View source for Semi-group of holomorphic mappings

Non-linear semi-group theory is not only of intrinsic interest, but is also important in the study of evolution problems (cf. also [[Evolution equation|Evolution equation]]). In recent years (as of 2000) many developments have occurred, in particular in the area of non-expansive semi-groups in Banach spaces. As a rule, such semi-groups are generated by accretive operators (cf. also [[Accretive mapping|Accretive mapping]]) and can be viewed as non-linear analogues of the classical linear contraction semi-groups (cf. also [[Contraction semi-group|Contraction semi-group]]). Another class of non-linear semi-groups consists of the semi-groups generated by holomorphic mappings. Such semi-groups appear in several diverse fields, including, for example, the theory of Markov stochastic branching processes, Krein spaces, the geometry of complex Banach spaces, control theory, and optimization. These semi-groups can be considered natural non-linear analogues of semi-groups generated by bounded linear operators (cf. also [[Semi-group of operators|Semi-group of operators]]). These two distinct classes of non-linear semi-groups are related by the fact that holomorphic self-mappings are non-expansive with respect to Schwarz–Pick pseudo-metrics (see, for example, [[#References|[a8]]], [[#References|[a7]]], [[#References|[a6]]]).

Recall that a function $h$ defined on a domain (open connected subset) $\textbf{D}$ in a complex [[Banach space|Banach space]] $X$ and with values in a complex Banach space is said to be holomorphic in $\textbf{D}$ if for each $x\in\textbf{D}$ the [[Fréchet derivative|Fréchet derivative]] of $h$ at $x$ (denoted by $Dh(x)$ or $h'(x)$) exists as a bounded complex-linear mapping of $X$ into the Banach space containing the values of $h$. (Cf. also [[Banach space of analytic functions with infinite-dimensional domains|Banach space of analytic functions with infinite-dimensional domains]].)

If $\textbf{D}$ and $\Omega$ are domains in complex Banach spaces $X$ and $Y$, respectively, then the set of holomorphic mappings from $\textbf{D}$ into $\Omega$ is denoted by $\text{Hol}(\textbf{D},\Omega)$. The notation $\text{Hol}(\textbf{D})$ is used to denote the set $\text{Hol}(\textbf{D},\textbf{D})$ of holomorphic self-mappings of $\textbf{D}$.

A family $\{S(t)\subset\text{Hol}(\textbf{D}\}$, where $t\in(0,T)$, $T>0$, is called a (one-parameter) continuous semi-group if

\begin{equation}S(s+t)=S(s)\circ S(t),s,t,s+t\in(0,T),\end{equation}

and

\begin{equation}\lim_{t\to 0^{+}}S(t)(x)=x,x\in D,\end{equation}

where the limit is taken with respect to the [[Strong topology|strong topology]] of $X$.

==Differentiability of semi-groups with respect to the parameter.==
Let $\{S(t)\}$, $t\in(0,T)$, be a continuous semi-group defined on $\textbf{D}$. If the strong limit

\begin{equation}g(x)=\lim_{t\to 0^{+}}\frac{1}{t}(x-S(t)(x))\end{equation}

exists for each $x\in \textbf{D}$, then $g\in\text{Hol}(\textbf{D},X)$ is called the (infinitesimal) generator of the semi-group $\{S(t)\}$. In this case the semi-group $\{S(t)\}$, $t\in(0,T)$, is said to be differentiable (or generated).

For the finite-dimensional case, M. Abate proved in [[#References|[a2]]] that each continuous semi-group of holomorphic mappings is everywhere differentiable with respect to its parameter, i.e., it is generated by a holomorphic mapping. In addition, he established a criterion for a holomorphic mapping to be a generator of a one-parameter semi-group. Earlier, for the one-dimensional case, similar facts were presented by E. Berkson and H. Porta in their study [[#References|[a4]]] of linear $C_0$-semi-groups of composition operators on Hardy spaces. E. Vesentini investigated semi-groups of fractional-linear transformations that are isometries with respect to the infinitesimal hyperbolic metric on the unit ball of a Banach space [[#References|[a18]]]. He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Note that, generally speaking, such semi-groups are not everywhere differentiable in the infinite-dimensional case.

