Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces

(s-t)(0.

(1.163)

48

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Let X be a Banach space with dual space X*, and let T> c X. As before, we use the pairing (x, j) to denote j(x), x G X, j G X*. Definition 1.19 A mapping T : V t-» X* is said to be monotone if for each u,v eV, Re(u - v, T(u) - T(v)) > 0.

(1.164)

It is called strongly monotone if for some c > 0, Re(u - v, T{u) - T{v)) > c\\x - y\\2.

(1.165)

A natural analogue of the above for mappings taking values in X is the following definition. Definition 1.20 A mapping T : V H-> X is said to be accretive if for all u, v G V and some j G J(u — v), Re(T(u) - T(v),j) > 0.

(1.166)

It is called strongly accretive if for some c > 0, (1.167)

Re(T(u) - T(v),j) > c\\x - yf.

Here J denotes the normalized duality mapping introduced earlier: for xGX, J(x) = {j G X* : (x,j) = \\x\\2 = \\j\\2}.

(1.168)

We note first that if X is a Hilbert space then X — X*, the classes of monotone (respectively, strongly monotone) and accretive (respectively, strongly accretive) mappings defined in X coincide, and (1.164) denotes the usual inner product. Thus, if X is specialized further to X = R, (1.164) becomes {u - v)(T(u) - T(v)) > 0. One connection between accretive mappings and nonexpansive mappings is immediate. If F : V H-> X is nonexpansive, then for T — I -F, x,yeV, and j G J(x - y), (T(x) - T(y),j) = {x-y=

(F(x) - F(y)),j)

\\x-y\\2-(F(x)-F(y),j)

> \\x - yf - \\F(x) - F(y)\\\\x - y\\ > 0. (1.169) Thus T is accretive. In addition, if F : V i-> X is a strict contraction, ||F(a;)-F(y)||
x,yeV,

0 < k < 1,

(1.170)

Mappings in Metric and Normed Spaces

49

then T = I — F is strongly accretive. On the other hand, not all accretive mappings are of the form I — F with F nonexpansive. A complete characterization of accretive mappings in metric terms has been given by Kato [Kato (1967)] (see also [Deimling (1974)]. Proposition 1.26 Let X be a Banach space, V C X, and T : V i-> X. Then T is accretive if and only if for each x,y € P and A > 0,

II* - 2/11 <\\x-y

+ \{T(x) - T(y))\\.

(1.171)

Thus a mapping T : T> i-> X is accretive if and only if the mapping J\ = (I + AT)" 1 (called the resolvent ofT) is nonexpansive on its domain for each positive A. Remark 1.2 Using the above, it is possible to extend the definition of accretivity in a natural way to the multivalued case. For a given subset B :xeB}. ofX, let \B\ = ini{\\x\\ A mapping T : V K-> 2X{V C X) is said to be accretive (respectively, strongly accretive) if there exists e > 0 (respectively, e > 0) such that for each x,y£T>, z£ T(x), w G T(y), and A > 0, (1 + e)\\x - 2/|| < \\x - y + X(z - w)\\.

(1.172)

Again, it can be shown that the mapping J\ = (I + AT)" 1 is singlevalued and nonexpansive (respectively, strict contraction) on its domain. If it is the case that the domain of J\ is all of X for some (hence all) A > 0, then T is said to be m-accretive (sometimes called hyperaccretive). The theory of accretive operators is extensive. The solvability of many equations involving partial differential operators (e.g., Laplacians) can be formulated as questions concerning accretive operators. We will not discuss the details here but instead refer the reader to [Browder (1976); Deimling (1992); Barbu (1976)], and [Martin (1973)]. Our motivation for including this notion here is simply to note the usefulness of fixed point theory for nonexpansive mappings in another context and to illustrate the dependence of each theory upon the other (see also [Kirk and Sims (2001)] and references therein). It is also not difficult to see that the resolvent J\ of an accretive mapping is firmly nonexpansive. As a matter of fact, F is firmly nonexpansive if and only if it is the resolvent (/ + A)~1 for some accretive mapping A C X x X.

Chapter 2

Differentiable and Holomorphic Mappings in Banach Spaces 2.1

Differentiable Mappings. Frechet Derivatives

The fundamental notions of abstract differentials, polynomials and power series were introduced by Maurice Prechet around 1909. The crux of the theory of functions between normed spaces is the question of differentiability. In this general situation the differentials of Prechet appear to be the most appropriate concepts. In the present chapter we develop Prechet differentials. Let X, Y be normed spaces and let U be an open set in X. As above, we denote by the same symbol || • || the norms of both X and Y. Definition 2.1 A mapping / : U H-> Y is said to be differentiable at x £ U if there exists a linear map A (= Ax) £ L(X, Y) such that / and the continuous affine linear map h £ X —> f(x) + Ah £ Y are tangent at x, i.e.,

a, !/<*+*>-/(->-*»!_„.

fc-o

M

\\h\\

If there is such a linear map Ax satisfying (2.1), then it is unique. We call A(= Ax) the Prechet derivative of / at x, and A will be denoted by T>f(x) or F'(x). The element f'(x)h £ Y is called the Prechet differential or the differential of / at x in the direction of h £ X. We note that differentiability depends only on the topologies of X and Y, and not on the particular norms used to define these topologies. If / : U i-> Y is differentiable at each point of U, then / is said to be differentiable on T>. In this case the mapping Vf : z € £/ H-> f'(x) £ L{X, Y) 51

(2.2)

52

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

is called the derivative (or differential) of / on U. If V is continuous on U, the mapping / is said to be of class C 1 on U. Note that the differential T>f is always vector-valued and not scalar-valued even when we may be interested in the differentiation of scalar-valued functions / : U i-» K (i.e., when Y = K). In fact, in this case we have Vf : U t-> L(X,K) = X*. Let X, Y be normed spaces and V a non-empty open subset of X. Then the notion of differentiability is a local property in the following sense. Proposition 2.1 If / : M H-> Y is differentiable on M, then f is differentiable on any open subset U of M, and W\u) Proposition 2.2

= (Vf)\u.

(2.3)

If M = [j Ui where each Ui is open in M, then f is i€l

differentiable on M if and only if it is differentiable on each Ui. Theorem 2.1 (Lipschitzian property) Let X and Y be normed spaces and U a nonempty open subset of X. If f : U — i > Y is differentiable at x € U, then there exist C > 0 and 6 > 0 such that Il/Cx) - / ( y ) | |
(2.4)

for y £ U, \\x — 2/|| < S. In particular, it follows that f is continuous at x. Theorem 2.2 (Chain rule) Let X,Y,Z be normed spaces, U an open i > V and g : V >—» Z. subset of X, and V and open subset of Y. Let f : U — If f is differentiable at x e U and g is differentiable at f(x) G V, then g o f : U f-> Z is differentiable at x and {9of)'{x)=g'{f{x))of'{x).

(2.5)

Corollary 2.1 Suppose F :T> >->Y is differentiable at a point XQ € V and assume that there exists a mapping G : Y — i > X defined on a neighborhood U of the point j/o = F(xo), and satisfying the relations GoF = Iv,

FoG = Iu

(2.6)

(we write in this case G = F~1). IfF~x is differentiable at yo = F(xo) G U, then the linear operator F'(xo) has a continuous inverse and [F'(xo)}-1 = (F-'Yiyo).

(2.7)

Differentiable and Holomorphic Mappings in Banach Spaces

2.1.1

53

Examples

Example 2.1 (Constant mappings) Let X and Y be normed spaces. Then a constant mapping is a mapping of the form f : x e X >-> b £Y for a fixed point b in Y. If / : X i-> Y is a constant mapping, then Vf = 0, i.e., f'(x) = 0 for all x e X. Example 2.2 (Linear mappings) If A € Lpf, Y), then PA is the constant mapping satisfying A'(x) = A for all x e X. Let / : £> t-> Y be differentiable a t i G O and let A& L(Y,Z). Then (Ao/)'(*) = A o / ' ( z ) .

(2.8)

Example 2.3 (Urysohn operator) Let K(t, s, x) be a function denned on a < t, s < b, \x\ < r, such that K(t, s, x) and K'x(t, s, x) are everywhere continuous. Let X = C[a, b\ be the space of all real- (or complex-)valued continuous functions on [a,b\ with the norm ||x|| = sup \x(t)\. Then one a
can define a mapping

/ : P H I , P = {XGI:

||:r|| < r}, by the formula

f(x(t))= [ K(t,s,x(s))ds. (2.9) Ja This mapping is usually called the nonlinear Urysohn integral operator. The operator / takes values in C[a, b] and is Frechet differentiable on the open ball ||x|| < r. Moreover, for any XQ € C[a, b] with ||:ro|| < r, and for every h 6 C[a, 6], we have

(f'(xo)h)(t)= f K'x(t,s,x0(s))h(s)ds.

(2.10)

Ja Note that the Urysohn operator (2.9) can be also considered on the space C[a, b] of all complex-valued continuous functions; then / is Frechet differentiable on C[a, b] if the kernel K(t, s, x) has continuous partial derivatives with respect to the variable x in the disk { x e C : | a ; | < r } o f the complex plane. Example 2.4 (Hammerstein operator) A particular case of the operator (2.9) is the Hammerstein operator denned by the equality

f(x(t))= [ K(t,s)f(s,x(s))ds:

(2.11)

Ja where the functions K(t, s), f(s, x) and fx(s, x) are continuous with respect to all variables for a < t, s < b, \x\ < r. Example 2.3 implies that the

54

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Frechet derivative of / is given by (f'(x)h)(t)= I K{t,s)f'x{s,x{s))h(s)ds.

(2.12)

Ja

Example 2.5 (Nemytski operator) Suppose that the functions (p(s,x) and •-> X = C[a, b] denned by (2.13)

*(x)(8):=
for each x in the ball D = {x G X : \\x\\ < r} of the space C[a, b]. The operator (2.13) is differentiable and (2.14)

{*'(xo)h)(s) =
Example 2.6 Let fi(x\..., xn),..., fm(xi, • • •,^n) be functions defined and on a domain in the Euclidean space K n = {x : x = (xi,...,xn)}, assume that fj, j = 1,..., n, and all their first order partial derivatives are continuous. Then the operator (2.15)

/=(/i,-..,/m):K"~Km is Frechet differentiable on the considered domain, and dxi

:

f'(x)h= \

dx\

•••

dxn

\

I

: dxn /

1

: N

\

.

(2.16)

"'

On the right hand side of (2.16) we have the action of the Jacobi matrix of / on the vector h = (hi,..., hn) S K n . 2.2

Holomorphic Mappings

Definition 2.2 A complex Banach space valued function h defined on a domain (open, connected subset) T> in a complex Banach space X is said to be holomorphic in V if for each x e V, the Frechet derivative of h at x (denoted by Vh(x) or h'(x)) exists as a bounded complex-linear mapping of X into the Banach space Y containing the values of h. If V and Q are domains in complex Banach spaces X and Y, respectively, then the set of holomorphic mappings from V into 0, is denoted by

55

Differentiable and Holomorphic Mappings in Banach Spaces

Hol(D,Sl). The notation Hol(V) is used to denote the set Hol(V,T>) of holomorphic self-mappings of V. Note that Hol(V) is a semigroup with respect to the composition operation. One of the most important properties of holomorphic mappings is that they have the Cauchy integral representation, which implies many special features of holomorphic mappings such as the maximum modulus principle, power series representation, the Schwarz Lemma and others. Here we will see that a substantial portion of basic complex function theory can be extended to normed spaces in the light of the Cauchy integral formula. 2.2.1

The Cauchy integral formula

The classical Cauchy integral formula represents a holomorphic function defined on a domain Q in the complex plane C and continuous on the closure of fi by its values on the boundary d£l of Q. Here is a simple version of this formula. Theorem 2.3 Let Cl be a simply connected domain in C with a smooth boundary dCl, and let f e Hol(fl,C) be continuous on the closure of Q. Then for each z € fl, K

2m Jm

w- z

v

'

Let now T> be a domain in a complex Banach space X. Since T> is open, for each x G V and y e X, there is r > 0 such that x + £y € T> for C 6 C, |C| < r. Theorem 2.4 Let X and Y be complex Banach spaces, and let D be a domain in X. Suppose that f € Hol(V, Y), x e T>, y € X and r > 0 are such that x + Cy € P /or C 6 C, |C| < r. Then -"" J\z\=r

C

(2.18)

For the finite-dimensional case, where V is a domain in C n , there are many different integral representations of holomorphic mappings on V depending on geometrical properties of V. For instance, if T> is the open unit ball in the Euclidian metric, the well-known representation of MartinellyBochner is a generalization of the Cauchy formula. Other representations, e.g., for the unit polydisk, the hypercone, convex and pseudoconvex domains, can be found in [Aizenberg and Yuzhakov (1983)]. In the framework

56

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

of functional analysis, however, most of the arguments may be presented by using a reduction to the one-dimensional case. This can be achieved by applying the following assertion which is based on the famous Dunford theorem (see, for example, [Khatskevich and Shoikhet (1994a)]. Theorem 2.5 Let X and Y be complex Banach spaces, let V be a domain in X, and let f be a Y-valued mapping on T>. Then the following are equivalent: (i) The mapping f G Hol(V,Y); (ii) for each x G V, y G X and r > 0 such that x + £y e P for ( s A r = {C G C : |C| < r}, the mapping g : Ar t-> Y defined by g(C)= f(x + £y) is holomorphic in A r ; (in) for each bounded linear functional £ G Y*, the mapping h = to f : T> <—> C, is holomorphic in T>. 2.2.2

Power series representation

As in classical complex analysis, the power series representation, Taylor's formula, plays a crucial role in the development of the theory. As a matter of fact, the theory of power series in normed spaces is a very rich subject developed in the middle of the 20th century by E. Hille and R. Phillips, L. Nachbin, and others. Further contributions were also made by J. A. Barroso, J. Bochnak, L. A. Harris and S. Dineen (see details in [Franzoni and Vesentini (1980)] and [Chae (1985)]). We do not intend to elaborate here on this nice theory; we just present some definitions and notations which are necessary for our further discussion. Theorem 2.6 (Power series representation) Let X and Y be complex Banach spaces, let T> be a domain in X and let f G Hol(X>, Y) be bounded. Then for each XQ G V and for each ball B CC V centered at XQ, the following representation holds: oo

f(x) = Y,pW(x0)(x-x0),

xeB,

(2.19)

k=0

where P^(xo),

k — 0 , 1 , 2 , . . . , are homogeneous polynomials of order k

andPM(xo) = f(xo). In addition, the convergence in (2.19) is uniform.

Differentiable and Holomorphic Mappings in Banach. Spaces

It is clear that P^fao) i.e.,f(xo)=P^(xo).

57

is the Frechet derivative of / at the point XQ,

Proposition 2.3 (The Cauchy inequalities) In the setting of the above theorem, let r be the radius of the ball B, and let ||/(a;)|| < M for all x E B. Then for each k — 0,1,2,... the following inequality holds: \\Pw(x0)\\<Mr-k.

(2.20)

Theorem 2.7 (The uniqueness theorem) Let f G Hol(P,y). Then the following assertions are equivalent: (i) PW(x0) = 0, k = 0,1,2,... for some x0 € V; (ii) f is identically zero in some neighborhood U C D . (Hi) f is identically zero in T>.

2.2.3

The maximum modulus theorem

One consequence of the Cauchy integral formula is the so-called maximum modulus principle. For a connected open set U in the complex plane, if / : U <-> C is holomorphic on U, then the classical maximum modulus theorem states that • if \f(z)\ has a maximum at a point of U, then \f{z)\ is constant on U. Furthermore, if\f(z)\ is a constant function, it follows that f(z) is constant too. This theorem has a generalization for Banach space valued functions, but, unlike complex-valued functions, we cannot infer that if ||/(z)|| is constant, then f{z) is itself constant (see an example below). The form of the maximum modulus theorem which claims that if ||/(z)|| has a maximum at a point in U, then f{z) is a constant on U is referred to as the strong maximum modulus principle. Now we generalize the weak maximum modulus theorem for mappings denned on Banach spaces. Theorem 2.8 Let U be a connected open subset ofC and f e H(U,Y). If ||/(2)|| has a maximum at a point in U, then ||/(z)|| is a constant on U. Proof. We use linear functional in this proof. Suppose that the theorem were false, and for some x € U, ||/(z)|| < ||/(a;)|| for all z £ U. We may assume that ||/(a;)|| 7^ 0, since otherwise the theorem holds trivially. There

58

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

exists y* e Y* such that ||y*|| = 1 and y*(f{x)) = \\f(x)\\. Since / is holomorphic, y*(f) : U i-» C is holomorphic and for all z £ U we have \y*(f(z))\ < ||/(2)|| < ||/(x)|| = |y*(/(z))|.

(2.21)

Now the classical maximum modulus theorem implies that y*(f) is a constant map on U, and hence y*(f(z)) = ||/(z)|| for all z e U. Since \\y*\\ = 1 and ||/(z)|| is not a constant map, there exists y 6 [/ such that

II/(W)II < ll/(*)l|. Thus ll/(*)ll=y^/fo))
(2-22) •

Corollary 2.2 Let U be a connected open subset of X and f £ Hol(£/, Y). If\\f(x)\\ ^ a s a maximum at a point in U, then ||/(:c)|| is a constant function on U. Proof.

Suppose that there exists a point £ in U such that

| | / ( 0 | | = sup{H/OOII : x G U } .

(2.23)

Since U is open, we can find r such that Br(£) C U. For x in Br(£), x ^ £, consider the holomorphic mapping g(z)

:=f

{(1-z)t

+ zx]

(2.24)

defined on an open set V which contains the unit disk of C. The mapping g takes its maximum modulus at 0; thus \\g\\ must be constant on V. In particular, \\g(0)\\ = \\g(\)\\ and hence ||/(x)|| = ||/(£)|| holds for all x in BriQ- We now consider the following set to complete the proof:

A={xeU:\\f{x)\\

= ||/(0ll}.

(2.25)

Then A is closed in U. If a G A, then, replacing in the above argument £ with a, we see that a is an interior point of A. This proves that A is both closed and open in the connected set U\ thus A = U. • We now present the strong maximum modulus theorem of Thorp and Whitley (1967). To do this, we recall the following definitions. A point x in a convex subset U of a vector space X is said to be a (real) extreme point of U if for each pair of points X\ and X2 in U such that x = axi + @X2, it follows that x\ = X2 = x whenever 0 < a,f3 and

a+ 0=1.

Differentiable and Holomorpkic Mappings in Banach Spaces

59

A point a; in a convex subset U of a complex vector space X is said to be a complex extreme point of U if there is no y ^ 0 in X such that the set {x + Xy : |A| < 1} is contained in U. Thus, if U is a convex subset of a complex vector space X, then each real extreme point U is also a complex extreme point. In particular, • Each point of the unit sphere of a complex strictly convex Banach space is a complex extreme point of its closed unit ball. Theorem 2.9 (The strong maximum modulus principle) Let U be a connected open subset ofC. Then every mapping f 6 Hol(f7, V) satisfies the strong maximum modulus principle if and only if every point in the unit sphere ofY is a complex extreme point of the closed unit ball. A proof of this theorem can be based on a lemma of L. A. Harris (see Subsection 2.6.5, Corollaries 2.6 and 2.7). Corollary 2.3 Let U be a connected open subset of a complex Banach space X and let f € Hol(£7, Y), where Y is a complex strictly convex Banach space. Then f satisfies the strong maximum modulus principle. Example 2.7 If / is a complex-valued function on U such that \f(z)\ is constant for all z 6 U, then / is a constant function on U. Therefore, for complex-valued functions, the strong maximum modulus theorem holds. However, this is not the case in general for vector-valued functions, as the following example shows. Let U be the open unit disk in the complex plane, and let V = C2 be normed by |M|=max{|xa|,|x 2 |}.

(2.26)

If / : U i-> y is defined by /(*) = (!,*),

(2-27)

then / is obviously a non-constant holomorphic mapping and ||/(«)|| = max{l>|z|} = l

(2.28)

for all z € U. This shows that the strong maximum modulus theorem fails to hold for the Banach space C2. Example 2.8 Let CQ be the Banach space of all complex sequences x = (xi,X2 • • •) such that xn —» 0 as n —> oo, with the norm ||x|| = max{|zi|, |x2|»- • . } . We define the mapping / : £7 H-> co by f(x) =

60

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

(l,X2,X3,...) on the open unit ball U of CQ- Then we have ||/(a;)|| = 1 for all x e U in spite of the fact that the mapping / is not constant. As a matter of fact, it can be shown that the boundary of U has no complex extreme point of the closure of U. Example 2.9 Since every complex (Lebesgue) space Y = Lp[0,1], 1 < p < oo, is a strictly convex Banach space, we see that every holomorphic mapping / G Hol(f/, Y) on a domain U of a Banach space X with values in Y satisfies the strong maximum modulus principle. Example 2.10 Although the complex space £x[0, 1] is not strictly convex, it can be shown (see, for example, [Pranzoni and Vesentini (1980)]) that each boundary point of its unit ball is a complex extreme point. Thus every holomorphic mapping / € Hol(£/, Y) on a domain U of a Banach space X with values in Y satisfies the strong maximum modulus principle whenever Y = Lp[0,1], 1 < p < oo. 2.3 2.3.1

Topologies in Hol(X>, Y) T-topology and compact open topology on Hol(X>, Y)

For / G Hol(T>, Y) and a subset K strictly inside V (we write X g P), we set \\f{x)\\K = sup ||/(:r)||.

