)\right)〉-\mu \parallel {\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)\parallel \le \hfill \\ & \le \mu \parallel {\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)\parallel -\mu \parallel {\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)\parallel =0.\hfill \end{array}$$

Hence, we conclude that the function $t\mapsto f\left(t\right)$ is decreasing; in particular, we have the contraction property $\parallel {\phi }^t\left(e_1\right)-{\phi }^t\left(e_2\right)\parallel \le \parallel {\phi }^0\left(e_1\right)-{\phi }^0\left(e_2\right)\parallel =\parallel e_1-e_2\parallel$ for $t\in I_{e_1}\cap I_{e_2}$. Assumption $W\left(0\right)=V\left(0\right)=0$ implies ${\phi }^t\left(0\right)\equiv 0$ with $I_0=[0,\infty )={\mathbb{R}}_{+}$. Hence, we also see that $\parallel {\phi }^t\left(e\right)\parallel =\parallel {\phi }^t\left(e\right)-{\phi }^t\left(0\right)\parallel \le \parallel e-0\parallel =\parallel e\parallel <1$ for all $e\in B_1\left(\mathbf{E}\right)$ and $t\in I_e$. This is possible only if $sup{I}_e=\infty$. Therefore, the maximal flow of W is defined for all (time) parameters $t\in {\mathbb{R}}_{+}$ and consists of $B_1\left(\mathbf{E}\right)$-contractions ${\phi }^t$.

It is well known that flows parametrized on ${\mathbb{R}}_{+}$ are automatically strongly continuous semigroups. The uniform continuity in our case is a consequence of the fact that $\parallel {\phi }^{t_2}\left(e\right)-{\phi }^{t_1}\left(e\right)\parallel \le {\int }_{t_1}^{t_2}\parallel \frac{d}{dt}{\phi }^t\left(e\right)\parallel dt={\int }_{t_1}^{t_2}\parallel W\left({\phi }^t\left(e\right)\right)\parallel dt\le {\int }_{t_1}^{t_2}4\mu dt$ $(0\le t_1\le t_2)$, which shows that $\omega (e,\delta )\le 4\mu \delta$ $(e\in B_1\left(\mathbf{E}\right),\delta \in {\mathbb{R}}_{+})$. □

Example 1.

Let $\mathbf{E}:=\mathbb{C}$ with $B_1\left(\mathbf{E}\right)=\Delta =\{\zeta \in \mathbb{C}:|\zeta |<1\}$ and let $V\left(z\right)\equiv z^2$. Since $|z_1^2-z_2^2|=|z_1-z_2|·|z_1+z_2|\le 2|z_1-z_2|$, we can apply Lemma 3 with $W\left(z\right):=z^2-2z$. For the flow $[{\phi }^t:t\in {\mathbb{R}}_{+}]$ of W, we obtain the holomorphic maps

$${\phi }^t\left(z\right)=\frac{2z}{\left(1-e^{2t}\right)z+2e^{2t}}(z\in \Delta ,t\ge 0).$$

Indeed, the solution of the initial value problem

$$\frac{d}{dt}x\left(t\right)=x{\left(t\right)}^2-2x\left(t\right),x\left(0\right)=z$$

(3)

is $x\left(t\right)=2z/\left[\left(1-e^{2t}\right)z+2e^{2t}\right]$, as one can check by direct computation. As for heuristics, we obtain a real-valued solution with real calculus for (3) with initial values $-1<z<1$, and the obtained formula extends holomorphically to Δ.

Theorem 2.

Given a complex Banach space $\mathbf{E}$ with symmetric or separable unit ball, there is a $C_0$-semigroup of non-linear holomorphic 0-preserving norm and Carathéodory isometries of the open unit ball of the function space $\mathcal{X}:=C_0({\mathbb{R}}_{+},\mathbf{E})$.

Proof. 

We can apply the construction of Proposition 1 with a semigroup $[{\phi }^t:t\in {\mathbb{R}}_{+}]$ obtained with the construction of Lemma 3 with any $\mathbf{E}$-polynomial polynomial vector field $V:\mathbf{E}\to \mathbf{E}$. □

Example 2.

Let $\mathbf{E}:=\mathbb{C}$ and $\mathbf{X}:=C_0({\mathbb{R}}_{+},\mathbb{C})$. Then, the maps

$${\Phi }^t\left(x\right):{\mathbb{R}}_{+}\ni \tau \mapsto \left[\frac{2x\left(0\right)}{\left(1-e^{2(t-\tau )}\right)x\left(0\right)+2e^{2(t-\tau )}}if\tau \le t,x(\tau -t)if\tau \ge t\right]$$

form a $C_0$-semigroup of non-linear holomorphic 0-preserving norms and Carathéodory isometries of the unit ball $B_1\left(\mathbf{X}\right)$.

Question 1.

Is any holomorphic norm-isometry of the unit ball of a complex Banach space automatically a Carathéodory isometry as well?