In fact, it can be shown (see for example, [[#References|[a14]]]), that a continuous semi-group $S(t)$ of holomorphic self-mappings of a domain $\textbf{D}$ in $X$ is generated if and only if the convergence in (2) is locally uniform on $\textbf{D}$ (cf. also [[Uniform convergence|Uniform convergence]]).

Moreover, if $\textbf{D}$ is hyperbolic (in particular, bounded; cf. also [[Hyperbolic metric|Hyperbolic metric]]), then $S(t)$ can be continuously extended to all of $\textbf{R}^+=[0,\infty)$ as the solution of the [[Cauchy problem|Cauchy problem]]

\begin{equation}\begin{cases}\frac{\partial u(t,x)}{\partial t}+g(u(t,x))=0,\\u(0,x)=x,\end{cases}\end{equation}

where $x\in \textbf{D}$ and $t\in \textbf{R}^+$, i.e.,

\begin{equation}u(t,x)=S(t)x,x\in \textbf{D},t\in[0,\infty)\end{equation}

(see, for example, [[#References|[a12]]], [[#References|[a13]]]). Thus, there is a one-to-one correspondence between locally uniformly continuous semi-groups and their generators. If $S(t)$ has a continuous extension to all of $\textbf{R}=(-\infty,\infty)$, then it is actually a one-parameter group of automorphisms of $\textbf{D}$.

==Exponential and product formulas.==
A holomorphic [[Vector field|vector field]]

\begin{equation}T_g=g(x)\frac{\partial}{\partial x}\end{equation}

on a domain $\textbf{D}$ is determined by a holomorphic mapping $g\in\text{Hol}(\textbf{D},X)$ and can be regarded as a linear operator $T$ mapping $\text{Hol}(\textbf{D},X)$ into itself, where $T_gf\in\text{Hol}(\textbf{D},X)$ is defined by

\begin{equation}(T_gf)(x)=Df(x)g(x),x\in\textbf{D}\end{equation}

The set of all holomorphic vector fields on $\textbf{D}$ is a [[Lie algebra|Lie algebra]] under the commutator bracket

\begin{equation}[T_g,T_h]=\bigg[g(x)\frac{\partial}{\partial x}, h(x)\frac{\partial}{\partial x}\bigg]:=\\=(Dg(x)h(x)-Dh(x)g(x))\frac{\partial}{\partial x}\end{equation}

(see, for example, [[#References|[a9]]], [[#References|[a15]]], [[#References|[a6]]]).

Furthermore, each vector field (6) is locally integrable in the following sense: for each $x\in\textbf{D}$ there exist a neighbourhood $\Omega$ of $x$ and a $\delta>0$ such that the Cauchy problem (4) has a unique solution $\{u(t,x)\}\subset\textbf{D}$ defined on the set $\{|t|<\delta\}\times\Omega\subset\textbf{R}\times\textbf{D}$.

A holomorphic vector field $T_g$ defined by (6) and (7) is said to be (right) semi-complete (respectively, complete) on $\textbf{D}$ if the solution of the Cauchy problem (4) is well-defined on all of $\textbf{R}^+\times\textbf{D}$ (respectively, $\textbf{R}\times\textbf{D}$), where $\textbf{R}^+=[0,\infty)$ (respectively, $\textbf{R}=(-\infty,\infty)$).

Thus, if $\textbf{D}$ is hyperbolic, then $T_g$ is semi-complete (respectively, complete) if and only if $g$ is the generator of a one-parameter continuous semi-group (respectively, group).

On the other hand, if $\textbf{D}$ is bounded and $\widetilde{\text{Hol}}(\textbf{D},X)$ is the subspace of $\text{Hol}(\textbf{D},X)$ consisting of all $f\in\text{Hol}(\textbf{D},X)$ that are bounded on each ball strictly inside $\textbf{D}$, then a semi-group (group) $\{S(t)\}$, $t\in \textbf{R}^+$ (respectively, $t\in\textbf{R}$), induces a linear semi-group (group) $\{\mathcal{L}(t)\}$ of linear mappings $\mathcal{L}(t):\widetilde{\text{Hol}}(\textbf{D},X)\to\widetilde{\text{Hol}}(\textbf{D},X)$, defined by

\begin{equation}(\mathcal{L}(t)f)(x):=f(S(t)x),\end{equation}

where $t\in\textbf{R}^+$ ($t\in\textbf{R}$) and $x\in\textbf{D}$.