(2.29)

x€K

We will mostly consider the subspace Hol(D,Y) of Hol{V,Y) consisting of all those holomorphic mappings which are bounded on each ball strictly inside V. The system of semi-norms (2.29) induces on Hol(T>, Y) the topology of locally uniform convergence over V (or briefly, T-convergence). Thus if a net {fj}jeA C Hol(D, Y) T-converges to / e Hol(D, Y), we write / = T-lim/,-.

(2.30)

jeA

(For more details see [Franzoni and Vesentini (1980)] and [Isidro and Stacho (1984)].) More generally, let M = {Ki}i€i be a family of subsets Ki C V which is closed with respect to finite unions, and for /o £ Hol(X>, Y), K G Ai and e > 0, define C/(/o, *-,£) = {/€ Hol(P, Y) : ||/ - / o | k < £}•

(2-31)

Differentiable and Holomorphic Mappings in Banach Spaces

61

Varying K € M and e > 0, we get a family {U(fo, K, e)} which defines a fundamental system of neighborhoods for a locally convex topology r^ of Hol(Z>,y). This topology is not necessarily Hausdorff. Thus the T-topology on Hol(2?, Y) is seen to be a rM-topology when M is the family of all finite unions of closed balls strictly inside T>. If M. is the family of all compact subsets of T>, then TM defines the topology of uniform convergence on compact subsets (the so-called compact open topology). The topology of local uniform convergence is finer than the compact open topology. They are one and the same if and only if X is finite dimensional. Proposition 2.4 The space Hol(P, Y) is complete with respect to the compact open topology. Corollary 2.4 The space Hol(X>, V) is complete with respect to the Ttopology (the topology of local uniform convergence over T>). 2.3.2

Montel's

theorem

The classical Montel theorem states that any bounded subset of Hol(D, C) is relatively compact for the compact open topology on T>. This assertion will still be true if we replace C by C n . However, the Montel theorem is not valid if Y is infinite dimensional. It is also not valid for the topology of local uniform convergence over T> if X is not finite dimensional. We mention here two analogs of the Montel theorem for the infinite dimensional case which are based on the classical Ascoli theorem (see, for example, [Yosida (1974)] and [Dunford and Schwarz (1958)]). Theorem 2.10 Let X and Y be Banach spaces and let V be a domain in X. The set TM = {/ e Hol(D,y) : ||/|| p < M < oo} is relatively compact with respect to the compact open topology on Hol(P, Y) if and only if the orbit J-M(X) = {f(x) '• f £ TM} is relatively compact in Y for each x G T>. Now, by the classical Alaoglu-Bourbaki theorem (see [Dunford and Schwarz (1958)]) we know that any bounded subset of a reflexive Banach space is relatively weakly compact. Hence, using the above theorem and Dunford's theorem on the equivalence of holomorphy and weak holomorphy (see Theorem 2.5) one can arrive at the following useful assertion. Theorem 2.11

Let Y be reflexive and let TM = {/ € Hol(£>, Y) :

62

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

\\f\\v < M}. Then for any sequence {/„} C TM, there is a subsequence {/nfc} which weakly converges to a holomorphic mapping f G Hol(Z?, Y), uniformly on each compact subset ofV.

2.3.3

Vitali's

theorem

In the one-dimensional case, Vitali's convergence theorem plays an important role in the study of analytic continuation. This theorem can be formulated as follows. Theorem 2.12 Let V be a domain in C, and let {/„} C Hol(X>, Y) be a sequence of uniformly bounded holomorphic mappings on T>. Then {fn} converges to f G Hol(P, Y), uniformly on each compact subset ofT> if and only if the sequence {fn(%)} converges at each point x in a subset UofD having a cluster point in T>. This assertion is no longer true if 2? is a domain in an infinite dimensional Banach space. Nevertheless, one can modify the assumptions of the above theorem and obtain an infinite dimensional version of Vitali's theorem. Theorem 2.13 Let V be a bounded domain in a Banach space X, and let = {/ G Hol(D, Y) : | | / | | D < M < oo}. Then the following assertions are equivalent:

TM

(i) The net {fj}j^A C Tu converges to f 6 Hol(P,Y) in the topology of local uniform convergence overV, i.e., / = T-lun/i.

(2.32)

(ii) There exists a ball B CC V such that the net {fj} converges to f € Hol(£?,Y), i-e., l i m | | / - / i | | B = 0.

(2.33)

jeA

Note that this theorem remains true if we replace the T-topology by the compact open topology on V or even by the pointwise convergence topology on V.

Differentiable and Holomorphic Mappings in Banach Spaces

2.4

63

Elements of Functional Analytic Calculus

2.4.1

Symbolic calculus on Banach algebras

For any complex Banach algebra E one can define formally different analytic function by using the algebraic and topological structures of E. For instance, the polynomials on E can be denned by setting m

<*i e C, A G E.

P(A) = ^ <*iAi,

(2.34)

t=0

The exponential function of an element AG E can be denned by

exp(A) = f^-An.

(2.35)

i=on-

More generally, we can define

f{A) = JT,anAn

(2.36)

n=0

for any function /(A) whose power series converges in a circle the radius of which exceeds the spectral radius p(A) of A. An equivalent (but more useful) approach to the definition of analytic functions on Banach algebras is to employ the classical Cauchy formula. The idea comes from the following simple, but very crucial assertion. For simplicity, we will consider E = L(X) — the Banach algebra of all the linear bounded operators on a Banach space X. Lemma 2.1 Suppose that A € L(X) and a G C\cr(^4). Let Q be any domain in C such that a £ 0, and cr(A) c fi c C, and let T be a positively oriented simple closed rectifiable contour around o~(A). Then, for all n = 0,±l,±2,..., - ^ / ( a - X)n(XI - A)~ld\ = (a/ - A)n.

(2.37)

This remarkable relation leads to the following notion. Note that if fl is an open set in C and Hol(fi, C) is the algebra of all holomorphic functions on fl, then La = {A € L{X) : a(A) c n} is an open subset of E ( = L(X)).

(2.38)

64

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

We define Hol(Ln,£0 to be the set of all £-valued functions / on La that arise from / € Hol(fi, C) by the formula

f(A) = ^jr

fWW - Ar'dX,

(2.39)

where F is any simple positively oriented contour in ft such that the interior domain of F contains cr(A). Remark 2.1 Formula (2.39) is usually called the Riesz-Dunford representation. It calls for some comments. (a) Since F stays away from cr(A) and the inversion operation in E is continuous, the integrand in (2.39) is continuous. So the integral (2.39) exists and defines f(A) as an element of E. (b) The integrand is a holomorphic E-valued mapping in the complement of cr(T). The Cauchy formula implies therefore that f(A) is independent of the choice ofT, provided only that T surrounds a {A) in Q. (c) If A = al and a € Cl, then the above lemma and (2.39) imply f(al) = f(a)I.

(2.40)

In most discussions of this topic, f(A) is written in place of f(A). The notation f is used here because it avoids certain ambiguities that may • > / the "shift" cause misunderstanding. We will sometimes call / — operation, and the function f is called the producing function of f. Theorem 2.14 The mapping f — i » / is a linear algebraic isomorphism o/Hol(f2,C) onto Hol(Lfi,.E) which is continuous in the following sense: If fn € Hol(fi, C) converge uniformly on compact subsets of Q to /eHol(Q,C), then f(A) = lim fn(A), n—>oo

A e L n,

(2.41)

in the operator topology on E. Proof. The integral representation (2.39) makes it obvious that f >-> f is a linear mapping. If / = 0, then f(a)I = f(al) (see Remark (c) above) implies / = 0. Thus / — i * f is one-to-one. The asserted continuity follows directly from formula (2.39), since ||(A/ — -A)"1!! is bounded on F. Thus, it remains to prove that / t-> / is multiplicative. Indeed, if / and g are elements of Hol(fi, C) and h(X) = /(A) • #(A), Aefl, one can show, by using Runge's

Differentiable and Holomorphic Mappings in Banach Spaces

65

approximation theorem (see, for example, [Shabat (1976)]) and continuity, that h(A) = f(A)g(A),

AzLQ.

(2.42)

Since Hol(f2, C) is obviously a commutative algebra, it follows that is also commutative. This seems to be surprising, because elements of E need not be commutative in general. However, f{A) and g(A) do commute in E for each A e LQ and f,g € Hol(Q, C), i.e.,

B.OI(LQ,E)

f{A)g{A) = g(A)f(A).

2.4.2

(2.4Q

The spectral mapping theorem,

One of the most important achievements of the functional calculus is the so-called spectral mapping theorem of Dunford. Theorem 2.15 /GHol(fi,C),

Let ft be a domain in C and let Ae LQ. Then, for any a(f(A)) = f(a(A)).

(2.44)

To prove this theorem, we need the following assertion. Lemma 2.2 Let A € LQ and let f e Hol(fi,C). Then f(A) is invertible inE ( = L{X)) if and only if f(A) ^ 0 for all A e a(A). Proof. If / has no zero in a(A), then g — 4 is holomorphic in a neighborhood fii c ft of a(A). Since fg = 1 in fii, it follows that f(A)g(A) = I and thus f(A) is invertible. Conversely, if f(a) = 0 for some a £ cr(-A), then there is h S Hol(fi,C) such that (X-a)h(X) = f(X),

Aefl.

(2.45)

This implies that (A - al)h(A) = f(A) = h(A)(A - a/).

(2.46)

Since a €
66

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

that f(a) — 0. If in this conclusion we use / - /3I in place of / , then we obtain that f(A) - (31 is not invertible if and only if there is a € cr(A) such that /(a) = (3. Thus we have /? £
Some *-algebras

Let A be a Banach algebra over the complex field C, i.e., A is a complex Banach space with the multiplication operation satisfying \\x-y\\ < \\x\\ • \\y\\. Recall that A is called an algebra with involution if there is an antilinear mapping x •-> x* of A into itself such that (a;*)* = x and (x • y)* — y*x*. The element x* is called the element adjoint to x. Definition 2.3 A complex Banach algebra A with involution is called a C*-algebra if ||a;||2 = \\x • x*\\ for all x £ A. Standard examples of C*-algebras are A — C, the complex plane with the conjugation z H-> Z, which is obviously an involution; A = Co (5), the algebra of all complex continuous functions vanishing at infinity on a locally compact Hausdorff space S; A= L(H), the algebra of all the bounded linear operators on a complex Hilbert space H with the inner product (•,•). For an element A e L(H), the adjoint operator A* is defined by the equality (Ax,y) = (x,A*y). At the same time, by the Gelfand-Naimark Theorem (see, for example, [Rudin (1973)]), each C*-algebra can be realized as a closed subalgebra of L(H) with a suitable H. Actually, for our purpose we can restrict ourselves to these subalgebras. An element x € A is said to be Hermitian (self-adjoint) if x* = x. The set of all hermitian elements will be denoted by AH- in particular, the elements Rex:= -(x + x*) and Imx := — (x - x*)

(2.47)

are Hermitian. Definition 2.4 An element x £ A is said to be positive if a; 6 AH and there is y £ AH such that y2 = x. It is known (see, for example, [Dixmier (1969]) that x £ AH is positive if and only if there is y £ A such that x = y*y.

(2.48)

Differentiable and Holomorphic Mappings in Banach Spaces

67

If A G L(TC), then (2.48) implies, in turn, that A G A is positive if and only if (AC, C) > 0 for all ( G W ,

(2.49)

or if and only if the spectrum of A a(A) c R + = [0,oo).

(2.50)

Now we turn to the class of J*-algebras which were introduced by L. A. Harris. This class consists of Banach spaces of operators mapping one Hilbert space into another. Definition 2.5 Let H and K, be complex Hilbert spaces and let L(H, K.) be the space of all bounded linear operators from H into K. A closed subspace il of L(H, K.) is called a J*-algebra if AA* A G it whenever A G il Of course, J*-algebras are not algebras in the ordinary sense. However, from the point of view of operator theory, they may be considered a generalization of C*-algebras (see [Harris (1981)]). Any open unit ball of a J*-algebra is a natural generalization of the open unit disk of the complex plane. In particular, any Hilbert space 7t may be thought of as a J*-algebra identified with L(H,C). Also, any C*-algebra in L(H) is a J*-algebra. Other important examples of J*-algebras are the so-called Cartan factors of type I, II, II, IV, which are the sets iti,il2,il3,il4, respectively, where ill = L(H,K), ih = {A e L(H) : At = A}, it3 = {A G L{H) : A1 = -A} (where Atx = A*x for a given conjugation x H-» X in H), and H4 is any closed complex subspace of L(H) such that both A* G H4 and A2 = XI for some complex number A whenever A € H4. (Cartan factors of type IV are variants of the spin factors.) Thus, the four basic types of the classical Cartan domains and their infinite dimensional analogues are the open unit balls of J*-algebras, and the same holds for any finite and infinite product of these domains. A crucial property of J*-algebras is that they have a kind of Jordan triple product structure and contain certain symmetrically formed products of their elements. In particular, for all elements A, B, C in a J*-algebra il, AB*C + CB*AeU.

(2.51)

Also, we observe that every J*-algebra is isometrically J*-isomorphic (see [Harris (1974a); Harris (1981)]) to a J*-algebra in L(H) for a suitable Hilbert space "H. our further considerations will be restricted to this case.

68

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Definition 2.6 We call a J*-algebra unital if the underlying Hilbert spaces are the same and if it contains the identity operator. Note that a closed subspace of L(H) which contains the identity operator is a unital J*-algebra if and only if it contains the squares and adjoints of each one of its elements. When a unital J*-algebra contains an operator, it contains any polynomial in that operator. 2.4.4

l-analytic functions on unital J*-algebras

Let it be a unital J*-algebra and let Q be a domain in the complex plane C. Consider the set £>n = {A e il : a (A) C f2}.

(2.52)

Since il is a closed subset of L(H), X>n is an open set in the topology of il induced by the sup-norm of L(H). For a function / e Hol(ft, C) we define the function / : VQ I-> L(H) by using the Riesz-Dunford integral: f(A) = ^

| /(A)(AI - A^dX,

(2.53)

where r C Dp is a positively oriented simple closed rectifiable contour such that the interior domain of F contains cr(A). Remark 2.2

Since U is closed, the equality An = — f Xn(\I - A^dX 2m JT

and (2.51) imply that f : V >-> il, i.e., the values of f{A) for are also in il. It is clear that f belongs to Hol(Z?n,it).

(2.54) AeVnCii

Definition 2.7 Let V be a domain in a unital J*-algebra H c L(H). A holomorphic mapping F : T> i-> it is said to be an l-analytic function if (i) there is a domain O c C such that V C "Dn, where Dn is defined by (2.52); (ii) there is a holomorhic function / S Hol(fi, C) such that F(A) = f(A) for all A G T>Q, where / is defined by (2.53).

69

Differentiable and Holomorphic Mappings in Banach Spaces

This function will be called the producing function of F ( = / ) . The set of all /-analytic functions on V will be denoted by Hol(D,il). We have already mentioned that the "lifting" mapping f >-> f denned by (2.53) is multiplicative, i.e., if / and g belong to Hol(fi, C) and h = f-g, then f(A)g(A) = h(A).

(2.55)

Remark 2.3 Although in general 11 is not an algebra in the ordinary sense, i.e., for each pair of elements, does not necessarily contain their product, condition (2.55) shows that the product of values of I-analytic functions of the same element is commutative and is also an element of ii. Also, if / G Hol(fi,C), Q\ is a domain in C such that f(Cl) C Cl\, and g 6 Hol(fti,C), then the equality h(X) = <7(/(A)) implies that h{A)=g(f(A))

(2.56)

for all A 6 VQ. Thus the "lifting" mapping / i-> / preserves the composition operation. Remark 2.4 If f : Q, — i > C is univalent, then it is clear that f(fl) is open and that f has a holomorphic inverse g which maps /(fi) onto Cl. In this situation, the composition law (2.56) shows that the l-analytic function f '• T^ci •-* / ( ^ n ) is biholomorphic and, in particular, that fi^Dn) is open. For more information on J*-algebras and Z-analytic functions see [Elin et al. (2002a)].

2.5 2.5.1

The Schwarz Lemma The classical Schwarz Lemma and Carton's uniqueness theorem

Throughout what follows we will denote by A the unit disk of the complex plane C, and by Hol(A) the set of holomorphic self-mappings of A. As we have mentioned, this set is a semigroup with respect to the composition operation. We begin with the classical Schwarz Lemma which plays a crucial role in geometric function theory. Theorem 2.16 (The Schwarz Lemma) Let F e Hol(A) be such that F(0) = 0. Then

70

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

(i) \F(z)\ < \z\ for all z G A;

(ii) |F'(0)| < 1. Moreover, if equality holds in (i) for at least one z G A, z ^ 0, or \F'(0)\ — 1, then F(z) = ellfiz for some ip G [0, 2TT], and thus equality holds in (i) for all z e A . Proof. Since F(0) = 0, the function F has the following Taylor representation:

F(z) = f^akzk.

(2.57)

fc=i

Therefore the function

G(z) = ^

= jrakzk-1 z

(2.58)

fe=i

is also holomorphic in A. Take any r G (0,1) and denote A r = {z G A : \z\ < r}. By the maximum modulus principle we get:

\G(z)\= ffl- < i , zeAr.

(2.59)

Letting r —> 1~ we obtain the inequality: |G(*)| < 1

(2.60)

for all z G A, which is equivalent to (i). Note also that G(0) = F'(0), whereby (ii) follows too. If, in addition, \G(z)\ = 1 for some z £ A, then, once again, by the maximum modulus principle, G(z) = etv for some ip G [0, 2TT]; hence F(z) = e^z for all z G A. • An elementary extension of the first part of this theorem to a Banach space can be formulated as follows. Let T> be the open unit ball in a complex Banach space X, and let Hol(P, X) be the set of all holomorphic mappings from T> into X. Theorem 2.17 Let F G Hol(X>,X) be such that F(0) = 0 and \\F(z)\\ < M for all xeV. Then (i) \\F(x)\\ < M\\x\\ for all x G V; (ii) ||F'(0)|| < M. Proof.

Apply Theorems 2.5 and 2.16.

•

71

Differentiable and Holomorphic Mappings in Banach Spaces

A higher dimensional analogue of the second part of the Schwarz lemma is known as Cartan's uniqueness theorem. Theorem 2.18 Let V be a bounded domain in a Banach space X and let 0 G V. Suppose that F £ Hol(P) (F : V *-* V) satisfies the following assumptions: (1) F(0) = 0 and (2) F'(0) = I — the identity operator on X. Then F = I/v,

i.e., F(x) = x for all

x€V.

As a matter of fact, a standard proof of this theorem shows that one can require slightly weaker assumptions to arrive to the same conclusions. Namely, let V be a domain (not necessarily bounded) in a Banach space X,T>3 0, and let F € Hol(P) be such that for a ball B C V centered at the origin, all iterates Fn of F are uniformly bounded, i.e., \\Fn\\B < M < oo, n = 0,l,2, F = I/v

(2.61)

If F satisfies conditions (1) and (2) of the theorem, then

Proof. Consider the power series expansion of the mapping Fn : V i-> V around zero: oo

n = 0,1,2,...,

Fn(x) = ^2PJl(x),

(2.62)

fe=0

where P°(x) = Fn(0) = 0, Pn{x) = x, and P* are homogeneous polynomials of order k > 2. Assume that for n = 1, Pj m is the first polynomial of order m > 1 which does not vanish identically, i.e.,

F(x)=x+f;P 1fc (x),

PT?0.

