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 $\mathcal{G}_0$ is the real Banach subspace of $\overline{\text{Hol}}(\textbf{D},X)$ consisting of transvections and $\mathcal{G}_+$ is the subcone of $\mathcal{G}$ such that for each $h\in\mathcal{G}_+$,

\begin{equation}\inf_{x'\in J\langle x\rangle}\text{Re}\langle h(x),x'\rangle\geq0,\:\text{for}\:\text{all}\:x\in\textbf{D}.\end{equation}

In other words, $f\in\mathcal{G}$ admits a unique representation

\begin{equation}f=g+h,\end{equation}

where $g=f(0)-P_{f\langle0\rangle}(x)$ is complete, $h\in\mathcal{G}_+$ and $h(0)=0$.

The natural examples of $\text{JB}^*$ triple systems are a complex Hilbert space $H$, the space $L(H)$ of bounded linear operators on $H$, and its subspaces $J$ such that $A\in J$ if and only if $AA^*A\in J$ (such subspaces are usually called $J^*$-algebras). In the latter case the general form of transvections on $\textbf{D}$ is $g(x)=a=xa^*x$, where $a\in J$ and $a^*$ is its conjugate. Thus, each semi-complete vector field on the open unit ball of a $J^*$-algebra has the form

\begin{equation}f(x)=f(0)-xf^*(0)x=h(x),\end{equation}

where $h\in\mathcal{G}^+$ and $h(0)=0$.

In particular, when $X=\textbf{C}$ is the complex plane and $\textbf{D}=\Delta$, the open unit disc in $\textbf{C}$, (33) becomes

\begin{equation}f(z)=f(0)-\overline{f(0)}z^2+zp(z),\end{equation}

where $p(z)\in\text{Hol}(\Delta,\textbf{C})$ and

\begin{equation}\text{Re}\:p(z)\geq0,z\in\Delta.\end{equation}

In 1978 E. Berkson and H. Porta [[#References|[a4]]], solving an entirely different problem, gave a parametric representation of generators on the unit disc $\Delta$ in the complex plane. More precisely, $g\in\mathcal{G}$ if and only if for some $\tau\in\overline{\Delta}$, $g$ has the representation

\begin{equation}g(z)=(z-\tau)(1-z\overline{\tau})p(z)$\end{equation}

with $\text{Re}\:p(z)\geq0$ everywhere. This point $\tau\in\overline{\Delta}$ is exactly the limit point of the semi-group generated by $g$ (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 $H$ be a complex Hilbert space with inner product $\langle.,.\rangle$ and let $\textbf{B}$ be its open unit ball. Let $\rho$ denote the Poincaré [[Hyperbolic metric|hyperbolic metric]] on $\textbf{B}$ [[#References|[a7]]] (cf. also [[Poincaré model|Poincaré model]]). A mapping $f:\textbf{B}\to H$ is said to be $\rho$-monotone if for each pair $x,y\in\textbf{B}$ and positive $r$ the following condition holds: $\rho(x+rf(x),r+rf(y))\geq\rho(x,y)$ whenever $x+rf(x)$ and $y+rf(y)$ belong to $\textbf{B}$. It was shown in [[#References|[a13]]] that if $H$ is separable and $f:\textbf{B}\to H$ is a bounded continuous mapping, then $f$ is $\rho$-monotone if and only if it generates a semi-group of $\rho$-non-expansive self-mappings of $\textbf{B}$. Note also that $\rho$-monotonicity can be equivalently described as follows:

\begin{equation}\text{Re}\bigg(\frac{\langle x,f(x)\rangle}{(1-||x||^2)}+\frac{\langle y,f(y)\rangle}{(1-||y||^2)}\bigg)\geq\\\geq\text{Re}\bigg(\frac{\langle f(x),y\rangle+\langle x,f(y)\rangle}{1-\langle x,y\rangle}\bigg),x,y\in\textbf{B}\end{equation}

For a bounded holomorphic mapping $f:\textbf{B}\to H$ and for an arbitrary $H$ the latter condition is a criterion for $f$ to be semi-complete. For the one-dimensional case, if $f(y)=0$, then this condition becomes the Berkson–Porta representation of semi-complete vector fields.

====References====
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