This semi-group is called the semi-group of composition operators on $\widetilde{\text{Hol}}(\textbf{D},X)$. If $\{S(t)\}$, $t\in\textbf{R}^+$ ($t\in\textbf{R}$), is $T$-continuous, (that is, differentiable), then $g\in\widetilde{\text{Hol}}(\textbf{D},X)$, $\{\mathcal{L}(t)\}$, $t\in\textbf{R}^+$ ($t\in\textbf{R}$), is also differentiable and

\begin{equation}\begin{cases}\frac{\partial\mathcal{L}(t)f}{\partial t}+T_g(\mathcal{L}(t)f)=0,\\\mathcal{L}(0)f=f,\end{cases}\end{equation}

for all $f\in\widetilde{\text{Hol}}(\textbf{D},X)$, where $g=-dS(t)/dt|_{t=0}$.

In other words, a holomorphic vector field $T_g$, defined by (6) and (7), and considered as a linear operator on $\widetilde{\text{Hol}}(\textbf{D},X)$, is the infinitesimal generator of the semi-group $\{\mathcal{L}(t)\}$. It is sometimes called the Lie generator. Thus, a holomorphic vector-field $T_g$ is semi-complete (respectively, complete) if and only if it is the Lie generator of a linear semi-group (respectively, group) of composition operators on $\widetilde{\text{Hol}}(\textbf{D},X)$. This follows from the observation that

\begin{equation}\mathcal{L}(t)I_{\textbf{D}}=S(t)\end{equation}

and

\begin{equation}T_gI_{\textbf{D}}=g,\end{equation}

where $I_D$ is the restriction of the identity operator to $\textbf{D}$.

Moreover, using the exponential formula representation for the linear semi-group,

\begin{equation}\mathcal{L}(t)f=\sum^{\infty}_{k=0}\frac{(-1)^kt^k}{k!}T^k_gf=\text{exp}[-tT_g]f\end{equation}

(see, for example, [[#References|[a19]]], [[#References|[a9]]], [[#References|[a12]]]), one also has

\begin{equation}S(t)=\sum^{\infty}_{k=0}\frac{(-1)^kt^k}{k!}T^k_gI_{\textbf{D}}=\text{exp}[-tT_g]I_{\textbf{D}}.\end{equation}

So, a locally uniformly continuous semi-group of holomorphic self-mappings can be represented in exponential form by the holomorphic vector field induced by its generator.

Another exponential representation on a hyperbolic convex domain can be given by using the so-called non-linear resolvent of $g$.

More precisely, let $\textbf{D}$ be a bounded (or, more generally, hyperbolic) convex domain. Then it was shown in [[#References|[a12]]] and [[#References|[a13]]] that $g\in\widetilde{\text{Hol}}(\textbf{D},X)$ is a generator if and only if for each $r\geq0$ the mapping $(I+rg)^{-1}$ is a well-defined holomorphic self-mapping of $\textbf{D}$.

Furthermore, if $\{G(r)\}$, $r\geq0$, is any continuous family of holomorphic self-mappings of $\textbf{D}$ such that the limit

\begin{equation}g(x)=\lim_{r\to0^{+}}\frac{1}{r}(x-G(r)x)\end{equation}

exists, then $g$ is a generator and the semi-group generated by $g$ can be defined by the product formula

\begin{equation}S(t)=\lim_{n\to\infty}G^n\bigg(\frac{t}{n}\bigg).\end{equation}

In particular,

\begin{equation}S(t)=\lim_{n\to\infty}\bigg(I+\frac{t}{n^g}\bigg)^{-n}\end{equation}

(exponential formula), where the limits in (17) and (18) are taken with respect to the locally uniform topology on $\text{Hol}(\textbf{D},X)$.

==Flow-invariance conditions.==
Let $\textbf{D}$ be a convex subset of a Banach space $X$ and let $g:\overline{\textbf{D}}\to X$ be a continuous mapping on $\overline{\textbf{D}}$, the closure of $D$. Then the following tangency condition of flow invariance

\begin{equation}\lim_{h\to0^{+}}\text{dist}\frac{(x-hg(x),\overline{\textbf{D}})}{h}=0,x\in\overline{\textbf{D}},\end{equation}

is a necessary condition for the solvability of the evolution equation (4). A result of R.H. Martin [[#References|[a11]]] shows that if $g:\overline{\textbf{D}}\to X$ is a continuous [[Accretive mapping|accretive mapping]] on $\overline{\textbf{D}}$, then (18) is also sufficient for the existence of solutions to the Cauchy problems (4). These solutions yield a continuous semi-group of contraction mappings on $\overline{\textbf{D}}$.