(2.63)

k—m

Then it can be shown by induction that oo

Fn{x) = x + nP^(x)+

^

p n(*),

(2-64)

fc=m+l

i.e., P™(x) — nP^n(x). But it follows by the Cauchy inequalities that ||PnmU<^,

m=l,2,...,

(2.65)

72

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

where r is the radius of the ball B. Thus we have

Wpr\\ < ^ hence P{n(x) = 0. This contradicts the choice of PJ". Consequently, F(x) = x, and we are done. 2.6

(2.66) •

Automorphisms

Let V be a domain in the complex Banach space X, and let Hol(X>, X) be the set of all holomorphic mappings from V into X. As above, by Hol(X>) we denote the family of all holomorphic self-mappings of V. Recall that this family is a semigroup with respect to the composition operation. Definition 2.8 A mapping F E Hol(X>) is said to be an automorphism of T> if F is biholomorhic self-mapping of T> such that F~x is well-defined on V and belongs to Hol(P). In other words, F € Hol(D) is an automorphism if and only if F is biholomorphic and F(D) = V. The set of all automorphisms of V will be denoted by Aut(X>). This set is a group with respect to the composition operation. 2.6.1

The unit disk

As above, by A we denote the open unit disk in the complex plane C. Let us fix a point a € A and consider the fractional linear transformation (the so-called Mobius transformation) of the unit disk denned by

(2.67) It is easy to see that the following inequality holds:

l-KWIa=(1-|,?ll)(^'Z|a)>0.

(2.68)

This inequality implies that |m o (z)| < 1 whenever z £ A and a is a fixed element of A. That is, ma maps A into itself and ma £ Hol(A). Now let h be an arbitrary element of Aut(A). We claim that h is of the form h(z) = Xma(z)

(2.69)

Differentiable and Holomorphic Mappings in Banach Spaces

73

for some unimodular number A. Indeed, if h(0) = b e A, then m_f,(b) = 0, and g := m-b o h

(2.70)

satisfies the conditions of the Schwarz Lemma, i.e., g £ Hol(A) and g(0) = 0. Hence \g(z)\ < \z\ for all z € A. On the other hand, g"1 = ft"1 o rrn, also belongs to Hol(A), and g~l{0) = 0. Therefore, I s " 1 ^ ) ! <\w\, w G A. Substituting here w = g(z) we get \g(z)\ > \z\, hence, in fact, |g(.z)| = \z . Consequently, g(z) = etvz for some

Z+

\ = ellfi

+ &

.

(2.71)

Setting a = e~lipb and A = elif> we prove our claim. Thus we have proved the following assertions. Proposition 2.5 Each Mobius transformation ma is an automorphism of A, and each automorphism of A is a composition of a Mobius transformation ma and a rotation rv : rip(z) = eltfiz,

(2.72)

ma o mb = rv o mmb(a)

(2.73)

maorv (ii)

with tp = 2arg(l + ab); (Hi) m~l = rri-a and (—m_ a ) - 1 = — m_ a . Let I denote the identity mapping on C, i.e., I{z) = z, z G C. For a fixed a 6 A, define a Mobius transformation Ma by the formula Ma(z) = -m-a(z):

Mo(z) = £ ^ . .

(2.74)

It is obvious that Ma satisfies the so-called involution property: Ma o Ma = I, i.e., M-1 = Ma.

74

2.6.2

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

The poly disk in C n

The open unit ball in the Chebyshev norm of C n is the polydisk A n defined by AB={z = (z1>...,zn)eCn:|zi|
i = l,...,n}.

(2.75)

Let us define functions /* : A i-» A by

(2.76) Since /it (A), A; = l , . . . , n are automorphisms of A, it is clear that the mapping / : A n •-> A n defined by f(zu ...,zn) = ( / i ( z i ) , . . . , /„(*„)) is an automorhism of A n . As a matter of fact, it turns out that each automorphism of the polydisk A n can be represented as a vector function consisting of automorphisms of the unit disk. More precisely: Proposition 2.7 If F : A n — i > A" is an automorphism of An, then there are automorphisms F\,..., Fn of A, and there is a permutation (ii, • • • ,in) of (1, ...,n) such that F(z1,...,zn)

= (F1(zil),...,Fn(ziJ).

(2.77)

The group Aut(A n ) is a Lie group of real dimension 3ra. 2.6.3

The Euclidean ball in Cn and the Hilbert ball

Let B = MH denote the open unit ball of a complex Hilbert space H with inner product (•, •). In particular, by l c » we denote the open unit ball in the Euclidean norm of C", i.e.,

BCn = | z = ( z l l . . . l z n ) e C B : ^ | z / y < l | .

(2.78)

The group Aut(5 n ) for n = 2 was described by Poincare (see Rudin (1980)). The case of arbitrary n is completely analogous to this particular case (c/., for example, Shabat (1976) and Rudin (1980)). The group Aut(S n ) is a connected Lie group of dimension n 2 + 2n. With each point a € B ( = B#) we may associate a unique automorphism ipa enjoying the following properties: 1. -*;

75

Differentiable and Holomorphic Mappings in Banach Spaces

3. (fa has a unique fixed point. This automorphism can be described by an explicit formula: r -z,

0 = 0,

(2.79) v

1 —" (2, <x)

Here, P a denotes the orthogonal projection onto the one-dimensional subspace C • a, and Qa the projection onto its orthogonal complement, i.e., Paz =

(2.80)

{^\ a, (a, a)

(2.81) (a, a) The automorphism ipi, o ipa carries the point a € B to the point b G B. Thus the group Aut(B) is transitive on B. Any automorphism of the ball B which fixes the origin is the restriction to B of a unitary operator. Any automorphism I/J G Aut(B) has a unique representation of the form QaZ

i> = U o<pa

= z-^La.

(respectively, ip = <pa o U),

(2.82)

where U is a unitary operator. The point a G B is uniquely determined by the equation ip(a) — 0 (correspondingly, V'(O) = a). We present two convenient formulas in which tp € Aut(B) and a — ip'1^):

>-™-«-H~ttt-t$y

If B = B c - , then also

detV.'(*)=

/

1-lla2 V + 1 " a • V|l-
**<*>• (2-83) (2.84)

For each a € B consider also the Mobius transformation defined by ma{x) = —-± r- (Vl-||a|| 2 <3a + Pa)(x + a), (2.85) i •+- \x,a) where x £ B, Pa is the orthogonal projection of H onto the subspace {Xa : A G C} spanned by a, and Qa = I - Pa. We note that m a (0) = a, m~l = m_ a , m 0 = 7, and that for every a, b G B the mapping mft o m_ a takes a to b. For the one-dimensional case H = C, the mapping m a becomes

76

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

ma(z) = f ^ as defined before. Again, as in the one-dimensional case, for the case of the Hilbert ball we have the following property: l-\\m-x{y)\\2=a{x,y),

(2.86)

where a(x v)

2.6.4

- (i-IMIa)(i-IMI2)

,2 87)

UnitalJ*-algebras

Let H be a complex Hilbert space and let L(H) be the (Banach) space of all the linear bounded operators on H. Recall that a linear subspace il C L(H) is said to be a unital 7*-algebra if AA*A belongs to il whenever A € il and I, the identity operator, belongs to il. The space L(H), in particular, is a unital J*-algebra. If V is the open unit ball in il and A £ T>, then the mapping MA : X> I-> il defined by MA(T) = (I-AA*)-*(T

+ A)(I + A*T)-1(I-A*A)*

(2.88)

is a biholomorphic self-mapping of V (see [Potapov (I960)], [Harris (1974a)]) and MA(0) = A. Since M^1 = M-A, the mapping MA is, in fact, an automorphism of T>. We call MA a Mobius transformation of T>. One can note that when H is one-dimensional, then MA becomes the standard form given in the literature for Mobius transformations of the unit disk A in C. We also observe that the operators defined by (2.88) can serve as automorphisms in a more general situation when il is the Banach space L(H, K) of bounded linear operators from one Hilbert space H into another Hilbert space K (see, for example, [Dineen (1989)]). Moreover, MA is a well-defined automorphism of the unit ball V of each closed subspace il of L(H, K) whenever V is homogeneous (or, which is the same, a bounded symmetric domain) (see [Harris (1974b)], [Upmeier (1986)] and [Dineen (1989)]). 2.6.5

The Schwarz-Pick lemma

The following assertion is the invariant form of the Schwarz Lemma.

Differentiable and Holomorphic Mappings in Banach Spaces

77

Theorem 2.19 Let F S Hol(A). Then for each pair z and w in A the following inequality holds: \m_F{w)(F(z))\

< \m-w(z)\

(2.89)

or, explicitly, F(z) — F(w) —— •• <

z- w

(the Schwarz-Pick inequality). (2.90)

Moreover, if equality holds in (2.90) for at least one pair z ^ w in A, then FeAut(A). Proof. Fix w S A and consider the mapping G = m_p(w) o F o mw. It is clear that G G Hol(A) and G(0) = 0. By the Schwarz Lemma, \G(z)\ < \z\, z £ A. Replacing z by m-w(z) we get the required inequality. If this inequality becomes an equality for some pair z ^ w, then |G(m_ w (z))| = |Tn_u,(2;)|. Again by the Schwarz Lemma, G(z) = elipz for some ip € [0,2TT] and for all z 6 A. Hence G belongs to Aut(A) and so does F = m,F(w) ° Gom-W. • Corollary 2.5 Let F e Hol(A,C). Then F maps A into its closure A if and only if it satisfies the condition

\n*)\<^nf-

(2-91)

Moreover, if F is not a constant and equality holds at one w £ A, then F e Aut(A). Proof. Indeed, if this inequality holds, then 1 - \F{z)\2 > 0 and \F(z)\ < 1. Conversely. Suppose that \F(z)\ < 1. If \F(z)\ = 1 for some z € A, then F is a constant and our assertion is trivial. Thus we may assume that 1^(^)1 < 1 for all z £ A. Now, rewriting the Schwarz-Pick inequality in the form F(z)-F(w) z— w

^ l-F(z)FH ~ 1 — zw

(2.92)

and letting w —> z we get the claimed inequality. To prove the last part of the corollary, consider the mapping G = m^F(w) ° F o mw and note that |G'(0)| = 1. Thus the assertion follows from the second part of the Schwarz D Lemma.

78

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

The following version of the Schwarz Lemma is a consequence of the Schwarz-Pick Lemma.

Corollary 2.6

Let F G Hol(A). Then |F(0) + \{F(z) - F(0)) \ < 1 for

allzZA

with \X\ < ^ g i .

Proof.

By the Schwarz-Pick Lemma,

(2.93) for all z in A. Since

\l-F@)F(z)\ < 1 - |F(0)|2 + \F(0)\\F(z) - F(0)| < 2(1 - |F(0)|) + \F(0)\F(z) - F(0)|

(2.94)

we see that \F(z) - F(0)| < 2| 2 |(1 - |F(0)|) + \z\ |F(0)||F(z) - f (0)|,

(2.95)

2\z\ |f(0)| + (1 - \z\ |F(0)|)|F(z) - F(0)| < 2|z|.

(2.96)

i.e., Since 1 - \z\ < 1 - |z| |F(0)|, we have 2|z| |F(0)| + (1 - |z|) |F(«) - F(0)| < 2|z|.

(2.97)

Hence if |A| < ±=$-, then |F(0) + \(F(z) - F(0)) | < |F(0)| + l ^ M |F(z) - F(0)| < 1. (2.98) Applying the Hahn-Banach theorem one can formulate a similar assertion in a more general setting. Corollary 2.7 ([Harris (1969)]) Let Xo denote the open unit ball of a Banach space X and let f e H{A,X0). Then ||/(0) + \(f{z) - /(0))|| < 1 for allzeA with |A| < ^ . The following corollary is another consequence of the Schwarz-Pick Lemma. Corollary 2.8

For each F £ Hol(A) and z E A, the estimate \F'(z)\ < j - 1 ^

holds.

(2.99)

Differentiable and Holomorphic Mappings in Banach Spaces

79

Remark 2.5 Observe also that for each r G (0,1), we get the following uniform Lipschitz condition: \F{z) - F(w)\ < y - ^ \z - w\,

(2.100)

whenever z and w belong to Ar = {z G A : ]z\ < r). Now if f G Hol(A,C) is bounded, i.e., \f(z)\ < M for all z G A, then F = jg f G Hol(A). Thus we have that each bounded holomorphic function on A is locally uniformly Lipschitzian with respect to the Euclidean distance in C. A similar assertion holds in general. Proposition 2.8 Let V be a bounded domain in a complex Banach space X and let F G Hol(O, Jf) be bounded, i.e., \\F(x)\\ < M for all x G V. Then for each xeV,

•"M'sas&w)-

<2-101)

Consequently, each bounded holomorphic mapping on V is locally Lipschitzian on D.

Chapter 3

Hyperbolic Metrics on Domains in Complex Banach Spaces 3.1

The Poincare Metric on the Unit Disk

In this section we review some elements of classical hyperbolic geometry on the unit disk. Additional details can be found in [Harris (1979)], [Franzoni and Vesentini (1980)], [Goebel and Reich (1984)] and [Jarnicki and Pflug (1993)]. Prom the point of view of geometric function theory on the unit disk, the Euclidean metric on the disk is inappropriate. Given a class of selfmappings of a domain, it is natural to look for a metric such that each mapping of this class becomes nonexpansive with respect to it. There are different ways to define such metrics. For example, one may use the invariant metric defined by the so-called pseudohyperbolic distance. The pseudohyperbolic distance between the points z and w of A is determined by the function d : A x A - » R+ = [0, oo) defined as follows: (3.1)

d(z,w) = \m..w(z)\.

It is easy to see that, actually, d is a metric on A that induces the usual Euclidean topology. Thus the Schwarz-Pick Lemma asserts the contractive property of a mapping F S Hol(A) with respect to this metric: d(F{z),F(w))
z,weA.

(3.2)

Moreover, F is an isometry with respect to d (i.e., d(F(z), F(w)) = d(z,w)) if an only if F € Aut(A). Also, it is easy to show that for each r £ (0,1), lim

inf d(z,w) = 1. 81

(3.3)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

However, from the classical Riemannian metric theory point of view, it is impossible to measure the "lengths" of vectors tangent to A to obtain d (see details below), it turns out that the only way to eliminate this deficiency is to define the so-called Poincare hyperbolic metric p = tanh" 1 d,

(3.4)

which, in spite of an additional level of computation, provides the needed properties in the construction of a non-Euclidean geometry in the spirit of Riemann. T h e o r e m 3.1 erties:

Let p : A x A t — * M . be a function

with the following

prop-

(i) for each F £ Aut(A) and each pair of points z,w in A, p(F(z),

F(w)) = p(z,w);

(3-5)

(ii) for each pair of numbers s,t G (0,1), s < t, p(0,t)=p(0,s)+p(s,t);

(3.6)

(Hi) lim e^l = i. 1

' t->o+

*

Then

p(^) = t a nh- d( ^)=ilo g i±|^M. Proof.

(3.7)

First we note that by (i), p(z, w) = p(m-z(z), m-z(w)) = p(0, m-z(w))

(3.8)

because m_ z € Aut(A). Also, using the rotation automorphism r^, r^{u) = ellfiu, u € A, with ip — — argu, we have p(0, u) = P M O ) , r^(u)) = p(0, |u|).

(3.9)

Comparing with (3.8), we get p(z,w) = p(0,\m-z(w)\).

(3.10)

So, it remains to show that for each r £ (0,1), p(0,r) = tanh- 1 (r).

(3.11)

Differentiable and Holomorphic Mappings in Banach Spaces

83

Indeed, for each 6 > 0 such that r + 6 < 1, we have, by (ii) and (i), p(0, r + S) - p(0, r) = p(r, r + 6) = p(m-r(r), m-r(r + 6))

=K°T^^)-

(3-i2)

Hence, by (iii), ^ dr

Similarly,

^)

=

= ^ . 1 — r2

(3.13)

_L_,

(3-14)

dr

1-r2

and we obtain (3.11): p(0,r) = ^ T - ^ - J = tanh-^r) Jo

*• ~

(3.15)

s

because p(0,0) = 0. It is also clear that the function tanh~ (|m_2(ti;)|) satisfies conditions (i)-(iii). D Remeirk 3.1 The following properties of the real function ip(t) — tanh^ x (t) are simple calculus exercises: (a)

1~; (c) V ( T ^ ) = V ( * ) + V ( * ) -

Theorem 3.2

The function p(-, •) defined by (3.7) is a metric on A.

Proof. It is clear tht p(z,w) > 0 for all (z,w) £ A x A, and since |m_z(u;)| = |m_ w (z)|, we have p(z,w) = p(w,z) and p{z,w) = 0 if and only if z — w. Now we need to show the triangle inequality p(z,w) < p(z,u) + p{u,w)

(3.16)

for each triple of points z, w, u in A. Indeed, let 7 : [0,1] H-> A be a curve joining the points z and w such that j'(t) is piecewise continuous. We will call such a curve an admissible curve. Consider the function 6(z, w) = inf{L7 : 7 is an admissible curve joining 2 and w}, (3.17)

84

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

where L 7 is the "length" of 7:

''-jfi^F*

<3'18)

We claim that 5(z,w) = p(z,w). Indeed, it is sufficient to show that 6(-, •) satisfies the conditions of Theorem 3.1. In fact, if F e Hol(A) and 7 is an admissible curve joining z and w, then F o 7 is an admissible curve joining F(z) and F(w). It follows by the Schwarz Lemma that

f'lF'jjmij'jt)]

LFO" = h

ri

\7'(t)\

i-iiW))i2 * - L T^HW dt = L^

(3J9)

Hence, 6(F(z),F(w))<6(z,w).

(3.20)

If now F e Aut(A), then applying (3.20) to F " 1 we get the equality 6(F(z), F(w)) = 6(z,w),

(3.21)

which proves (i). To prove (ii) and (iii), it is sufficient to evaluate 6(0, s) for 0 < s < 1. If 7 joins 0 and 5, then so does 0 = Rej. Hence r1 \j{t)\dt 1

r1 (3>{t)dt

Jo 1-I7WP-./0 i - [ / W /

Jo

2 =tanh- 1 (s),

i- — r

(3.22)

i.e., 6(0, s) > tanh - 1 (s). On the other hand, for ^(t) = ts, L1 = tanh - 1 (s). Consequently, we have J(0,s)=tanh~ 1 (s).

(3.23)

Now it follows, by properties (b) and (c) (see Remark 3.1), that the function 6 satisfies conditions (ii) and (iii) of Theorem 3.1. Thus, 5(z,w) = p(z,w).

(3.24)

Now it can be seen directly that (3.16) is a consequence of the definition of *(-,•)• • As a matter of fact, we have established somewhat more in our proof.

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Differentiable and Holomorphic Mappings in Banach Spaces

Theorem 3.3 Each mapping F G Hol(A) is nonexpansive with respect to the metric p, i.e., (3.25)

p(F(z)), F(w)) < p(z,w). Moreover,

if equality

in (3.25) holds for some pair of points

z,w £ A , then

F G Aut(A). Actually, the conclusion of this theorem is also a direct consequence of the Schwarz-Pick inequality and (3.4). This metric, the existence of which we have just established, is called the Poincare hyperbolic metric on A and will be denoted by p. Theorem 3.4 The pair (A, p) is a complete unbounded metric space, and p defines on A the same topology as the original Euclidean topology on A. Proof. We list several topological properties of the Poincare metric which prove our assertion. (a) lim p(0,zn) = oo if and only if \zn\ —> l~. Indeed, by (3.9) we have n—>oo

p(0, zn) = p(0, \zn\) = t a n l T 1 \zn\, and the assertion follows. (b) Each p-ball B{a,R) = {z G A : p(a, z) < R) is the disk Ar(a) by the formula B(a,R) = {ze A:\z-sa\

< t-t&nhR},

(3.26) defined (3.27)

where .

l-(tanhfl)2 l-(tanhfl)2|a|2'

1-H2 l-(tanhfl)2|a|2 '

(328) K

'

In particular, if a = 0, then B(0,R) = {z G A : \z\ < tanhil} is a disk centered at zero. (c) B(a,R) = ma(B(0,r)), i.e., each p-ball is the image of a disk centered at zero under the corresponding Mobius transformation. Now, it is clear that if the sequence {zn} C A converges to z G A in the sense of the /9-metric, i.e., p(zn, z) —> 0, then \zn - z\ < \zn — sz\ + (1 - s)\z\ —> 0 because s —> 1~ when R —> 0 (see (3.28)). Conversely, if \zn — z\ —> 0 for zn, z G A, then there is r G (0,1) such that zn G A r = {z G A : \z\ < r). Hence, p(zn, z) = tanh" 1 \m-z(zn)\

< tanh" 1 ' ^ ~ *} -> 0. 1 — rz

(3.29)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Finally, note that each p-ball is bounded away from the boundary of A; this fact implies the completeness of the metric space (A,/j).

•

3.2

The Infinitesimal Poincare Metric and Geodesies

Let z be any point of A and let u e C be such that z + u S A. If p is the Poincare metric on A, then p(z,z + u)=

'7

i - \z\

(l + e(ti)),

(3.30)

where e(u) —» 0 as it —> 0. Formula (3.30) shows that the linear differential element dpz in the Poincare metric is denned by the formula dp'

(3-31)

= T^M2-

The form a{z,u)=

\u\ ' ' ,

zeA,

uGC,

(3.32)

is called the infinitesimal (or differential) Poincare hyperbolic metric on A. The following properties are an immediate consequence of the definition: (a) a(z, u)>0, z £ A, u e C . (b) a(z,tu) = \t\a{z,u), t e C . A consequence of the Schwarz-Pick Lemma is that each F G Hol(A) is a contraction for the infinitesimal Poincare metric. If F e Hol(A), then dpF(z) < dpz, or equivalently, a(F(z), dF{z)) < a(z, dz).

(3.33)

If F G Aut(A), then equality in (3.33) holds for all z e A. Moreover, if equality in (3.33) holds for at least one z G A, then F € Aut(A). This notion allows us, using Riemann integration, to define the "length" of any admissible curve in A.

Differentiable and Holomorphic Mappings in Banach Spaces

87

Let 7 : [0,1] •—> A be an admissible curve in A joining two points z and win A. Then the quantity

L 7 (= Ly(z, w)) = J dPy{t) = £ ~r^2dt

(3-34)

is called the hyperbolic length of 7. We already used this notion in the previous section and saw that the hyperbolic length is greater than, or equal to, the hyperbolic distance between its end points, i.e., (3.35)

p(z,w) < Ly(z,w).

A curve 7 joining the points z, w in A is called a geodesic segment in A if its length is equal to the hyperbolic distance between its end points z and w, i.e., (3.36)

L7(z,w) = p(z,w).

Proposition 3.1 For each pair of points z and w in A, there is a unique geodesic segment joining z and w and it is either a linear segment (if z and w lie on a diameter of A) or a segment of the circle in C which passes through z and w and is orthogonal to dA, the boundary of A. Proof. Indeed, for the points 0 and s, 0 < s < 1, the curve 71 (t) = ts is the unique curve joining 0 and s such that

'

M

=/ , * " = / ^ W

=

to"h-'w-

(3'37)

i.e., 71 (t) is the geodesic segment joining 0 and s. If z and w are arbitrary points in A, then the automorphism g = rtp om_ 2 , where m-z is a Mobius transformation and r v is the rotation with tp = — argm- z (w), takes z into 0 and w into s = |ra_ z (iy)|. If we now define 71 (t) as before, that is, 7i(t) = f|m_z(u>)|

(3.38)

i{t)=g-1{n{t)),

(3.39)

and

we obtain 7(0) = 5 -1 (7i(0)) = g'1^) = z and 7(1) = 5 -1 (7i(l)) = p~1(|m_z(u;)|) = mz(ei'p\m-z(w)\) — (mzom-z)(w) = w. Thus j(t) is an

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

admissible curve joining z and w. In addition,

r = f1 W\dt 7

=

Jo 1-I7WI 2

f1 Kr'Yhim • |7j(t)| Jo

i-lff-H-nC*))! 2

(3.40) But p{z,w) = p(0,m-z(w)) = L 7l and hence we obtain relation (3.36). The second part of our assertion is a direct consequence of Proposition 3.1.

•

3.3

The Poincare Metric on the Hilbert Ball and its Powers

Let B denote the open unit ball of a complex Hilbert space H. Recall that the Mobius transformation ma on B is denned by

ma(x)=

]

1 -(- [x, a)

(^l-\\a\\Qa + Pa)(x + a),

(3.41)

where i s l , P a is the orthogonal projection of H onto the subspace {Xa : X e C}, and Qa = I - Pa- The Poincare metric on the Hilbert ball B is the function PB • B x B —> 1R+ given by pM(x,y) = tanh- 1 \\m^(y)\\ = \ log \+_ | | ^ ^ | | .

(3.42)

Note that 1 — jlm—^d/)]!2 = a(x,y), where

(3.43) Applying this property of Mobius transformations we get the following more explicit formula for p, namely, pM(x,y) = tznh-1(l-<7(x,y))i.

(3.44)

It is clear that each ma is a pi-isometry and has a norm continuous injective extension from 1 onto itself. In general, if Bi and B2 are the open unit balls in complex Hilbert spaces Hi and if2, respectively, p\ and P2 are the Poincare metrics assigned to Bi and B2, respectively, and f € Hol(Bi,B2), then P2(f(x),f(y))
(3.45)

Differentiable and Holomorphic Mappings in Banach Spaces

89

In particular, each holomorphic self-mapping of B is nonexpansive with respect to p&. It is not difficult to see that for the Cartesian product B n of n Hilbert balls, n > 2, the function PB* : B n x Bn i-> R+ given by P*»(x,y) = max {pTiiixj, yj)},

(3.46)

l<j
where x = ( n , . . . , xn) and y = (j/i,. • •, j/ n ) belong to B n , defines a metric on B n which we call the Poincare metric on B n . Again, each holomorphic self-mapping of B n is nonexpansive with respect to this metric. We will see below that the Poincare metric pB" (as well as, of course, pjg itself) is a particular case of the so-called Caratheodory and Kobayashi pseudometrics assigned to any domain in a complex Banach space. 3.4

The Caratheodory and Kobayashi Pseudometrics

We are now in a position to define a pseudometric on an arbitrary domain T> which generalizes the Poincare metric on the unit disk, the Hilbert ball and on the powers of the Hilbert ball. Prom now on we let D b e a domain in an arbitrary complex Banach space X. 3.4.1

The Caratheodory

pseudometric

For x £ V we set BT(x) := {y £ X : \\y - x\\ < r} C T> and for y £ Br(x), y ^ x, we let 0 : A —> V be denned by

{\) =x + r ,{y ~ X\ A, A G A,

(3.47)

\\y-x\\ where A is the open unit disk in the complex plane. If g £ Hol(T>, A), then g o cf> £ Hol(A) and consequently

PA (W(0)), 5 ^ ( J ^ J [ ) ) ) < PA(O, t ^ ) ,

(3.48)

where p& is the Poincare metric on A. Hence, PA(g(x), g(y)) < tanh-1 (^y^j

•

(3.49)

If x and y are arbitrary points in V then, by connectedness, we can find xn} of points of V such that x 0 = x,xn— y, 5 > 0 and a chain {xo, x\,...,

90

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and xi+i G Bs(xi) C T> for i = 0 , . . . , n - 1. By (3.49), we have

PA(
( 3. 5 0 )

for all g G Hol(£>, A). Consider the nonnegative function C (= Cv) on V xT> defined as follows: Cv(x,y)

= sup{PA(f(x),

f(y)) : f G Hol(P,A)}.

(3.51)

By (3.50), Cv{x,y) < oo for all x,y G V. It is easily checked that Cv is a pseudometric on T>. Definition 3.1 The function C (= Cv) is called the Caratheodory pseudometric on an arbitrary domain T>. Theorem 3.5 IfC\ andC2 are the Caratheodory pseudometrics assigned to the domains T>\ and T>2, respectively, and f G Hol(Pi,Z>2), then (3.52)

C2(f(x)J(y))
In particular, each holomorphic self-mapping of a domain V is nonexpansive with respect to the Caratheodory pseudometric C-p • Proof. Let / G Hol(X>i,X>2) be given. If h G Hol(P 2 , A), then h o f G Hol(X>i, A) and so for any x, y G £>i, we have (3.53)

pA{h(f(x)), h(f(y))) < d{x,y) and C2(f(x), f(y)) = sup{pA(h(f(x)),


h(f(y)) : h G Hol(2>2, A)}

(3.5g

IfV = A, then CA = PA is the Poincare metric on A.

Proof.

Using the mapping f(z) = z in A, we see that C&(z,w) > for z,w G A. Since holomorphic mappings from the disk into itself are nonexpansive with respect to the Poincare metric, we have CA(Z, W) < P&(z, w), and so equality holds. • PA(Z,W)

We will see below that also for the Hilbert ball B the Caratheodory pseudometric CB assigned to B coincides with the Poincare metric ps on B. Even in the one-dimensional case, however, it may happen that C-p is not a true metric. If, for example, T> = C, then Liouville's theorem implies that

Differentiable and Holomorphic Mappings in Banach Spaces

91

every / in Hol(C, A) is a constant mapping and hence pA(f(z), f{w)) = 0 for all z, w G C, and consequently Cp = 0. Nevertheless, if V is a bounded domain in an arbitrary complex Banach space X, then C-p is, in fact, a metric on T>. Proposition 3.3 Let V be a bounded domain in a complex Banach space X. Then Cv is a metric on V which induces the original topology on V defined by the norm topology of X. Furthermore, the metric space (D,C-p) is complete. We will prove this assertion later in a more general setting. 3.4.2

The Kobayashi

pseudometric

As the Caratheodory pseudometric on a domain V is defined by means of mappings from T> into A it is natural to look at mappings from A into T> to see if they also define a pseudometric on T> which generalizes the Poincare metric for the case of the unit disk. We will present two equivalent definitions of the Kobayashi pseudometric Kx> on V. The first definition is, in fact, that of the Lempert function S ([Lempert (1982)]; see also [Dineen (1989)]). In this case we can define a pseudometric on V after a minor adjustment to obtain the triangle inequality. For points x and y in a domain V we let 5v(x,y) = ini{PA{z,w)

: f G Hol(A,D), f(z) = x, f(w) = y). (3.55)

This definition presupposes the existence of a mapping / G Hol(A, V) whose range contains both x and y. This can be shown by applying the vectorvalued Stone—Weierstrass theorem. If V — A is the open unit disk in the complex plane, then using the homogeneity of the unit disk we see that 6&(z, w) = 6A(W, Z) for all z, w G T>. In addition, for z,w € A, the identity mapping shows that S&(z,w) < PA(Z> W), where p& is the Poincare metric on A. If u, v G A and / G Hol(A) is such that f(u) = z and f(v) = w, then we have that PA(Z, W) < PA(U, V) and hence PA(Z, W) < 8A{Z, W). SO we get the following assertion. Lemma 3.1 IfT> = A is the open unit disk in the complex plane, then 5A = PA is the Poincare metric on A. Remark 3.2 Unfortunately, some examples show that 5t> does not always satisfy the triangle inequality. Therefore it cannot serve as a pseudometric generalizing the Poincare metric. Indeed, if V (= T>r) = {(21,22) G C2 :

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

\zi\ < 1, |z 2 | < 1, \z\Z2\ 0 sufficiently small and points x,y inV (= Vr) such that Sx>(z,O) + 6-p(0,w) < 8v{z,w). To avoid this deficiency, we define the Kobayashi pseudometric which is smaller than the Lempert function 8-p • Definition 3.2 fined by

The Kobayashi pseudometric Kp on a domain V is deKv(x,y)

=

inflj25v(xi>xi+i):n<=N,{x

= x1,...,xn+1

=y}cT>\.

(3.56)

Suppose that x, y and z belong to V, and {x = x\, X2, •. •, xn+i = y} and {y = 2/i 12/2, • • •, J/m+i = z} are finite subsets of V. Then n

m

Kv(x,z) < ^5v{xi,Xi+i)

+ Y^st>(yj,yj+i)

i=l

(3.57)

j=l

and, upon taking the infimum of each sum on the right-hand side independently, we see that Kv(x, z) < K-D(X,

Z)

+ Kv(y, z).

(3.58)

It is clear that K-p < Sx>- It is also clear that if 6x> does satisfy the triangle inequality then, in fact, Sv — KT>- AS a matter of fact, we have a more general assertion. Lemma 3.2 If p is any pseudometric on T> such that p < 5x>, then p < Kx>. In particular, C-r>(x,y) < Kt>{x,y). Proof.

If {x = xi, X2,. • •, xn+i n

= y} is a subset of V, then n

p{x,y) < Y^P(xu Xi+\) < ^foixi, i=l

x i + i),

(3.59)

i=l

and upon taking the infimum of the right-hand side, we find that p(x, y) < Kv{x,y). • We will see below that for a bounded convex domain the Caratheodory and Kobayashi metrics actually coincide. Meanwhile, we just mention this fact for the one-dimensional case. Proposition 3.4

IfV = A, then K& = PA is the Poincare metric on A.

Differentiable and Holomorphic Mappings in Banach Spaces

93

Suppose now that T>\ and X>2 are arbitrary domains and h £ Hol(A, T>{) with h(z) = x and h(w) = y for some z and u; in A. Let / belong to H o l ( P i , P 2 ) . T h e n fohe

H(A,V2)

a n d f(h(z))

= f(x),

f(h(w))

= f(y).

Hence <5p2(/(x), f(y)) < p&{z,w); taking the infimum of the right-hand side of this inequality over the appropriate set of holomorphic mappings, we see that W/0»0>/(y))<*Pi(s.v)-

(3-60)

Thus we arrive at the following assertion. Theorem 3.6 / / K\ and K-z are the Kobayashi pseudometrics assigned to the domains V\ and T>i, respectively, and f e Hol(X>i,X>2), then K2(f(x),f(y))
(3.61)

In particular, each holomorphic self-mapping of a domain V is nonexpansive with respect to the Kobayashi pseudometric K-p. In order to present the second definition of the Kobayashi pseudometric, we need the Kobayashi infinitesimal pseudometric which is a particular case of the so-called Finsler infinitesimal pseudometric considered in the next section. 3.5

Infinitesimal Finsler Pseudometrics

In this section we describe an infinitesimal approach to the definition of a pseudometric on a domain. It is based on the idea mentioned in previous sections regarding the measurement of the lengths of smooth curves. Definition 3.3 An infinitesimal Finsler pseudometric on a domain 23 in a Banach space X is a nonnegative function a : T> x X —> 1R such that (a) a(x,tv) = \t\a(x,v) for all (x,v) eV x X and t € C; (b) a is upper semicontinuous. If, in addition, a(x, v) > 0 for v ^ 0, we call a an infinitesimal Finsler metric. For two points x and y in V, let 7 : [0,1] —> V denote a parametrized curve, with a piecewise continuous derivative, joining x and y. We call 7 an admissible curve joining x and y. If a is an infinitesimal Finsler pseudometric and 7 : [0,1] —> T> is an admissible curve, then the function

94

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

t € [0,1] —> a(7(i),7'(£)) is a bounded measurable function and the length La(i) of 7 is defined by L a ( 7 ) - / a( 7 (t), i(t))dt. Jo

(3.62)

a(a;, iu) = ta(x, v) for ( i , u ) e P x X and t € R +

(3.63)

a(x, v) = a(x, -v) for (x,v) eVx X

(3.64)

The properties

and

are clearly implied by condition (a) and are equivalent, respectively, to the fact that curve length is independent of the parametrization and of the orientation of the curve. The integrated form of a is the function da :T> xT> —> R + defined by da(x,y) = inf{L Q (7) : 7 is an admissible curve joining x and y}. (3.65) It is easily verified that da is a pseudometric on V. If oti and OL-i are infinitesimal Finsler pseudometrics on T>\ and 2?2, respectively, and / : T>\ —> 2?2 *s a C 1 function, then / is called nonexpansive (respectively, an isometry) with respect to a.\ and c*2 if a2(f(x),f'(x)(v))
(3.66)

a2(f(x)J'(x)(v))=a1(x,v),

(3.67)

respectively,

for all (x,v)

eVxX.

Lemma 3.3 Let «i and a2 be infinitesimal Finsler pseudometrics on V\ andT>2, respectively, and let di andd2 be the integrated forms ofct\ anda2, respectively. If f : V\ —> V2 is a smooth function such that f : (T>i,ai) —> — > {T>2,d2) is also nonexpansive, (T>2,Oi2) is nonexpansive, then f : (T>i,di) 1 i.e., d2{f{x),f{y))
(3.68)

Differentiable and Holomorphic Mappings in Banach Spaces

95

Proof. Let 7 be an admissible curve in T>\ joining x and y. Then / o 7 is an admissible curve in X>2 joining f(x) and f(y)- Hence

< W/°7) = I «j((/o7)W.(/»7)'(tP ./o

=

f1a2(f(l(t)),f'(j(t))h'(t)))dt Jo

< I ai( 7 (t), -/(*))* = £O1(7) Jo

(3-69)

and "2(/(a;)) /(j/)) < inf{^ Ql (7) : 7 is an admissible curve joining x and j/} (3.70)

= d1(x,y).

D 3.5.1

Examples

Example 3.1 (The infinitesimal Poincare metric) Let a(z, v) = t_yLa for (z,v) € A x C. The integrated form of a is the Poincare metric p on A since inf{LQ(7) : 7 is an admissible curve joining z and u>} = tanh^ 1 \m-z(w)\, where "»—•

(3-71)

This function a is called the infinitesimal Poincare metric on A. Example 3.2 (The infinitesimal Caratheodory pseudometric) The infinitesimal Caratheodory pseudometric on a domain T> at the point (x, v) in V x X is defined by CD(X,V) = sup{|/'(z)(t,)| : / G Hol(2>, A)}.

(3.72)

If V is a domain in a complex Banach space X and / € Hol(X>, A), then f'(x) G X* and hence cv{x,\v)

= \X\cv{x,v) for all A G C and (z,u) e 23 x X.

(3.73)

The Cauchy estimates show that c-p is finite and locally bounded. If V = A is the open unit disk in the complex plane C, then it is clear that CA is

96

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

the infinitesimal Poincare metric; hence the integrated form of CA is the Poincare metric on A. If T>\ and X>2 are domains in complex Banach spaces X\ and Xi, respectively, and / € Hol(£>i,r>2), then cv2 (f{x), f'(x)(v))

< oDl(x,v)

(3.74)

for all (x,v) £ V\ x X\. Thus holomorphic mappings are nonexpansive with respect to the integrated forms d\ and cfo of cvx and cp 2 , respectively. In finite dimensions the definition of the pseudometric dc(x, y) = inf {Z/Cl)(7) : 7 is an admissible curve joining x and y} (3.75) is due to Reiffen [Reiffen (1965)] and the pseudometric is sometimes called the Caratheodory-Reiffen-Finsler (or simply CRF) pseudometric on a domain T>. Observe also that the infinitesimal Caratheodory pseudometric cv is a continuous infinitesimal Finsler pseudometric on the domain V. Finally, it can be shown (see, for example, the next section) that for any domain P in a complex Banach space, Cv < dc,

(3.76)

where C© is the Caratheodory pseudometric on V. We remark in passing that a strict inequality holds for some domains. Example 3.3 (The infinitesimal Kobayashi pseudometric) The infinitesimal Kobayashi pseudometric on a domain T> at the point (x, v) in T> x X is defined by kv(x,v)

= inf{77 > 0 : 3 / <E Hol(A,D), /(0)=x, f'(0)(r))=v}.

(3.77)

In contrast with the Caratheodory pseudometric, the integrated form dk(%, y) — inf{£fcD(7) : 7 is an admissible curve joining x and y} (3.78) of the infinitesimal Kobayashi pseudometric coincides with the Kobayashi pseudometric K-p defined above for any domain V, i.e., Kv(x,y) = dk(x,y), x,y£V.

(3.79)

Differentiable and Holomorphic Mappings in Banach Spaces

97

Therefore the last formula can serve as the second definition of the Kobayashi pseudometric.

3.6

Schwarz-Pick Systems of Pseudometrics

We are now in a position to define the so-called Schwarz-Pick systems of pseudometrics introduced by L. A. Harris which include the Caratheodory and Kobayashi pseudometrics as the smallest and the largest pseudometrics, respectively. Definition 3.4 A system which assigns a pseudometric to each domain in each normed linear space is called a Schwarz-Pick system if the following conditions hold: (i) the pseudometric assigned to A is the Poincare metric; (ii) (the Schwarz-Pick inequality) if p\ and p2 are the pseudometrics assigned to T>\ and Z>2> respectively, and / £ Hol(T>i,X>2)) then P2(f(x),f(y))
(3.80)

for &llx,y e D i . Note that condition (ii) implies that all biholomorphic mappings are isometries. A further obvious but useful consequence of (ii) is the following fact: if T>\ C 2?2 and p\ and pi are the pseudometrics assigned to V\ and T>z, respectively, then p2(x,y) < p\{x,y) for all x,y £ T>\. Proposition 3.5 The Caratheodory pseudometrics form the smallest Schwarz-Pick system. Proof. Let pz> be assigned to the domain T> by any Schwarz-Pick system. Let x, y € V be arbitrary. If / G Hol(£>, A), then PA{f{x), f{y))
(3.81)

Hence Cv{x,y) = sup{pA(f(x),f(y))
: f e Hol(D, A)} (3.82)

•

98

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

We have already seen that the Kobayashi pseudometrics {K-D} form a Schwarz-Pick system. Another immediate consequence of Lemma 3.2 is the following one. Proposition 3.6 The Kobayashi pseudometrics form Schwarz-Pick system.

the largest

Proof. Suppose that p-r> is assigned to the domain V by any SchwarzPick system of pseudometrics and suppose that x,y € T>. If / € Hol(A,D) and z, w S A are points such that x = f(z) and y = f{w), then (3.83)

Pv(x,y) < PA(Z,W).