For the class of holomorphic mappings, an analogue of Martin's theorem was given in [[#References|[a3]]]; namely, if $g\in\text{Hol}(\textbf{D},X)$ has a uniformly continuous extension to $\overline{\textbf{D}}$, then it is a semi-complete vector field if and only if it satisfies the boundary flow invariance condition (18).

However, there are many examples of semi-complete vector fields that have no continuous extension to $\overline{\textbf{D}}$. In particular, if $F\in\text{Hol}(\textbf{D})$, then $g=I-F$ is semi-complete (see [[#References|[a12]]]).

For absolutely convex domains, interior flow invariance conditions can be given in terms of their support functionals.

Let $X'$ be the dual of $X$ (cf. also [[Duality|Duality]]; [[Adjoint space|Adjoint space]]). For $x\in X$ and $x'\in X;$, the pairing $(x,x')$ will denote $x'(x)$. The duality mapping $J:X\to2^{X'}$ is defined by

\begin{equation}J(x):=\{x'\in X': \text{Re}\langle x,x'\rangle=||x||^2=||x'||^2\}\end{equation}

for each $x\in X$.

If $\textbf{D}$ is the open unit ball in $X$ and $g$ maps $\overline{\textbf{D}}$ into $X$, then (18) is equivalent to the condition

\begin{equation}\inf_{x'\in J(x)}\text{Re}\langle g(x),x'\rangle\geq0,x\in\partial\textbf{D}\end{equation}

For the Euclidean ball $\textbf{D}$ in $X=\textbf{C}^n$, a certain condition in this direction was established by Abate [[#References|[a2]]]. Namely, he proved that $g\in\text{Hol}(\textbf{D},\textbf{C}^n)$ is a semi-complete vector field if and only if it satisfies the estimate

\begin{equation}2\big[||r(x)||^2-|\langle r(x),x\rangle|^2\big]\text{Re}\langle r(x),x\rangle+\\+(1-||x||^2)^2\text{Re}\langle g'(x)g(x),r(x)\rangle\geq0,\end{equation}

where

\begin{equation}r(x)=(1-||x||^2)g(x)+\langle g(x),x\rangle x.\end{equation}

For $n=1$ this condition becomes

\begin{equation}\text{Re}\:g(z)\overline{z}\geq-\frac{1}{2}\text{Re}\:g'(z)(1-|z|^2),\end{equation}

where $z\in\Delta$, the open unit disc in the complex plane $\textbf{C}$, and $g\in\text{Hol}(\Delta,\textbf{C})$.

Despite the usefulness and simplicity of condition (23) it is not clear how (21) can be derived from (23) when $g$ has a continuous extension to $\overline{\Delta}$.

Note also that in the one-dimensional case it follows from the [[Maximum principle|maximum principle]] for harmonic functions that (21) implies the following interior condition:

\begin{equation}\text{Re}\:g(z)\overline{z}\geq\text{Re}\:g(0)\overline{z}(1-|z|^2),z\in\Delta.\end{equation}

Conversely, it is clear that (21) does result from (24) if $g$ has a continuous extension to all of $\overline{\Delta}$. It turns out that an analogue of (24) is a necessary and sufficient condition for $g$ to be semi-complete [[#References|[a1]]]: Let $\textbf{D}$ be the open unit ball in a complex Banach space $X$. Then $g\in\text{Hol}(\textbf{D},X)$ is a semi-complete vector field on $\textbf{D}$ if and only if it is bounded on each subset strictly inside $\textbf{D}$ and one of the following conditions holds:

a) For each $x\in\textbf{D}$ there exists an $x'\in J(x)$ such that

\begin{equation}\text{Re}\big{\langle} g(x)-g(0)(1-||x||^2),x'\big{\rangle}\geq0;\end{equation}

b) $\inf_{x'\in J\langle x\rangle}\text{Re}\langle2||x||^2g(x)+(1-||x||^2)g'(x)x,x'\rangle\geq0$, $x\in\textbf{D}$;

c) For each $x\in\textbf{D}$ and for each $x'\in J(x)$,

\begin{equation}\text{Re}\bigg{\langle}\frac{1-||x||}{1+||x||}g'(0)x+(1-||x||^2)g(0),x'\bigg{\rangle}\leq\\\leq\text{Re}\langle g(x),x'\rangle\leq\\\leq\text{Re}\bigg{\langle}\frac{1+||x||}{1-||x||}g'(0)x+(1-||x||^2)g(0),x'\bigg{\rangle}.\end{equation}

Furthermore, equality in one of the conditions a), b) or c) holds if and only if it holds in the other conditions and $f$ is complete.