Taking the infimum of the right-hand side over all such / we get pz>(x, y) < 8t>{x,y). By Lemma 3.2, this proves our assertion. • In proving the finiteness of the Caratheodory pseudometric we used the mapping : A —> BT(x) C V

A-. + rJ^A.

(3.84)

\\y~x\\ The same mapping shows that

Kv(x,y) < Sv(x,y) < pA U, MzA\ = tanh"1 (^^)

(3.85)

for any domain T> with Br(x) C T>. We summarize the previous results by listing the relations we have established between the various pseudometricss on a domain T>: (3.86)

Cv
Since the Kobayashi pseudometrics form the largest Schwarz-Pick system, (3.85) implies the following proposition. Proposition 3.7 then

If {Pv} is a Schwarz-Pick system of pseudometrics,

pv(x,y) is a continuous pseudometric on T>.

x,y€V.

(3.87)

Differentiable and Holomorphic Mappings in Banach Spaces

99

The question that arises at this point is: does p induce the initial topology of T>1 The following theorem shows that for any bounded domain T>, a pseudometric px> assigned to I? by a Schwarz-Pick system is locally equivalent to the norm || • || of X. We denote by r{x) = dist|,.,| {x, dV) = inf {\\x - z\\ : z £ dV}

(3.88)

the distance in X between the point x and the boundary &D of the domain V. We also set R{x) = s\xp{\\x - z\\, z £ V). Theorem 3.7 then

(3.89)

If V is a bounded domain in a complex Banach space X,

tanh-1 ( ^ j ^ p ) < Pv(x, y)

(3.90)

for all x,y £ T>, and pv(x,y)

rCta.nh-1^^

(3.91)

whenever \\x — y\\ < dist||.||(a;,92)). Proof. The second inequality is a direct consequence of the previous proposition. Regarding the first inequality, it is enough to prove it for the Caratheodory pseudometrics since they form the smallest Schwarz-Pick system. So let Cv(x,y) = snp{pA(f(x), f(y)) : / £ Hol(P,A)}

(3.92)

be the Caratheodory pseudometric on T>. For x and y in T>, let I £ X* be a bounded linear functional on X of norm 1 such that (x — y,£) = \\x — y\\. Consider the holomorphic mapping / : V —» A defined by

(3.93) Clearly, f(x) = 0. Now it follows from the definition of the Caratheodory pseudometric Cv(x,y)

that

Cv(x,y) > PA(f(x),f(y)) = pA(0,/(y)) = ^nh'1 (\f(y)\)

(3.94)

100

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Definition 3.5 A domain V in a complex Banach space X is said to be /3-hyperbolic (or just hyperbolic) if there exists a metric p assigned to V by a Schwarz-Pick system of pseudometrics which is equivalent to the norm of X. Corollary 3.1

Each bounded domain is hyperbolic.

Definition 3.6 A domain P in a complex Banach space X is said to be homogeneous if for each pair x and y in T> there exists an automorphism F of V such that F{x) = y. A simple consequence of the above theorem is the following assertion which we will use in the sequel. Corollary 3.2 Let X be a complex Banach space such that its open unit ball B is a homogeneous domain. Then all Schwarz-Pick systems of pseudometrics (actually metrics) on B coincide. Moreover, if for x € B we denote by Fx an automorphism of B such that Fx{x) = 0, then

pB(x,y) = tuab.-1(\\Fx(y)\\).

(3.95)

Proof. Let p (= ps) be any metric assigned to B by a Schwarz-Pick system. Then for each z £ B we have, by the above theorem, that p(0,«)=tanh- 1 (||z||).

(3.96)

Consequently, if Fx is an automorphism of B such that Fx(x) = 0, then p(x,y) = p{0,Fx(y)) = tanh^dlF^y)!).

(3.97)

• In particular, it is well known that if X is a J*-algebra, i.e., X is a closed subspace of L(H), the space of bounded linear operators on a complex Hilbert space H, such that for each x € X, xx* x £ X,

(3.98)

and V is the open unit ball of X, then, given x £ V, one can define the generalized Mobius transformation Mx by Mx(y) = (I-xx*)-l{x-y)(I-

x*y)-\l - x*x)i,

(3.99)

101

Differentiable and Holomorphic Mappings in Banach Spaces

where / is the identity operator in L(H). Thus each metric of a SchwarzPick system on D can be represented by p(x, 2 /)=tanh- 1 (||M : c ( 2 /)||).

(3.100)

Remark 3.3 Note that if X is a complex Banach space such that its open unit ball B is a homogeneous domain, then B is a bounded symmetric domain in X. The converse is also true: each bounded symmetric domain can be realized as the open unit ball of a complex Banach space. Obviously, this ball is a homogeneous domain. Since any Schwarz-Pick system is preserved under a biholomorphic mapping it follows that all Schwarz-Pick systems of pseudometrics (actually metrics) on bounded symmetric domains coincide. Actually, it was shown by S. Dineen, R. M. Timoney and J.P. Vigue [Dineen et al. (1985)] (see also [Lempert (1982)] for the finite dimensional case) that this fact holds in a more general setting. Theorem 3.8 If D is a convex domain in a complex Banach space X, then Cx> = Kx>, i.e., all Schwarz-Pick systems of pseudometrics on T> coincide. If, in addition, T> is bounded, then, as we already know, this pseudometric is, in fact, a metric on T>. This unique metric is sometimes called the hyperbolic metric on T>. Finally, we observe that a convex domain T> in Cn (which is not necessarily bounded) is p-hyperbolic if and only if it does not contain a complex affine line (see, for example, [Dineen (1989)]).

3.7

Bounded Convex Domains and Metric Domains in Banach Spaces

In this section we consider some special properties of bounded convex domains with respect to the hyperbolic metric. We show that this metric is compatible with the convex structure of a domain. Lemma 3.4 Let V be a bounded convex domain in a complex Banach space X endowed with the hyperbolic metric p. Then each p~ball in the metric space (T>,p) is bounded away from the boundary ofT>. Proof. We may suppose without loss of generality that 0 S V. It suffices to show that Br := {x € V : p(0, x) < r} is bounded away from the

102

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

boundary. Let £ e X* be such that supRe^(x) < 1.

(3.101)

Let /(x) = 2 ^ j for x £ V. Then /(0) = 0 and |/(x)| < 1 for all x G V. Hence / e Hol(D, A). Now if x € Br, then tanh-^l/Cx)!) = /9A(/(0), /(*)) < p(0,x) < r.

(3.102)

Hence ||/||B r < tanh(r) < 1 and there exists a, 0 < a < 1, such that Re^(x) < a for all x £ Br. Consequently, we get that Re^(x) < 1 for all x £ Br + -piy (1 — a)Xo, where XQ is the open unit ball of X. Since x = 0 is an interior point in V, it follows that there is M > 0 such that \\£\\ < M for all t satisfying (3.101). Hence by the Hanh-Banach separation theorem, we have that Br + -Ar, (1 - a)X0 c T> for some £ £ X*. Hence BT is bounded • away from the boundary of V. Thus we see that any p-Cauchy sequence is bounded away from the boundary. This yields the following important assertion. Theorem 3.9 A bounded convex domain V in a complex Banach space X endowed with the hyperbolic metric p is complete and p-hyperbolic. Morebounded away from the boundary of V if and only over, a subset KofVis if it is p-bounded. If a subset K of T> is bounded away from the boundary of T> we will sometimes also say that K lies strictly inside T>. We need the following simple lemma, which will be useful in the sequel. Lemma 3.5 Let T> be a bounded convex domain in a complex Banach space X, and let z be a point in V. Then for a fixed s € [0,1), the set K = {sx + (1 - s)z : x € V)

(3.103)

is strictly inside T>. Moreover, if r = d\st{z, dD), then 5 — dist(K, &D) > (1 - s)r. Proof. Let u S X be such that ||u|| < r. Then the element z + u belongs to T>. Since T> is convex, it follows that for all x € T> the elements sx + (1 - s){z + u) = sx + (1 - s)z + (1 — s)u also belong to V. This • completes the proof. Proposition 3.8 Let V be a bounded convex domain in a Banach space X, and let p be the hyperbolic metric on T>.

Differentiable and Holomorphic Mappings in Banach Spaces (i) Ifx,yeT>

and s,t€ [0,1], then p(sx + (1 - s)y, tx + (l-

(ii) Ifx,y,ztV

103

t)y) < p(x, y).

(3.104)

and s € [0,1], then p(sx + (1 - s)z, sy + (l- s)z) * diamP + ^ d i a t ^ a P ) **>V)'

(Hi) Ifx,y,w,z

(3J°5)

€ V and s € [0,1], then

p(sx + (1 - s)y, sw + (1 - s)z) < max[/o(x, w), p(y, z)]. (3.106) Proof. Assertions (i) and (iii) follow directly from the definition of the Lempert function and the Kobayashi pseudometric on a domain. To prove assertion (ii), let usfixz 6 P and s G [0,1). Consider the set K — {sx+(l — s)z : x E T>} and let r = dist(z, dV). Since V is bounded, one can find e > 0 such that e\\x-y\\ < ( l - s ) r < S = dist(K,&D) (say, e = £ * ^ ) for all x, y in V. Hence all elements of the form sx + (1 - s)z + se(x — y) also belong to V. Fix any y 6 V and consider the affine (hence holomorphic) mapping / defined as follows: fix) = sx + (1 - s)z + se(x - y).

(3.107)

Let p be the (unique) hyperbolic metric on V. Then p can be represented as the integrated form of the infinitesimal Caratheodory pseudometric on the domain V defined at the point (x, v) in T> x X by cv(x,v)=sup{\f'(x)iv)\:f£Eol(V,A)},

(3.108)

i.e.,

r

r

1 p(x,y) = m£\ L cc ( 7 ) = / d,( 7 (t) f -/(t))dt: Jo I 7 is an admissible curve joining x and y >.

(3.109)

Since the mapping / is a holomorphic self-mapping of V, we have that cv(f(x), f'(x)(v)) < c-oix, v) for all x e V and veX. At the same time, noticing that f'(x) = (1 -f e)s/ (/ is the identity mapping on X) does not depend o n i e l and f(y) = sy + (1 - s)z, we

104

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

see that cv{sy + (1 - s)z, (1 + e)slv) < cv(y, v), or cv(sy+(l-s)z,

(3.110)

slv) < Y^cv(y,v).

Since y is arbitrary, we can replace it with any element in V. Let now 7 : [0,1] —> T> be any admissible curve joining x and y. Then 71 = sj+ (1 — s)z is an admissible curve joining sx + (1 — s)z and sy + (1 — s)z. We have co(7i(*).7i(*)) = cu(s 7 (i) + (1 - s)z,

sf'(t))


(3.111)

Consequently, p(sx + (1 - s)2, 51/ + (1 - s)2;) < j - ^ p(x, y). Now substituting e = - ( 1 - ~ 3 ^ ( p ' 9 P ) we get assertion (ii).

(3.112) •

Definition 3.7 We say that V is a metric domain in X if there exists a metric p on T> such that (i) for each x 6 V and for each 0 < 5 < dist(a;,3D), there are positive numbers L (= L(6)), r(= r(S)) and m(= m(S)) such that p{x,y) < £||^ — 2/|| whenever ||a; — y\\ < S,

(3.113)

and p(x,y) > m\\x — y\\ whenever p(x,y) < r;

(3.114)

(ii) each p-ball is strictly inside V. A bounded convex domain in a complex Banach space with a metric assigned to it by the Schwarz-Pick system is a metric domain. It is clear that if V is a metric domain, then the metric space (V, p) is complete. For a bounded convex domain in a real Banach space such a metric can be induced by the complexification of X and by using a Schwarz-Pick metric p on the direct product of T> by itself in the complex sense. Other constructions of such domains can be given by using Hilbert's projective metric or Thompson's metric on a cone associated with a convex bounded domain V in X (see [Nussbaum (1994)]).

105

Differentiable and Holomorphic Mappings in Banach Spaces

Definition 3.8 Let D be a metric convex domain in a Banach space X with a corresponding metric p. We say that the metric p is compatible with the convex structure of T> if the following conditions hold: (i) If x,y,w,zGT>

and s £ [0,1], then

p(sx + (1 - s)y, sw + (l-

s)z) < m&x[p(x, w), p(y, z)]; (3.115)

(ii) for each z £ V, there is a real function (p : [0,1) —> [0,1) such that 1—s limsup T-T- < oo, s^il-y(s)

(3.116)

and for each pair of points x,y S T> and s 6 [0,1], the following inequality holds: p(sx + (1 - s)z,

sy + (l-

s)z) <
(3.117)

The following assertion is a direct consequence of the above propositions. Proposition 3.9 For each bounded convex domain V in a Banach space X, there is a metric p onT> such that (T>,p) is a complete metric space, T> is a metric domain, and the metric p is compatible with the convex structure

ofV.

Proof. Choose p as the hyperbolic metric on V and use inequality (3.112) with e = dfamp'9P^ • Since e —> 0 + as s —» 1~, the last inequality proves our assertion. •

Chapter 4

Some Fixed Point Principles

4.1

The Banach Principle

Let V be a topological space and assume that F is a self-mapping of V, i.e., F(V) C V. Definition 4.1 if F(z) = z.

(4.1)

A point z € V is called a fixed point of the mapping F

Condition (4.1), which is referred to as the invariance of the set T> under the mapping F, enables us to define the iterations Fn of F by the recursive relations F1 = F, Fn+1 = Fn o F, where "o" denotes the composition operation of mappings. The subset of 23 consisting of all the fixed points of a mapping F :T> —> Z> will be denoted by ¥\X.T>(F) (or simply by Fix(F)). Fixed point theory is mostly concerned with the following basic issues: (a) (b) (c) (d)

Existence (i.e., is Fix(F) ^ 0 ?). Uniqueness (or whether the fixed points are isolated). Approximation (i.e., devising algorithms for locating fixed points). Properties (e.g., structure, connectedness, etc.) of Fix(F). The following properties of F and its iterates Fn are obvious:

(i) Fix(F) C Fix(F n ) for all n > 1; (ii) if F is continuous and Fn(x) —> y G V as n —» oo, then y € Fix(F); (iii) {Fn}n>i has a natural one-parameter semigroup structure: Fn+m = FnoFm. Recall that if (M,p) is a metric space and F : M —> M is a mapping, then we say that F is nonexpansive if p(F(x),F(y)) < p(x,y) for all 107

108

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

x,y € M. If, moreover, there exists k, 0 < k < 1, such that < kp(x,y) for all x,y € M , then we call F a sirict conp(F(x),F(y)) traction or a fc-contraction. It is clear that nonexpansive mappings are continuous. The following principle is among the more widely used fixed point theorems. Theorem 4.1 (Banach's Contraction Principle) Let (M,p) be a complete metric space and let F : M —> M be a strict contraction. Then F has a unique fixed point in M, and for each x0 € M, the sequence of iterates {Fn(xo)} converges to this fixed point. Proof. Select XQ G M and define the iterative sequence {xn} by xn+i = F(xn) (equivalently, xn = Fn(x0)), n = 0 , 1 , 2 , . . . . Observe that for any indices n, p £ N, P(xn,xn+P)

= p{Fn(x0),Fn+P(x0))

= p{Fn(x0),F"

oF*(xo))


+•••+

p{Fp-\x0), Fv{xo))}


+ k+--- + kp-1)P(x0,

F(x0))

(4.2) This shows that {xn} is a Cauchy sequence, and since M is complete, there exists x e M such that lim xn = x. To see that x is the unique fixed point n—too

of F, observe that

x = lim xn = lim xn+i = lim F(xn) = F(x) n—>oo

n—>oo

(4.3)

n—>oo

and, moreover, x = F(x) and y = F(y) imply .

p(x,y) = p(F(x), F(y)) < kp(x, y),

(4.4)

yielding p(x, y) = 0. Finally, letting p —> oo in (4.2), we obtain a rate of convergence: p(xn, x) = P(Fn(x0), x) < - ^ - p(xo,F(xo)). i-k

(4.5) n

109

Some Fixed Point Principles

The Banach fixed point theorem fails for nonexpansive mappings. A simple standard example is the shift of the real axis M. defined by F(x) = x + l. There is a natural class of mappings which falls properly between the class of strictly contractive mappings and nonexpansive mappings. A mapping F : M —> M is called contractive if p(F(x),F{y))

< p(x,y),

(4.6)

x,y € M, x^y.

Obviously, a mapping of this type can have at most one fixed point. The mapping F : R —> R defined by F(x) = 1 + l n ( l + ex) provides a simple example of a fixed point free contractive mapping. (In fact, \x — F(x)\ > 1 for all x € K.) However, in compact spaces such mappings always have fixed points. Theorem 4.2 Let (M, p) be a compact metric space and let F : M —* M be contractive. Then F has a unique fixed point in M, and for any xo € M, the sequence {F n (xo)} of iterates converges to this fixed point. Proof. The function

= p(Fn+i(x0),Fx)

< p(Fn(x0),x)

= an, (4.7)

{an} is a decreasing sequence of nonnegative real numbers and so has a limit, say a. Again by compactness, {Fn(x0)} has a convergent subsequence {Fn"(x0)}, say lim Fnk(x0) = z. Obviously, p{z,x) = a. If a > 0, then fe—*oo

we obtain the contradiction: a=limop(Fn"+1{xo),x)=p(F{z),x)

= p(F{z),F(x)) < p(z, x) = a.

Thus a = 0. Therefore any convergent subsequence of {Fn(x0)} converge to x, so by compactness, lim Fn(xo) = x.

(4.8) must n

n—>oo

The theorem is generally attributed to Edelstein [Edelstein (1962)] (see also [Edelstein (1964)]), who actually proved the following slightly more general result (cf. [Goebel and Reich (1984)]).

110

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Theorem 4.3 Let M be a metric space and F : M —> M a contractive mapping. If there is a point z in M such that a subsequence of {Fn(z)} converges to y, then y is the unique fixed point of F, and lira Fn(z) = y. n—>oo

Finally, we give an example which shows that additional conditions for the existence of fixed points are necessary even if M is bounded. Example 4.1 Let X = c0 and let M be the closed unit ball of X. The mapping F defined by F(x) = ( a , x x , . . . , x n . . . ) ,

0 < a < l ,

(4.9)

is an affine isometry of M, but it is fixed point free.

4.2

The Theorems of Brouwer and Schauder

Brouwer's theorem is very useful in applications in Analysis and its discovery has had a tremendous influence in the development of several branches of mathematics, most notably algebraic topology. The simple formulation of Brouwer's theorem belies the fact that it seems to require a 'nonelementary' (nonmetric) proof. All known proofs require facts not commonly used in metric fixed point theory. On the other hand, attempts to find other versions of Brouwer's theorem have given rise to several interesting questions of metric type which will be discussed later.

Theorem 4.4 (Brouwer's Fixed Point Theorem) // F : V -> T> is a continuous self-mapping of a compact convex subset V of C n into itself, then F has a fixed point. One of the most important fixed point principles is the well-known principle due to J. Schauder, which is a generalization of the finite-dimensional fixed point principle of Brouwer. Theorem 4.5 (J. Schauder) Let F be a mapping which maps a closed convex subset V of a Banach space X into itself. If F(D) is contained in a compact subset of T>, then F has at least one fixed point in T>. All the proofs of this theorem we know about rely on topological and geometric considerations. As a rule, these proofs use Brouwer's theorem and finite-dimensional approximations (see, for instance, [Trenogin (1980)] and [Krasnoselskii and Zabreiko (1984)]).

Some Fixed Point Principles

111

Note also that the Schauder principle has no constructive features; it fails to indicate the number of fixed points, as well as methods for their approximation. In the following section we will consider another class of mappings, namely, those mappings which are either contractive or nonexpansive with respect to the hyperbolic metric.

4.3

Holomorphic Fixed Point Theorems

Let 23 be a domain in a complex Banach space X, and let Hol(X>), as above, denote the family of all holomorphic self-mappings of V. Among the fixed point principles for a mapping F £ Hol(P) in general Banach space there are two useful classical criteria which ensure the existence and uniqueness of a fixed point of F. Recall that a subset S is said to lie strictly inside T> if there exists e > 0 such that the open ball Be(x) centered at x of radius e is a subset of V whenever x G V. In other words, inf{||x-y|| :ye&D, x € S} >e.