Also, if $g\in\text{Hol}(\textbf{D},X)$ is a semi-complete vector field, then $g$ is, in fact, complete if and only if its derivative at zero, $g'(0)$ is a conservative linear operator, i.e.,

\begin{equation}\text{Re}\langle g'(0)x,x'\rangle=0\end{equation}

for all $x\in X$ and $x'\in J(x)$ (see [[#References|[a10]]], [[#References|[a17]]]).

==Parametric representations of generators.==
It is well-known that a complete vector field $g$ on the open unit ball $\textbf{D}$ in a Banach space $X$ is a polynomial of degree at most $2$ (see, for example, [[#References|[a15]]], [[#References|[a6]]]). More precisely, $g$ has the form

\begin{equation}g(x)=\alpha+Ax+P_{\alpha}(x),\end{equation}

where $\alpha$ is an element of $X$, $A$ is a conservative operator on $X$ and $P_{\alpha}$ is a homogeneous form of the second degree such that $P_{i\alpha}=iP_{\alpha}$.

Suppose now that a complex Banach space $X$ is a so-called $\text{JB^{*}}$ triple system. This is equivalent to saying that its open unit ball $\textbf{D}$ is a homogeneous domain, i.e., for each pair $x,y\in\textbf{D}$ there exists a holomorphic automorphism $F$ of $\textbf{D}$ such that $F(x)=y$ (see, for example, [[#References|[a15]]], [[#References|[a6]]]). Then it is well-known that for each $\alpha\in X$ there exists a homogeneous polynomial $P_{\alpha}(x)$ such that $P_{i\alpha}=iP_{\alpha}$ and the mapping $g:\textbf{D}\to X$ defined by

\begin{equation}g(x)-\alpha-P_{\alpha}(x)\end{equation}

is a complete vector field on $\textbf{D}$, which is called a transvection of $\textbf{D}$ (cf. also [[Transvection|Transvection]]).

The cone $\mathcal{G}$ of semi-complete vector fields on $\textbf{D}$ admits the decomposition

\begin{equation}\mathcal{G}=\mathcal{G}_0\bigoplus\mathcal{G}_+,\end{equation}

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230228.png" /> is the real Banach subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230229.png" /> consisting of transvections and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230230.png" /> is the subcone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230231.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230232.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230233.png" /></td> </tr></table>

In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230234.png" /> admits a unique representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230235.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a25)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230236.png" /> is complete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230238.png" />.

The natural examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230239.png" /> triple systems are a complex Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230240.png" />, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230241.png" /> of bounded linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230242.png" />, and its subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230243.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230244.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230245.png" /> (such subspaces are usually called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230247.png" />-algebras). In the latter case the general form of transvections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230248.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230249.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230250.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230251.png" /> is its conjugate. Thus, each semi-complete vector field on the open unit ball of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230252.png" />-algebra has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230253.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a26)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230254.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230255.png" />.

In particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230256.png" /> is the complex plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230257.png" />, the open unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230258.png" />, (a26) becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230259.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a27)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230260.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230261.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a28)</td></tr></table>