(4.10)

The following theorem, originally due to Earle and Hamilton [Earle and Hamilton (1970)], may be viewed as a holomorphic version of the Banach contraction principle. We give a slightly more general formulation of it [Harris (2003)]. Theorem 4.6 (Earle—Hamilton) Let X> be a nonempty domain in a complex Banach space X and let F : V —> V be a bounded holomorphic mapping. If F{V) lies strictly inside V, then F has a unique fixed point in V. Moreover, the sequence of iterates {Fn(x)}^=1 converges to this point uniformly on each bounded subset of V. Proof. We use a pseudometric p, called the Caratheodory-ReiffenFinsler pseudometric (CRF-pseudometric), with respect to which F is a strict contraction. Let A be the open unit disk of the complex plane. Define a(x,v) = sup{\g'(x)v{ : g : V —» A holomorphic}

(4-11)

for x G T> and v e X, and set

L ( 7 ) = f a( 7 (t),7'(*))* Jo

(4.12)

112

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

for 7 in the set T of all curves in T> with piecewise continuous derivative. Clearly a specifies a seminorm at each point of T>. Define a pseudometric p : V x V -» R+ by p(x, y) = inf {L( 7 ) : 7 G T, 7 (0) = x, 7 (1) = y}

(4.13)

for x, y € T>.

Let x € 2? and » e l . By the chain rule, (g o F)'(x)v = 5'(F(a;))F'(a;)u

(4.14)

for any holomorphic function g : V —» A. Hence, a(F(x),F'(a;)i;)
(4.15) < L(j) (4.16)

holds for all x,y GT>. Now by hypothesis, there exists an e > 0 such that B£(F(x)) C P whenever x £ T>. We may assume that V is bounded by replacing V with the subset U{B£(F(x)) : i £ D } . Fix t with 0 < t < e/S, where 6 denotes the diameter of F(D). x € T>, define F(y) = F{y) + t[F(y) - F(x)}

(4.17) Given (4.18)

and note that F : V —> V is holomorphic. Given x 6 V and v € X, it follows from /"(*)<; = (1 + *)*"(*)«

(4.19)

and (4.15) with F replaced by F that a(F(x),F'(x)v)<-L-a(x,v). L -j~ t

(4.20)

Integrating this as before, we obtain p(F(x),F(y))<^-tp(x,y)

(4.21)

Some Fixed Point Principles

113

for all x, y £ V. Since V is bounded, (D, p) is a complete metric space. Now our assertion is seen to follow from by the Banach fixed point theorem. • The Earle-Hamilton theorem still applies in some cases where the holomorphic mapping does not necessarily map its domain strictly inside itself. In fact, Theorem 4.8 below is a generalization of the Earle-Hamilton theorem for convex domains. The following well known principle follows directly from a result of Krasnoselskii and Zabreiko [Krasnoselskii and Zabreiko (1984), Theorem 21.5] and a special property of holomorphic mappings [Krasnoselskii and Zabreiko (1984), Theorem 23.5]. Theorem 4.7 Let V be a convex domain in X, and let F € Hol(P) be continuous on T>. If F(T>) is contained in a compact subset ofT> and x £ F(x)

(4.22)

for all x S dV, then F has a unique fixed point in V. Theorem 4.8 (Khatskevich-Reich-Shoikhet [Khatskevich et al. (1995a)]) Let T> be a nonempty bounded convex domain in a complex Banach space and let F :T> —» T> be a holomorphic mapping having a uniformly continuous extension to T>. If there exists an e > 0 such that \\F(x) — x\\ > e whenever x 6 &D, then F has a unique fixed point in T>. Of course, the hypotheses of this theorem are satisfied on some homothety of T> under the assumptions of the Earle-Hamilton Theorem. The hypothesis ||F(a;) — a;|| > e for all x £ dV is also satisfied when F(T>) is contained in a compact subset of V and condition (4.22) holds. If F is not necessarily compact, this condition is also satisfied when T> contains the origin and S»P»<1. xedv INI

(4.23)

The proof we present here is due to L. A. Harris [Harris (2003)]. Proof. by

Given 0 < t < 1 and x£~D, define a holomorphic map / ( : V -> V

ft(y) = (l-t)x + tF(y)

(4.24)

114

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and let 6 > 0 be such that Bs(x) C V. To show that ft(V) lies strictly inside T>, take e = (1 - t)6. Let y G V and let w G BE(ft(y)). Then

z=^m

(4.25)

is in 2? since z G -B*(x), so w = (l-t)x

+ tF{y) G P .

(4.26)

Hence B£(ft(y)) C £> for all j , e D . By the Earle-Hamilton theorem, ft has a unique fixed point gt(x) in £>. Since the hyperbolic CRF-metric is continuous, the proof of the Banach contraction principle shows that the iterates of ft at a chosen point yo € T> are holomorphic and locally uniformly Cauchy in x. Hence the limit mapping gt : T> —» T> is holomorphic. Now a point x S Z> is a fixed point of gt if and only if a; is a fixed point of F. Thus, by the Earle-Hamilton theorem, it suffices to show that gt(T>) lies strictly inside V for some t > 0. Since F has a uniformly continuous extension to T>, by hypothesis there exist e > 0 and 6 > 0 such that ||-F(a;) — x\\ > e whenever x €T> and dist(a:,dX>) = inf{||a; - y\\ : y € dV} < 6.

(4.27)

Since V is bounded, there is an M with ||:r|| < M for all x € V. If x G T>,

F(gt(x)) - gt(x) = (1 - t)[F( 5t (i)) - z],

(4.28)

so \\F(gt(x))-gt(x)\\< 2(1-t)M.

(4.29)

Choose t close enough to 1 so that 2(1 — t)M < e. If d(gt(x),dT>) < S for some x G T>, then e<\\F(gt(x))-gt(x)\\, a contradiction. Thus, B&(gt(x)) C P for all x G P , as required.

(4.30) •

Example 4.2 The hypotheses on the behavior of F on dV cannot be omitted in Theorem 4.8. This follows by considering a translate of the shift operator. Specifically, let X = CQ and define F(x)=(^,xux2...)

(4.31)

Some Fixed Point Principles

115

for x 6 X. Clearly, F is an afRne isometry on X and F maps the ball Br(0) into itself for each r > | . However, if F(a;) = x, then -=Xl=x2

= ...,

(4.32)

contradicting the requirement that x belongs to CQ. Thus F has no fixed point in X. 4.4

Fixed Points in the Hilbert Ball

Despite Example 4.2, it is still an open problem whether if B is the open unit ball of a separable reflexive complex Banach space and F : B —> B is a holomorphic mapping with a continuous extension to B, then F must have a fixed point in B. However, Hayden and Suffridge [Hayden and Suffridge (1976)] have proved that e%eF has a fixed point in B for almost every 9. Also, Goebel, Sekowski and Stachura [Goebel et al. (1980)] have solved the problem in the affirmative for the case where X is a Hilbert space and this has been extended by Kuczumow [Kuczumow (1984); Kuczumow (1985)] to the case where X is a finite product of Hilbert spaces (with the max norm). An example of Kakutani [Kakutani (1941)] shows that holomorphy is essential in the hypotheses since he exhibited a fixed point free homeomorphism of the closed unit ball of any infinite dimensional Hilbert space. A related problem is to weaken the hypotheses of the Earle-Hamilton theorem by showing that if F : B —> B is a holomorphic mapping such that the sequence of iterates {Fn(x)} lies strictly inside B for some x £ B, then F has a fixed point is B. This has been established when X is a Hilbert space (see also [Goebel and Reich (1984)]) and when X is a finite product of Hilbert spaces in [Kuczumow (1985)] (see also [Kryczka and Kuczumow (1997)]). These results have been extended to bounded convex domains in a more general class of reflexive Banach spaces by Budzyriska [Budzynska (2004)]. The example presented above gives a counterexample for the general case. Let B be the open unit ball of a Hilbert space H. Applying the asymptotic center method due to M. Edelstein [Edelstein (1972)] and the properties of the Poincare metric p on the Hilbert ball we get the following fixed point theorem. Definition 4.2

Let F : B —> B be a p-nonexpansive mapping. We

116

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

shall call a sequence {yn} =0. lim p(yn,Fyn)

C B an approximating sequence for F if

n—KX>

Theorem 4.9 (see [Goebel and Reich (1984)]) Let F : 1 -+ B be a pnonexpansive mapping. Then the following are equivalent: (a) F has a fixed point; (b) There exists a point x in B such that the sequence of iterates {Fnx} p-bounded; (c) The sequence of iterates {Fnx} is p-bounded for all x in M; (d) There exists a p-bounded approximating sequence for F.

is

The following proposition can also be found in [Goebel and Reich (1984)]. Proposition 4.1 B such that {Fnx} point.

Let F : B —» B be p-nonexpansive. If there exists x in converges weakly to a point in B, then F has a fixed

Theorem 4.10 ([Goebel (1982b)], see also [Goebel and Reich (1984)] and [Shafrir (1992b)]) / / a norm continuous mapping F : B —> B is pnonexpansive on M, then it has a fixed point in B.

4.5

Fixed Points in Finite Powers of the Hilbert Ball

It is not difficult to observe that the hyperbolic metric in the Cartesian product B n of n open unit balls B is given by Pi"(X,y) = max pB(zj,Vj)-

(4.33)

In this section N(Mn) will denote the class of all pBn-nonexpansive selfmapping on B™. The class of those mappings in N(En) which have a continuous (in norm) extension to B" will be denoted by CN(Mn). It will also be convenient to consider the slightly more general class of mappings N(Mn) which consists of all norm continuous mappings / : B n —+ B" such N(Mn) for all 0 < t < 1 [Kuczumow (1985)] and [Shafrir that tf\Bn£ (1992b)]. Note that when / e iV(B") it may happen that f(x) £ d(Mn) for x £ Bjf. But in this case, if f(x) = v with ll«jill = ••• = »«*. 11 = 1.

(4-34)

Some Fixed Point Principles

117

then f(y)h = vh. • • •. f(v)h = vik

(4-35)

for all y 6 1". The following simple generalization of Theorem 4.9 uses induction with respect to n and is based also on the asymptotic center method. Theorem 4.11 (c/. [Kuczumow et al. (2001a)]) Let F : B n -> B" be a holomorphic mapping or more generally, a p%,n-nonexpansive mapping. Then the following statements are equivalent: (i) F has afixedpoint; (ii) there exists i £ l " such that {Fk(x)} lies strictly inside B n (this means that {Fk(x)} is pun-bounded); (Hi) there exists a ball B(x,r) in (Bn,pBn) which is F-invariant; (iv) there exists a nonempty, pBn-bounded, pmn-closed and convex subset of B" which is F-invariant. Theorem 4.12 ([Kuczumow (1985)] and [Shafrir (1992b)]) If F € JV(I"), then Fix(F) ^ 0. Results on common fixed points of commuting families of mappings in JV(B~") can also be found in [Kuczumow (1984)] and [Shafrir (1992b)]. Remark 4.1 Note that Theorems 4.11 and 4.12 do not explain what happens when one tries to approximatefixedpoints by simply iterating F. converge to a fixed point of In other words, do the iterations {Fn(x)}^1 F (locally or globally) under the conditions of these theorems? This question arises, in particular, in the context of Theorem 4.12 when FixB"(F) = 0. For the one-dimensional case, when n = 1 and B = A is the open unit disk in the complex plane C, a complete answer to this question is given by the classical theorem of Denjoy and Wolff which asserts that if F € Hol(A) has no interior fixed point in A, then the sequence {Fn{z)}™=1 converges to a unique boundary point r G dA for all z G A. This result has given a powerful thrust to the study of the asymptotic behavior of iterates in different situations. In the next section we will describe some generalizations of the DenjoyWolff Theorem as well as point out its crucial connections with the classical Julia Lemma, the Schwarz-Wolff boundary theorem, and geometric function theory in complex spaces.

Chapter 5

The Denjoy-Wolff Fixed Point Theory

Theorem 5.1 (Denjoy-Wolff) Let A be the open unit disk in the complex plane C. If F € Hol(A) is not the identity and is not an automorphism of A with exactly one fixed point in A, then there is a unique point a in the closed unit disk A such that the iterates {Fn}'^L1 of F converge to a, uniformly on compact subsets of A. Let D be a domain in a complex Banach space X, and F S Hol(£>). The main goal of this chapter is to discuss the behavior of the iterates of F in the spirit of Theorem 5.1. Definition 5.1 We say that a mapping F S Hol(P) is power convergent to a mapping h € Hol(Z>, X) if the sequence of iterates {-F"}£Li converges to h uniformly on each ball strictly inside T>. If ft = a is a constant mapping, then the point a S V will be called a locally uniformly attractive fixed point of F. Recall that for V = A an automorphism F € Aut(A) which has exactly one fixed point is called elliptic. So, the Denjoy-Wolff theorem asserts that F € Hol(A) is power convergent if and only if F is not an elliptic automorphism. 5.1 5.1.1

The One-Dimensional Case Iterates of holomorphic self-mappings of A with an interior fixed point

In this section we prove the Denjoy-Wolff theorem for the case when F € Hol(A) has an interior fixed point. 119

120

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

The Schwarz-Pick inequality provides us with information regarding the invariant behavior of a holomorphic self-mapping of a domain V when F € Hol(2?) has an interior fixed point in T>. Namely, if p is a pseudometric on T> assigned to 2} by a Schwarz-Pick system, then F(Q(a)) C Q(F(a)),

(5.1)

where Cl(a) is any />-ball centered at a. Consequently, Fn(fl(a)) C n(Fn(a))

(5.2)

for all n = 0,1,2, In particular, for the one-dimensional case where T> = A is the open unit disk of the complex plane C, using a Mobius transformation, one can see that the p-ball Q of radius r (= fir(a)) can be written in the form n r (a) = {w G A : \m-a(w)\ < tanhr},

0 < r < oo.

(5.3)

Solving this inequality we get that, in fact, the p-ball ilr(a) is the disk Qr(a) = {w G A : \w - sa\ < dt},

(5.4)

where d = tanhr, S=

T^W

(5-5)

IHW

Thus we have that for each F € Hol(A), F(fi r (a))C[) r (F(«)),

r €(0,(50),

a e A,

(5.6)

i.e., F maps the disk Qr(a) centered at sa into the disk Qr(F(a)) centered at sF(a) with the same radius td. However, for a fixed d G (0,1) this radius tends to zero when a tends to the boundary. We consider this case later. The next question which naturally follows is: does the sequence {Fn(z)} converge to a fixed point a of F for each z G A? The answer is "no" if F is an elliptic automorphism. But for the other cases the answer is affirmative. Proposition 5.1 Suppose that F G Hol(A) is not an elliptic automorphism and has a fixed point a G A. Then the iterates Fn of the mapping F converge uniformly on compact subsets of A to a holomorphic mapping tp G Hol(A). Moreover, if F is not the identity, then

121

The Denjoy-Wolf Fixed Point Theory

Proof. If F is not an elliptic automorphism of A and it has a fixed point a in A, then the Schwarz-Pick Lemma (Proposition 1.1.3) implies that \F'(a)\ < 1.

(5.7)

Since F'(z) is continuous in A there is a disk A(a, e) C A centered at a with radius e > 0 such that \F'{z)\ < 1

(5.8)

for all z e A(a,e), the closure of A(a,e). In turn, (5.8) implies that F satisfies the Lipcshitz condition (5.9)

\F(z) - F(w)\ < q\z - w\,

where q = max{\F'(z)\, z 6 A(a, e)}. In addition, from (5.9) we have that F maps A(a, e) into itself. So, F is a g-contractive self-mapping of A(a, e), and it follows from the Banach Fixed Point Theorem that {Fn(z)} converges to a for all z 6 A(a,e). Using the Vitali property, we get our assertion. D 5.1.2

Iterates of holomorphic self-mappings interior fixed point

of A with no

By several technical transformations one can show that

nP(o) = jiu £ A : j ^ 0 - < K} ,

(5-10)

where d = tanhr and K = i = ^ " j , (see formulas (5.4) and (5.5)). Now suppose that \a\ = 1. For an arbitrary K > 0 define the domain V(a,K) by the same formula as (5.10):

V(a,K) = jz e A : ! i f ^ £
(5.11)

It is not difficult to see that T>(a,K) is also a disk in A centered at the point j — • a e A with radius j ^ < 1, i.e.,

%i q ={ 2 £ A:| Z - I La|<^ I }.

(5.12)

This disk is internally tangent to the boundary of A at the point a and it is called a horocycle.

h

The following assertion establishes a property of the horocycles T>(a, K), with respect to the family Hol(A), which is analogous to the property of the domains Clr(a). Theorem 5.2 (Julia's lemma) Let F e Hol(A) and let a e dA be a unimodular point. Suppose that there exists a sequence {an} c A converging to a such that the limits

a=limlH£M

(6.13)

6 = lim F{an)

(5.14)

an->a. 1 - \an\

and n—*oo

exist. Then the following inequality holds for each z G A ; |i-5F(*)| 3 1 - \F(z)\> Proof.

a

|i-a«|» i - |*|»

(5-15)

It is clear that |6| = 1. For z,w G A we define the function

(5.16) It is a simple exercise to show that the Schwarz-Pick Inequality is equivalent to (5.17)

a(z,w)
(5.18)

o-{z,an)
l

°-y; •

The following assertion obtained by J. Wolff [Wolff (1926c)] (see also [Wolff (1926a)] and [Wolff (1926b)]) is often called the Wolff-Schwarz Boundary Lemma.

The Denjoy-Wolf Fixed Point Theory

123

Theorem 5.3 (Wolff) Let F € Hol(A) have nofixedpoint in A. Then there is a unique unimodular point a e 9A such that for each K > 0, the horocycle

V(a, K) = {* e A : ^ - g £ < if} ,

(5.20)

internally tangent to dA at a, is F-invariant. Moreover, there is a number a 6 (0,1] such that Fn(V(a, K)) C V{a, anK) for alln =

(5.21)

0,1,2,....

Proof. It is sufficient to show that there is a sequence {dnY^Li C A converging to a, which satisfies the conditions of Julia's lemma, i.e., the limits a =lunilM

(5.22)

on->» 1 — \an\ and lim F(an) = a exist. Moreover, we will show that a < 1. Then n—*oo

our assertion will be seen to be a consequence of Julia's lemma. Take an arbitrary positive sequence {rn} increasing to 1 and consider the mappings Fn = rnF. It is clear that Fn is continuous on A rn = {z £ A : \z\ < rn} and that it maps this set into itself. Thus Fn has a fixed point an G A rn C A. Passing to a subsequence, we may assume that {an} converges to a point a € A. If a € A, then, by continuity, we have a = lim an = lim rnFn(a) = F(a), (5.23) n—»oo

n—>oo

contradicting the assumption that F has no fixed point in A. Consequently, \a\ = 1. In addition,

idSsf«_lziwsl 1 - \dn\

(5.24)

1 - \an\

for all n € N. Once again, perhaps by passing to a subsequence, we conclude that a in (5.22) exists and is less than or equal to 1. It is clear that F(an) — ^~an converges to a and by inequality (5.15) of Julia's Lemma, we obtain 1 - \F(z)\*

S

1 - |*|* '

(b-2b)

124

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

which implies (5.21).

•

Definition 5.2 Given a point a G dA and a real number k > 1, a nontangential approach region at a is the set T{a,k)

= {z£ A : \ z - a \ < jfc(l - | z | ) } .

(5.26)

Definition 5.3 Given a function h G Hol(A,C), we say that h has the angular limit L at a boundary point a G dA if h(z) —> L as z —> a, z G F(a,/c) for each fc > 1. We write in this case L = Z lim

ft(z).

z—>a

(5.27)

Remark 5.1 /£ is easy to see that h has the angular limit L at a point a G dA if and only if f(z) —> L as z —> a for each angle region S — Iz G A : | arg(l - az)| < /?, |z - a| < r, /3G(0,|),

rG(0,2cos^)},

(5.28)

which is a sector in A bounded between two straight lines in A that meet at a and are symmetric about the radius to a. This set is usually called a Stolz angle at a. Definition 5.4 A mapping F G Hol(A) is said to have an angular derivative at a G dA if the angular limit Z. lim

F(2)

~ b := ZF'(a)

z~*a z - a

(5.29)

exists for some b £ dA. The following essential contribution to the Denjoy-Wolff Theory, which is a complement of the Julia Lemma, was made by C. Caratheodory [Caratheodory (1929)] and [Caratheodory (1954)]. Theorem 5.4 (Julia-Caratheodory theorem) Let F G Hol(A) and a G dA. The following are equivalent: (i) S = S(F) = liminf ^Luf^ < °°> where the limit is taken unrestrictedly in A. (it) F has a finite angular derivative at a. (in) F has the angular limit b G dA and Z lim F'(z) = Sab. z—>a

125

The Denjoy-Wolf Fixed Point Theory

We will prove this theorem in the next section in a more general setting. An important consequence of this theorem and Theorems 5.2 and 5.3 is the following version of the so-called Julia-Wolff-Caratheodory theorem (c/. [Shapiro (1993)] and [Cowen and MacCluer (1995)]). Theorem 5.5

Let F £ Hol(A). The following are equivalent;

(i) F has no fixed point in A. (ii) There is a boundary point a £ dA such that 0 < lim F'(z) < 1. z—*a

Such a point a £ dA, which satisfies the conclusion of Theorem 5.3 of Wolff, will be called a sink point of F on dA (or Wolff's point). A mapping F £ Hol(A) with a sink point a 6 dA is said to be of hyperbolic type if the number a in (5.22) (Julia's number) is strictly less than 1. Otherwise (a — 1) the mapping F is said to be of parabolic type. Of course, if 0 < a < 1, then F is power convergent. The question is whether this point is also attractive when a = 1. The affirmative answer to this question is given in the next assertion, following Wolff and Denjoy [Wolff (1926a); Wolff (1926b); Wolff (1926c)], [Denjoy (1926)]. Theorem 5.6 If F £ Hol(A) has no fixed points in A, then there is a unique unimodular point a £ dA which is a sink point of F and the iterates {Fn} converge locally uniformly on A to a constant mapping ip(z) = a. Proof. If F £ Hol(A) is of parabolic type, and an automorphism of A, then the assertion is an exercise requiring just simple calculations. Thus we may assume that F £ Aut(A). Since {Fn} is a normal family, there is a subsequence {Fn->} which converges to a mapping

= lim Fn'+1 = tp. j—too

j—>oo

(5.30)

Since

126

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

fixed point in A. Hence h is the identity. At the same time, g o F = lim Fq* oF= j—>oo

lim FPi =1=

j —»oo

lim F o Fqi = F og.

j—»oo

(5.31)

Thus, i*1 is an automorphism. Having reached a contradiction, we conclude that (fi(z) = a is a constant after all. Now, if there is a subsequence {nj} C N such that {Fni(z)} converges for z e A to a constant 6 6 A different from a, then one can find K > 0 such that the closure of the horocycle T>(a, K) does not contain b. Since Fn^(V(a,K)) C V(a,K), it is impossible that for z € V(a,K), the sequence {Fn'{z)} converges to b. Hence {Fn(z)} converges to a for any z £ A. • Thus the classical Denjoy-Wolff theorem is, in fact, a summary of the following three assertions due to Denjoy, Wolff and Julia. Each one of them has been extended to different situations. • If F € Hol(A) is not an automorphism of A and has a fixed point c in A, then this point is unique in A, and the sequence {Fn}n°=l converges to c, uniformly on compact subsets of A. • (The Wolff-Schwarz Lemma) If F e Hol(A) has no fixed point in A, then there is a unique unimodular point a S dA such that every disk T>a in A, internally tangent to <9A at a, is F-invariant, i.e., F{Va) C Va.

(5.32)

• If F 6 Hol(A) has no fixed point in A, then there is a unique unimodular point b G dA such that the sequence {Fn}^=1 converges to b, uniformly on compact subsets of A. The limit point of the sequence {Fn(z)}'^L1, z £ A, will be called the Denjoy-Wolff point of F. The point a and the point b in the last two assertions are, of course, one and the same. In other words, the sink point of F is also the Denjoy-Wolff point of F. However, this is not always the case in higher dimensional situations. In fact, there are many situations in the higher dimensional case when a holomorphic fixed point free mapping has a sink point, but is not power convergent (see the next section). Returning to the one-dimensional case, we refer the reader to the paper by R. Burckel [Burckel (1981)] and to the books [Shapiro (1993)] and

127

The Denjoy-Wolf Fixed Point Theory

[Cowen and MacCluer (1995)] for a modern interpretation of the DenjoyWolff Theorem and its applications. Here we only mention the following observation concerning this case. Remark 5.2 By using the Schwarz Lemma, Proposition 5.1 can be rephrased in the following manner: • Let F € Hol(A) have afixedpoint a £ A. IfF is not the identity, then F is power convergent if and only if \F'(a)\ < 1. It turns out that by using the notion of the derivative and its spectral properties, one can also study power convergent mappings in higher dimensional spaces. See Sections 5.4 and 5.5 below. We are now in a position to formulate several generalizations of Theorems 5.2 and 5.3. 5.2

The Unit Hilbert Ball

Let if be a complex Hilbert space with the inner product (•, •), and let B be the open unit ball in H. The following generalization of the Wolff-Schwarz Boundary Lemma (Theorem 5.3) is due to K. Goebel [Goebel (1982b)]. For the finite dimensional case, H = C n , this result was independently obtained by B. MacCluer [MacCluer (1983)] and G. Chen [Chen (1984)]. Theorem 5.7 IfF € Hol(B) has nofixedpoint, then there exists a unique point a e dB such that for each 0 < R < oo, the set E(a, R) = \x 6 B : | 1 ~ ^ a '

I

! ~~ INI

< R)

(5.33)

J

is F-invariant. Geometrically, the set E(a, R) is an ellipsoid the closure of which intersects the unit sphere dM at the point a. It is a natural analogue of the horocycle P(a, R). As a matter of fact, Theorem 5.7 holds in a more general setting (cf. Theorem 25.2 in [Goebel and Reich (1984)]). Theorem 5.8 Let p be the Poincare hyperbolic metric on B, and let F : B —> B be a p-nonexpansive mapping. If F is fixed point free, then there is a unique boundary point a e 9 B such that all the ellipsoids E(a, k), k > 0,

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

are F-invariant.

In other words, (5.34)

<Pa(F{x)) < if>a(x), where _ |l-(a,a)|2 ^

a W

~

i _ ||_i|2

•

(i).6b)

To prove this theorem we use the following generalized version of Julia's Lemma. Lemma 5.1 Let F : B —> B be a p-nonexpansive self-mapping of M. Suppose that for some sequence {an}'^L1 c B which converges to a boundary point a s dB, the following conditions hold: 6{F)

= lim i f J I ^ H < oo

n-oo l - | | a n | | and b = lim^oo F(an) € 3 1 . Then for each xeM,

(5.36)

(5.37)


Since F is /o-nonexpansive, we have p(F(x),F(an))
(5.38)

a(F(x),F(an))>o-(x,an),

(5.39)

or, equivalently,

where

(5.40) x,yeM. Writing the latter inequality explicitly we get

(1 - \\F(x)f

)(1 - \\F(an)f)

\l-(F(x),F(an))\2

>

( 1 - N | 2 ) ( l - Hanil2) |l-(^,an)|2

(5.41)

or

ll-(F(*),F(a n ))l 2 l-|lf(on)|| a \l-(x,an)\2 i-\\F(x)\\* ~ l - K I I 2 " i-INI 2 •

(

^

129

The Denjoy-Wolf Fixed Point Theory

Proof of Theorem 5.8 Let F : B —>IB be a p-nonexpansive self-mapping of I . Consider a mapping Ft : B -> B denned by Ft = tF, 0 < t < 1. We have />(#(:«;),Ft(i/)) < tp(F(x),F(y)) < tp(x,y).

(5.43)

So, for each t G [0,1) the mapping Ft : B —> B is a strict contraction with respect to the metric p. Hence it has a unique fixed point xt = Ft(xt). In addition, xt - F{xt) = Ft(xt)

- F{xt) = (t- l)F(xt).

(5.44)

So ||xt — -F(a;t)|| —> 0 when t tends to 1. Thus, if F has no fixed point in B we must have that lim_ ||a;t|| = 1.

(5.45)

On the other hand, since B is weakly compact, there is a sequence tn —> 1~ such that {xtn} weakly converges to an element a e B. But if we assume that ||a|| < 1, then we have p(F(xtn),F(a))
(5.46)


(5.47)

or, equivalently,

where a(x v)

- (i-INI2)(i-llg|la)

|i-(z,y>|2 Explicitly the latter inequality implies a{x>y)-

(5 48)

•

l-4ll^H 2 .(l-HI 2 )ll-^K,F(a))l 2 ~ 1-IKII 2 - (l-||F(o)P)|l-|2 •

(5"48)

^

y ;

Letting n —> oo, we get a(a,F(a)) > 1. Hence, a = F(a). This contradiction shows that a must lie on the boundary of B, i.e., \\a\\ = 1. It follows that the convergence of Xtn to a is, in fact, strong, that is, \\xtn — a|| —»• 0 as n —> oo.

(5.50)

In addition, the strong limit lim F{xtn) = lim — xtn = a

n—>oo

n—+oo t n

(5.51)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and (by passing to a subsequence, if necessary)

(5.52) Setting an = xtn and a = b in the above version of Julia's Lemma, we obtain the required inequality: <Pa(F{x)) < <pa(x).

(5.53)

Finally, we claim that a point a G <9B satisfying <pa(F(x)) < (fa(x) must be unique. Indeed, assume that there is b G dM such that ipb(F(x)) < 0 (large enough) such that T> := E(a, k) D E(b, k) ^ 0 and lies strictly inside B. Then for all n = 0,1,2 . . . , Fn(V) C V. Hence for a point x G V, the sequence {Fn(x)} is bounded away from the boundary of B. It follows that F must have a fixed point in B. A contradiction. Theorem 5.8 is proved. D Now Theorem 4.10 (see Chapter 4) follows immediately: Corollary 5.1 Let F : B —» B be a p-nonexpansive self-mapping If F has a continuous extension to B, then it has a fixed point a Moreover, if F has no interior fixed point in B, then its sink (Wolff's) a must be a boundary fixed point of F. In addition, the approximating {a;t}o
o/B. G IB. point curve (5.54)

converges to a as t goes to 1. R e m a r k 5.3 As a matter of fact, it can be shown (see [Goebel and Reich (1982)], [Goebel and Reich (1984)] and [Shafrir (1992b)]) that for each y G B the approximating curve xt = {l-

t)y + tF(xt)

(5.55)

is well defined and converges to a fixed point of F. If F has no interior fixed point in B, then for each j / £ l , this curve converges to the same (Wolff's) sink boundary point a G dM. Now the question is whether the sink point a in Theorem 5.7 is also the Denjoy-Wolff point of F, i.e., is it attractive?

The Denjoy-Wolf Fixed Point Theory

131

For the finite dimensional case (B is then the open Euclidean ball in H = Cn) and when F £ Hol(B) the affirmative answer was given by B. MacCluer [MacCluer (1983)]. Theorem 5.9 Let H = Cn be a Euclidean complex space (finite dimensional Hilbert space) and let B be the open unit ball in H. If F e Hol(B) has no fixed point in B, then there is a point a of norm I (a € dM) such that the iterates Fn(z) converge to a for all z £ B, uniformly on compact subsets ofM. For infinite dimensional Hilbert balls, A. Stachura [Stachura (1985)] has constructed a counterexample which shows that this convergence result fails even for biholomorphic self-mappings. Nevertheless, some restrictions on a mapping from Hol(B) lead to a generalization of Theorem 5.6. The following result was obtained by C.-H. Chu and P. Mellon [Chu and Mellon (1997)]. Theorem 5.10 Let B be the open unit ball in a Hilbert space H, and let F € Hol(B) be a compact mapping with no fixed point in B. Then the sink point a in Theorem 5.7 is attractive, i.e., the sequence {Fn} of iterates of F converges locally uniformly on B to the constant mapping taking the value a. Although this theorem contains Theorem 5.6 (as well as the abovementioned finite-dimensional result of B. MacCluer), it does not apply to the automorphisms of B because of the compactness restriction. In this connection we have to mention a result by T. L. Hayden and T. J. Suffridge [Hayden and Suffridge (1971)] which complements our information (see also [Suffridge (1974)]). Theorem 5.11 Let B be as above, and let F E Hol(B) be an automorphism of B with exactly two fixed points on dM. Then one of them is an attractive sink point of F. Remark 5.4 It is known that each automorphism o / B can be extended to an automorphism of B and that it has a fixed point in B (see [Hayden and Suffridge (1971)]). Moreover, if it does not have a fixed point in M, then it has one or two fixed points on the boundary. For holomorphic self-mappings of the Hilbert ball the following versions of the Julia-Caratheodory and the Julia-Wolff-Caratheodory theorems are very important complements to the Denjoy-Wolff theorem.

132

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

We need some additional notions which are well-known in the finitedimensional case (see, for example, [Rudin (1980)]). Definition 5.5 A curve A : [0,1) —> B is said to be asymptotically normal at a point T € B if

(5.56) M ^ < M < o o ,

0<.
(5.57)

where X(s) is the ortogonal projection of A(s) onto the complex line through 0 and T: (5.58)

X(S) = (A(S),T)T.

Let h be a holomorphic function on B with values in the complex plane. We say that h has a restricted limit L at r € dB if h has the limit L along every curve which is asymptotically normal at r. Definition 5.6 Let / : B —> H be a holomorphic mapping on B, and let r € dB. We say that / has a finite angular derivative at r if for some element y C B, the function / i : B —» C defined by h{X}-

(5.59)

l-(x,r) (5>59)

has a finite restricted limit at r. We denote this limit

Zf'(r).

Theorem 5.12 Let B be the open unit ball in the finite dimensional Hilbert space H = Cn, let F be a holomorphic self-mapping of B, and let a be a boundary point. The following are equivalent: (i) S(F) = lim inf ~|}.AyH < oo, where the limit is taken as x approaches a unrestrictedly in B. (ii) F has finite angular derivative at a; (Hi) F has a restricted limit b e 5B and (F'(x)a,b) has a restricted limit K ata£ dB. Moreover, if one of these conditions holds, then 5(F) = K. We will call the number S(F) the Julia number of F at a. Theorem 5.13 Let B be the open unit ball in the finite dimensional Hilbert space H = Cn, and let F be a holomorphic self-mapping ofM with

133

The Denjoy-Wolf Fixed Point Theory

no fixed point in B. Then there is a unique boundary point a € 3B such that

0 < S(F) = flim inf 1 ~ l l ^ ( f l l > ) - Z F » < 1. Y

x—>a

1 — ||x|| /

(5.60)

Remark 5.5 As a matter of fact, a similar assertion also holds for the infinite dimensional case if we replace the angular derivative by the so-called radial derivative of F at its boundary sink point

T F'{a)=j^_

^ y ^ .

(5.6D

Moreover, rather surprisingly, this derivative turns out to exist even if F is not necessarily holomorphic, but only p-nonexpansive. Theorem 5.14 Let B be the open unit ball in a complex Hilbert space H and let F : B —> B be a p-nonexpansive self-mapping of B with no fixed point in B. Assume that a € dM is the sink point of F and 5 = 6(F) is the Julia number at a. Then the radial derivative T F>{a) l= l i m r—»1~

l-WaW

(5.62)

1— T

exists and equals 8. Proof.

First, by the generalized Julia's Lemma 5.1, we have |l-(F(ra),a)l2 h

(1-rf

1-r

(5.63)

This implies that lim_(F(ro),o) = l.

(5.64)

Now we calculate l-|(F(ra),a))|

(l-|(F(ra),q)|)2

1-r

{l-r)(\-\{F(Ta),a)\)

<

H W f

-1-|^(~).->|2

l + |
r

'

( ] (5.63)

134

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Therefore it follows from (5.63) that l

l - | < F1(-r ra )1+r ,a)| ' '

+

|(F(m),q)|

Thus, by using (5.64), we obtain lim

r—i-

l-\(F{ra),a)\ ' \ v h / [ < S.

(5.67)

1 - r

v

'

On the other hand,

1-II^MII < l-KF(ra),a)| _ 1-r

(5.68)

1-r

Since ^hminf1-"^", x-»a 1 — ||x||

(5.69)

where the limit is taken when x approaches a unrestrictedly, we get that hm

r—i-

l-||F(nO|| 1-r

=

n m l-\{F{ra),a)\ r->i1-r

=&

(5.70)

Returning to inequality (5.63), we can rewrite it as follows:

|i-<W,,LJ£WE.

( , 7I)

Letting r tend to 1~ and using (5.70), we get |l-(F(ra),o)|

lim •!

i—»i-

But again, |l — (F(ra),a)\ by (5.70) and (5.72),

N V

;

1— r

'

/[

<<5.

> 1 — |{F(ra),a)| and therefore we have,

aJi-W,.)!^

r-»i-

(5.72)

1— r

(5.73)

Combining now (5.70) and (5.73), we get

ll-(F(ra), a )| r^i- l-|(F(ra),a)|

=

(5.74)

The Denjoy-Wolf Fixed Point Theory

135

It follows that arg(l — (F(ra),a)) tends to 0 as r tends to 1~. Thus we obtain I h n r-i-

- ^ ' ^ . 1— r

1

The theorem is proved.

(5.75) •

Now, using the Lindelof principle (see [Rudin (1980)]) and Theorem 5.14, we obtain Theorem 5.13 as well as Theorem 5.12 by applying an appropriate rotation of the unit ball. Combining now Theorem 5.14 with the proof of Theorem 5.8, we obtain a stronger version of the latter theorem; it is a generalization of the JuliaWolff-Caratheodory theorem. Theorem 5.15 Let F : B —» B be a p-nonexpansive self-mapping o / B . If F has no fixed point in B, then there is a unique point a s dM such that fa(Fn(x))

< 6n<pa(x),

(5.76)

where

0 « - r-»ito I 1 '^"'•"I-Umbf •-'??»
(5.77)

is the radial derivative of F at this point a.

Definition 5.7

A mapping F s 7VP(B) is said to be of hyperbolic type

if it is fixed point free and its radial derivative at its sink (boundary Wolff's) point is strictly less than 1. Corollary 5.2 Each p-nonexpansive mapping of hyperbolic type is power convergent to its boundary sink point.

5.3

Convex Domains in C n

In 1941 M. H. Heins [Heins (1941)] extended the Denjoy-Wolff Theorem to a finitely connected domain bounded by Jordan curves in C. His approach is specific to the one-dimensional case. Another look at the Denjoy-Wolff Theorem is provided by a useful result of P. Yang [Yang (1978)] concerning a characterization of the horocycle in

136

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

terms of the Poincare hyperbolic metric in A. More precisely, he established the following formula: Urn [p(X, /i) - p(0, /x)] = | log '* ~ ^

.

(5.78)

So, in these terms the horocycle T>a in A can be described by the formula

V(a,R) = {z € A : lim (p(z,u) - p(0,w)) < i logfl).

(5.79)

Since a hyperbolic metric can be defined in each bounded domain in C n , one can try to extend this formula and use it as the definition of the horosphere in a domain in C". Unfortunately, in general the limit in (5.79) does not exist. Therefore, the new idea of M. Abate [Abate (1988a)] was to study two kinds of horospheres. More precisely, he defined the small horosphere Ezo(x, R) of center x, pole z0 and radius R by the formula

EZ0(x,R) = \z € V : \imsup[Kv(z,w)

- KT>(ZO,W)} < - logR,}, (5.80)

and the big horosphere of center x, pole ZQ and radius R by the formula

FZ0(x,R) = {z e V : limmf [Kv(z,w) - Kv(z0,w)) < i logi?},(5.81) where D is a bounded domain in C" and Kx> is its Kobayashi metric. For the Euclidean ball in C n , EZo(x,R) = FZo(x,R) (see [Abate (1988a); Mercer (1992); Mercer (1993)]). Thus each assertion which states for a domain V in C n the existence of a point a e &D such that fn(Ez(a,R))cFz(a,R)

(5.82)

for all z G T>, R > 0, / e Hol(£>) and n = 1,2,... is a generalization of Wolff's Theorem 5.3. This is true, for example, for a bounded convex domain in C n . A more general result was established by M. Abate in another work [Abate (1989b)]. Theorem 5.16 Let V be a bounded complete hyperbolic domain with a simple boundary, and let z e V. Suppose that for some f € Hol(Z?) the sequence of iterates {/"} is compactly divergent. Then there is a £ &D such that for all z €T>,R> 0 and n = 1,2,... the inclusion (5.82) holds.

The Denjoy-Wolf Fixed Point Theory

137

In fact, the assumptions of Theorem 5.16 are not sufficient to ensure a convergence result. Indeed, in order to generalize the notion of a sink point we give the following definition. Let V be a domain in a Banach space X and let F G Hol(P). We will say that a point x G dT> is a boundary sink point for F if there exist two sets of neighborhoods {Ua} and {Va}, a £ A {& directed set), in V such that the following conditions hold: (i) (ii) (iii) (iv) (v)

Ua cVa C P; xeUa; HaeAUa = naeAVa = 0; F(Ua) C Va; For ax < a2, ax,a2 G A, Ua2 C Uai and Va2 C Vai.