In 1978 E. Berkson and H. Porta [[#References|[a4]]], solving an entirely different problem, gave a parametric representation of generators on the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230262.png" /> in the complex plane. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230263.png" /> if and only if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230264.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230265.png" /> has the representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230266.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230267.png" /> everywhere. This point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230268.png" /> is exactly the limit point of the semi-group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230269.png" /> (that is, its Denjoy–Wolff point, cf. [[Denjoy–Wolff theorem|Denjoy–Wolff theorem]]). The Berkson–Porta formula has also been successfully exploited in other fields; for example, in the classical functional equations of E. Schröder and N.H. Abel (see [[#References|[a5]]] and [[Functional equation|Functional equation]]; [[Schröder functional equation|Schröder functional equation]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230270.png" /> be a complex Hilbert space with inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230271.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230272.png" /> be its open unit ball. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230273.png" /> denote the Poincaré [[Hyperbolic metric|hyperbolic metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230274.png" /> [[#References|[a7]]] (cf. also [[Poincaré model|Poincaré model]]). A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230275.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230277.png" />-monotone if for each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230278.png" /> and positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230279.png" /> the following condition holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230280.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230281.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230282.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230283.png" />. It was shown in [[#References|[a13]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230284.png" /> is separable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230285.png" /> is a bounded continuous mapping, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230286.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230287.png" />-monotone if and only if it generates a semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230288.png" />-non-expansive self-mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230289.png" />. Note also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230290.png" />-monotonicity can be equivalently described as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230291.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230292.png" /></td> </tr></table>

For a bounded holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230293.png" /> and for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230294.png" /> the latter condition is a criterion for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230295.png" /> to be semi-complete. For the one-dimensional case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130230/s130230296.png" />, then this condition becomes the Berkson–Porta representation of semi-complete vector fields.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Aharonov,   S. Reich,   D. Shoikhet,   "Flow invariance conditions for holomorphic mappings in Banach spaces"  ''Math. Proc. Royal Irish Acad.'' , '''99A'''  (1999)  pp. 93–104</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abate,   "The infinitesimal generators of semi-groups of holomorphic maps"  ''Ann. Mat. Pura Appl.'' , '''161'''  (1992)  pp. 167–180</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Aizenberg,   S. Reich,   D. Shoikhet,   "One-sided estimates for the existence of null points of holomorphic mappings in Banach spaces"  ''J. Math. Anal. Appl.'' , '''203'''  (1996)  pp. 38–54</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Berkson,   H. Porta,   "Semi-groups of analytic functions and composition operators"  ''Michigan Math. J.'' , '''25'''  (1978)  pp. 101–115</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.C. Cowen,   B.D. MacCluer,   "Composition operators on spaces of analytic functions" , CRC  (1995)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Dineen,   "The Schwartz lemma" , Clarendon Press  (1989)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Goebel,   S. Reich,   "Uniform convexity, hyperbolic geometry and nonexpansive mappings" , M. Dekker  (1984)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.A. Harris,   "Schwarz–Pick systems of pseudometrics for domains in normed linear spaces" , ''Advances in Holomorphy'' , North-Holland  (1979)  pp. 345–406</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.M. Isidro,   L.L. Stacho,   "Holomorphic automorphism groups in Banach spaces: An elementary introduction" , North-Holland  (1984)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S.G. Krein,   "Linear differential equations in Banach spaces" , Amer. Math. Soc.  (1971)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.H. Martin Jr.,   "Differential equations on closed subsets of a Banach space"  ''Trans. Amer. Math. Soc.'' , '''179'''  (1973)  pp. 399–414</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S. Reich,   D. Shoikhet,   "Generation theory for semi-groups of holomorphic mappings in Banach spaces"  ''Abstr. Appl. Anal.'' , '''1'''  (1996)  pp. 1–44</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S. Reich,   D. Shoikhet,   "Semi-groups and generators on convex domains with the hyperbolic metric"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1997)  pp. 231–250</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  S. Reich,   D. Shoikhet,   "Metric domains, holomorphic mappings and nonlinear semi-groups"  ''Abstr. Appl. Anal.'' , '''3'''  (1998)  pp. 203–228</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  H. Upmeier,   "Jordan algebras in analysis, operator theory and quantum mechanics" , ''CBMS-NSF Reg. Conf. Ser. in Math.'' , '''67''' , Amer. Math. Soc.  (1987)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  E. Vesentini,   "semi-groups of holomorphic isometries"  ''Adv. Math.'' , '''65'''  (1987)  pp. 272–306</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  E. Vesentini,   "Krein spaces and holomorphic isometries of Cartan domains"  S. Coen (ed.) , ''Geometry and Complex Variables'' , M. Dekker  (1991)  pp. 409–413</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  E. Vesentini,   "Semi-groups of holomorphic isometries"  S. Coen (ed.) , ''Complex Potential Theory'' , Kluwer Acad. Publ.  (1994)  pp. 475–548</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1968)</TD></TR></table>