The following example is due to C.-H. Chu and P. Mellon [Chu and Mellon (1997)]. Example 5.1 Let V be the open unit bidisk in C 2 , i.e., V = A x A, and let h € Hol(A) be a fixed point free mapping with the Denjoy-Wolff point C, € dA. Consider the mapping F G Hol(P) denned as follows: F(z, w) = (eilfiz, h(w)),

(5.83)

where 0 < (p < 2TT. It is clear that if F has a sink point a £ dV, then a — (0, £) G dT>. At the same time, the sequence of iterates {Fn} does not converge to any boundary point of V. Nevertheless, the convergence result does hold for bounded strongly convex C 2 domains, and for strongly pseudo-convex hyperbolic domains with a C 2 boundary [Abate (1988a); Abate (1988b)]. To generalize these facts, M. Abate denned the following notion [Abate (1989b)]: A domain D c c C is said to be F-convex at x G dV if for all z G V and R > 0, Fz(x,R)DdV={x}.

(5.84)

The domain V is said to be F-convex if (5.84) holds for each x G &D. His result is the following one: Theorem 5.17 Suppose that in addition to the assumptions of Theorem 5.16 the domain V is F-convex. Then there is a point a G dV which is attractive for f.

138

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Certain types of domains are known to be F-convex. We mention, for example, strictly pseudo-convex domains with a C 2 boundary, and the so-called m-convex domains (see [Abate (1988a); Abate (1988b); Mellon (1996)]).

5.4

Domains in Banach Space.

Unfortunately, our knowledge does not include positive results regarding extensions of Theorems 5.3 and 5.6 to general Banach spaces. Indeed, a simple example shows that there is a situation where even a sink point does not exist. Example 5.2 Consider the Banach space CQ of all complex sequences x — (x\,X2, • • •,xn,...) such that xn —> 0 as n tends to infinity, with the max norm, and the affine (hence holomorphic) mapping F : Co —> Co defined by F(x) = (a,X\,X2-- •), where o ^ O , \a\ < 1. It is clear that F maps the open unit ball V of Co into itself and that it is continuous on V. If F had a sink point on dT>, then it would necessarily be a fixed point of F in T>. But F has no fixed point there. However, the question is still open for reflexive Banach spaces. Moreover, if V has a strictly convex boundary and F is compact, then the convergence result holds [Kapeluszny et al. (1999a) and (1999b)]. Proposition 5.2 Let X be a strictly convex Banach space, and let F be a holomorphic compact self-mapping of the open unit ball T> of X without a fixed point in V. Then F is power convergent on T>. The situation is more fully understood when F has a fixed point in the domain. From now on we will assume that T> is a bounded domain in a Banach space X and that F 6 Hol(2?) has an interior fixed point a £ T>. Of course, as simple examples show, one cannot expect that a is always an attractive point even if F is not an automorphism (consider Example 5.1 with h(w) — w2). Nevertheless, rephrasing the Denjoy-Wolff Theorem 5.6 in the form of Remark 5.2, one can point out several generalizations of this result. Proposition 5.3 ([Vesentini (1983); Vesentini (1985); Khatskevich and Shoikhet (1984)]). Let V be a bounded domain in X and let F € Hol(£>). Suppose that F has an interior fixed point a GT> and let F'(a) be its Frechet

The Denjoy-Wolf Fixed Point Theory

139

derivative at this point. Then a is a locally uniformly attractive point of F if and only if the spectral radius of F'(a) is strictly less than 1. Proof. The necessity of the assertion immediately follows from the Cauchy inequality if we note that (Fn)'(a) = [F'(aW

(5.85)

by the chain rule. To establish sufficiency, denote F'(a) by A. If p(A) < 1, then it follows from the Rutickii theorem that there is a norm || • ||i equivalent to the original norm of X such that ||A||i(= sup ||Aa;||i) < 1. Thus there is a V

IWIi=i

'

neighborhood (a ball) U CC V of the point a such that sup||F'(a;)||1=q
(5.86)

x€U

The mean-value theorem implies that for all x and y £U, \\F(x)-F(y)\\1
(5.87)

Now, by the Banach Fixed Point Principle, we get that the sequence { F " } ^ ! converges uniformly to a on U. Using the Vitali Theorem we obtain our assertion. • Remark 5.6 As a matter of fact, we will see below that this assertion is a consequence of a much more general assertion due to E. Vesentini (see Theorem 5.18). Combining Proposition 5.3 with the Earle-Hamilton [Earle and Hamilton (1970)] fixed point theorem, one can formulate a geometrical characterization of the attractive fixed point. Proposition 5.4 ([Khatskevich and Shoikhet (1984); Khatskevich and Shoikhet (1994a)]). Let V be a bounded domain in X and let F e Hol(X>). Then F has an attractive fixed point in V if and only if there exist a domain P C D and an integer n > 0 such that Fn maps V strictly inside itself. If a is an attractive fixed point of F, then by definition it is unique and F is power convergent. For P c C , the converse is also true: If F € Hol(I>) is not the identity and has an interior fixed point, and F is power convergent, then a is unique.

140

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

However, this is not always the case in higher dimensions. Indeed, the following example exhibits a function F, mapping the bidisk B 2 in C2 into itself, which is power convergent, but itsfixedpoint set consists of an infinite number of points (actually it is a submanifold of V of dimension 1). Example 5.3 Let V = B2 = A x A and F = (zi, § {z\ + z\)). Then F is a power convergent mapping, but all the points of the form (zi, 1 - ^1 — zi) are fixed points of F. A full description of such a situation was obtained by E. Vesentini [Vesentini (1983); Vesentini (1985)]. Theorem 5.18 (Vesentini) Let V be a bounded convex domain in a Banach space X, and let F belong to Hol(P). Suppose that F has a fixed point a e V, and denote the spectrum of the linear operator F'(a) by a(F'(a)). Then F is power convergent if and only if the following two conditions hold: (i) a(F'(a)) C AU{1}; and (ii) I is a pole of the resolvent of F'(a) of order at most 1. Comments.

Condition (ii) is actually equivalent to the condition Ker(J - F'{a)) 0 Im(/ - F'(a)) = X

(5.88)

(see, for example, [Gohberg and Markus (I960)] and [Lyubich and Zemanek (1994)]). It is also known that conditions (i) and (ii) are equivalent to F'(a) being power-convergent to a projection P onto Ker(/ — F'(a)). So, if R £ Hol(P) is the limit point of {Fn} under these conditions, then R = a is constant if and only if P = 0. In general, there is a neighborhood U of a such that R2 — Ron U, that is, R is an idempotent of the algebraic semigroup Hol(t/). In this case the mapping R is said to be a local holomorphic retraction, and its image R(U) is called a holomorphic retract of U. In our setting this means that FixF n U = R(U) is a submanifold of U tangent to Ker(/ - F'(a)) (see [Cartan (1986)]). This fact describes the general structure of the fixed point set. Actually, only condition (ii) or (5.88) is sufficient for FixF to be a local retract (see, for example, [Shoikhet (1986); Shoikhet (1993)], [Vigue (1986); Vigue (1991a)], [Mazet and Vigue (1991); Mazet and Vigue (1992)]). Moreover, if V is convex, FixF is a global retract of T>, and hence a connected complex submanifold of T>. To prove this assertion, as well as Theorem 5.18, we need the following lemma.

141

The Denjoy-Wolf Fixed Point Theory

Lemma 5.2 Let F be a holomorphic mapping defined on a neighborhood V of the origin and suppose F(0) = 0. Assume that all the iterates {Fn}^=0 are well defined on V and bounded, i.e., \\Fn{x)\\ < M < oo,

(5.89)

xeV.

Uf Ker(J - F'(0)) 0 Im(/ - F'(0)) = X,

(5.90)

then there is a neighborhood U C V such that F'ix(F) nU is a complex submanifold ofU tangent to Ker(7 — F'(0)). Moreover, there exists a holomorphic mapping * defined on U such that \I>2 = * and Fixtf (¥) = Fixu(F).

(5.91)

Proof. Let P be a linear projection onto Ker(7 — F'(0)) and Q its complement, i.e., P + Q = I. Let U\ be a neighborhood of the origin in the space PX, and consider any holomorphic mapping / defined on U\ with values in KerP (= Im(7 - F'(0)). We claim that if / satisfies the condition ||/(u)|| =o(||u||),

(5.92)

PF{u + /(«)) = u

(5.93)

then

for all u £ U\ such that u-\- f(u) S U. Indeed, fix such u G Ui and consider the vector-functions (5.94)

<^(A) = PFn(Xu + f(Xu))

with A e A = { A G C : |A| < l } . By our assumption the sequence {ipn} is uniformly bounded, i.e., |K(A)||<||P||.M,

„ = 0,1,2,...,

AeA.

(5.95)

Since ipn : A —> PX is holomorphic, one can represent it in the form fn(A) =
(5.96)

where hmrl £ X is the first nonvanishing coefficient in the power series expansion of
(5.97)

142

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Denoting S = F — F'(0), we get by a direct computation,
(5.98)

Similarly, by using an induction argument we obtain Y?n(A) = Xu + nhmiXmi + . . . . Thus we get that mn = mi and hmn = n • hmi, n = 1,2, other hand, it follows from the Cauchy inequalities that \\hmj

< \\P\\ • M.

(5.99) But on the (5.100)

This inequality shows that hmi must be zero, and therefore
(5.101)

Setting A = 1, we obtain
(5.102)

Since u is arbitrary, our claim is proved. Now consider the set FixF n U defined by the equation x = F(x),

x€U.

(5.103)

Using the projections P and Q, one can rewrite this equation in the form f u = PF(u + v) i \v = QF{u + v)

(5.104)

where u = Px and v = Qx. Since the operator I—QF'(0) is invertible on Im(7—.F'(O)), it follows by the implicit function theorem that the second equation of the latter system has a unique solution v = f(u) for all u sufficiently small. It is also easy to see that ||/(u)|| = O(||M||)- By using our previous claim we have that the first equation is satisfied with the substitution v = f(u), i.e., u = PF(u + f{u)).

(5.105)

In other words, all the solutions of the equation x = F(x) can be represented in the form x = u + f(u).

(5.106)

The Denjoy-Wolf Fixed Point Theory

143

Thus, shrinking U if necessary, we obtain that Fix(F) f~l U is a complex submanifold of U. Finally, if we define the mapping *:£/—» X by V(x) = Px + f(Px),

(5.107)

we get * 2 = * and Fix(#) n U = Fix(F) n U. This concludes our proof.D Proof of Theorem 5.18 First we note that by Lemma 5.2 and the Vitali property of holomorphic mappings in the topology of local uniform convergence over V, we can assume that V = U is a convex domain in X. Then there is a retraction ip : D —> Fix(F) which satisfies the condition 7poF = ip.

(5.108)

In addition, we have shown (see also the H. Cartan Theorem [Cartan (1986)]) that in a neighborhood U of the fixed point a of F we can find a local chart g : U —> V such that g{a) = 0 and such that gotpog~1 = P

(5.109)

is a linear projection. Now consider the mapping G = goFog~1,

(5.110)

defined on some neighborhood W of zero, together with its iterates Gn = goFnog~1. (Indeed, by the boundedness of {Fn} this sequence is uniformly Lipshitzian in some neighborhood of a. Hence, since Fn(a) = a,we can find a neighborhood W such that Fn(g~1(W)) C U.) We now have PG = got/jog-lgoFog-1

= gotjjog-1 = P.

(5.111)

In addition, G'(0) = g'{a) o F'(a) o [p'(a)]- 1 and therefore
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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

However, once again, a retraction onto the fixed point set can be obtained by this simple iteration method only if condition (i) holds, i.e., only if F'(a) has no spectrum points on the unit circle except, possibly, 1. This brings us to another interesting aspect of fixed point theory: the study of the holomorphic retracts of a domain. 5.5

Holomorphic Retracts and the Structure of the Fixed Point Sets.

Let, as above, X be a complex Banach space, and let T> be a domain (open connected subset) in X. Lemma 5.2 in the previous section provides us with information on the local structure of the fixed point set of a holomorphic self-mapping of T> under a certain condition of regularity. To elaborate on this result in more detail we give the following definition. Definition 5.8 Let F £ Hol(£>). A point a € Fix(F) is said to be quasi-regular if the following condition holds: (*)

K e r ( / - F ' ( G ) ) © I m ( / - F ' ( a ) ) = X.

(5.112)

If, in addition, Ker(7 — F'(a)) = {0}, i.e., the linear operator / — F'(a) is invertible, then we say that a is a regular fixed point of F. By the implicit function theorem, it is clear that a regular fixed point is an isolated point of the set Fix(F), and that in the case of a finitedimensional X each fixed point is quasi-regular (or, in particular, regular). Thus, in these terms Lemma 5.2 can be reformulated as follows: Theorem 5.19 Let V be a domain in a complex Banach space X, and let F € Hol(P) have a quasi-regularfixedpoint a e Fix(F) c V. If F(D) is bounded, then there is a neighborhood U C V with a £ U such that Fix(F) n U is a complex submanifold of U tangent to Ker(7 — F'(a)). If, in addition, V is a bounded convex domain, then it can be shown (see Theorems 5.22 and 5.23 below) that Fix(i?) is a connected subset of V. This result for finite dimensional Banach spaces was obtained by J.P. Vigue [Vigue (1986)] and for the general case by P. Mazet and J.P. Vigue [Mazet and Vigue (1991)] (see also [Shoikhet (1986)]) by using a retraction method.

The Denjoy-Wolf Fixed Point Theory

145

Earlier, W. Rudin [Rudin (1978)] (see also [Rudin (1980)]) established a surprising result regarding the global structure of holomorphic selfmappings of the Hilbert ball. See also [Goebel and Reich (1984), p. 121]. Theorem 5.20 Let B be the open unit ball in a complex Hilbert space H, and let F € Hol(B). Then Fix(F) is an affine submanifold ofM. Proof. First we note that by direct calculation one can easily verify that each automorphism of the unit ball translates an affine subset of B onto another affine subset of B. Therefore it is sufficient to prove the theorem under the assumption that Fix(F) contains the origin. In other words, we assume that F(0) = 0. We show that in this case Fix(F) = Fix(A) n B, where A is the linear operator defined by the Frechet derivative of F at the origin, i.e., A = F'(0). Indeed, it follows by the Schwatz Lemma that ||F(i)|| < ||x||,

xeB,

(5.113)

and \\A\\ < 1.

(5.114)

Further, one can write each x G B in the form x = ru, where u G dM and 0 < r < 1. Let A be the open unit disk in the complex plane C and consider the holomorphic function g on A defined by g(X) = (F(Xu),u),

(5.115)

where (•, •) denotes the inner product in H. Clearly, g(0) = 0 and g'(0) — (Aii.u). If now x (= ru) € Fix(F), then g(r) = r and, by the one-dimensional Schwarz Lemma, g(X) = A for all A G A. Consequently, g'(0) = (Au,u) = 1 which implies Au = u. Hence Ax = x. Conversely, assume that x (= ru) G Fix(A). Then again g'(0) = (Au, u) = 1 and, by the Schwarz Lemma , g(X) = A G A. In particular, g(r) (= (F(ru),u)) = r. On the other hand, we have that ||F(ru)|| < ||r||. Setting y = r - 1 F ( r u ) , we have ||y|| < 1 and (y,u) = 1. Therefore y = u. This means that F{ru) = ru, or, which is the same, F(x) = x. The theorem is proved. • The simple example from the previous section (Example 5.3) shows that this result does not hold even in C2 if we replace the Euclidian ball by the

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

polydisk (i.e., the unit ball of C2 in the Chebyshev norm). Nevertheless, as we have already mentioned, the global structure of the fixed point set can be described by using the retraction method and H. Cartan's theorem. Definition 5.9 A mapping

), i.e., ip2 = ip. In other words, if we let V = ) is a holomorphic retraction of T>, i.e., ) = Fix(t^). If now a £ V, then it follows by the chain rule that P = '(a) satisfies condition (5.112) of Definition 5.8. That is, a € V is a quasiregular fixed point of ending at these points, we get that 7 = tp(T) C V is also continuous with 7(a) = a and 7(6) = b. Thus we have proved the following assertion: Proposition 5.5 ([Cartan (1986)]) A holomorphic retract of a domain V in a complex Banach space is a connected complex analytic submanifold of V. In the context of the study of Fix(.F) = {x € T> : x = F(x)} one can ask several questions. The first one is: if T — Fix(F) ^ 0, when is it a holomorphic retract of V? This question in its turn leads to the second one: if T = Fix(F) is a holomorphic retract of T>, how can one determine a retraction ofT> onto T? The second question mentioned above is also of great interest. This is because the construction of a retraction onto T = Fhc(F) is equivalent to finding a method (explicit, implicit or approximate) for solving the equation x = F(x). A holomorphic retraction may not exist in general even in the one-dimensional case. Example 5.4 ([Mazet and Vigue (1991)]) The mapping F defined by F(z) = \ takes the annulus V = {z G C : \ < \z\ < 2} into itself and has two different fixed points z = 1 and z = — 1 in V.

The Denjoy-Wolf Fixed Point Theory

147

Even if the retraction onto Fix(F) exists, it need not be the limit of the iterates of the operator, even in the case of a linear operator A such that {0} ± Ker(J -A)± Ker(J - A)2. We will now mention several results concerning both these questions. Recall that a net {FJ}J&A C Hol(P, X) is said to converge to a mapping F £ 1101(1?, X) in the topology of locally uniform convergence over T> (or, briefly, T-converge) if for every ball B c c D , lim supllFjfc) - F(a;)|| = 0. "

j€Ax€B"

(5.116)

In this case we also write F = T-lim

Fj.

(5.117)

It is obvious that if F € Hol(P) is power convergent, then T = Fix(i?) ^ 0 and tp = T — lim Fn is a retraction onto T. So, by using a result due n—»oo

to J. J. Koliha [Koliha (1974)] Vesentini's theorem (Theorem 5.18) can be reformulated as follows. Theorem 5.21 Let V be a bounded convex domain in X, and let F G Hol(P) with T = Fix(F) ^ 0. If a € Fix(F) is a quasi-regularfixedpoint and A = F'(a), then F is power convergent to a retraction

T = Fix(ir) if and only if A is power convergent in the norm topology of L(X) to a projection P : X —> Ker(/ — ^4). Moreover, we will see below that if the domain V is bounded, then the set T consists of only quasi-regular (or regular) points. The converse, of course, is not true. To see this, consider, for example, a rotation F(x) = eiex, 0 < 9 < 2TT, of the open unit disk A in C. This mapping has a unique and regular fixed point (the origin) in A, but F is not power convergent. Remark 5.7 Let o-(A) denote the spectrum of the linear operator A : X —> X, and let A be the open unit disk in C. As above, assume that T = Fix(F) ^ 0 for some F € Hol(P) and that aeV. Setting A = F'(a) and using the Cauchy integral formula, the chain rule and the boundedness ofD, we see that the powers of A are uniformly bounded, i.e., \\An\\ < M < oo. This implies that o-(A) C A.

(5.118)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

It is clear that if in the setting of the above theorem cr(A) c A, then a G J- is a regular point. In addition, it was shown that such a point is the unique fixed point of F in V. As a matter of fact, as we will see in the sequel, for a bounded convex domain, the regularity of a point a G Fix(F) is already sufficient for its uniqueness, i.e., if a is regular, then Fix(F) = {a}. R e m a r k 5.8 Vesentini's theorem is also true for bounded domains which are not necessarily convex, but satisfy the following maximum modulus principle: For each f G Hol(£>, T>) such that f(D) D &D ^ 0, it follows that f(V) c &D. However, we will mainly consider convex domains in X because this is all that is needed in order to describe Fix(F) locally in each bounded T>. Indeed, it has been shown by P. Mazet [Mazet (1992)] that if V is bounded, then for each F G Hol(D) and each a G F\x(F), there is a convex neighborhood U
In addition, we need the convexity to consider different types of mean ergodic procedures. As a matter of fact, if the conditions of Theorem 5.20 hold for at least one point of T, then they hold for all the points of T. Moreover, if 1 G cr(A), then T contains infinitely many points because it is a retract of T>, hence a connected submanifold of V tangent to Ker(7 — A). We will see below that the latter fact is true whenever condition (5.112) holds, even if the spectrum a(A) contains other points on the boundary dA of the open unit disk A different from 1. But in this case, of course, by Vesentini's theorem, F is not power convergent and therefore the question of approximating its fixed points is still open. Nevertheless, if V is a bounded convex domain in X, and F G Hoi(T>) has at least one quasi-regular fixed point in V, then there is another mapping

) with Fix(F) = Fhc(. For the finite dimensional case this was established by J.-P Vigue [Vigue (1986); Vigue (1991a)], and in the general case by P. Mazet and J.-P Vigue [Mazet and Vigue (1991)]. They used nonlinear analogues of mean ergodic constructions. More precisely, they considered the Cesaro averages 1 "~1

Cn = - V Fk

(5.119)

The Denjoy-Wolf Fixed Point Theory

149

and proved the following results. Theorem 5.22 Let V be a bounded convex domain in X = Cn, and let F e Hol(X>) with T = Fix(F) ^ 0. Then there is a subsequence {Cnk} of {Cn} which T-converges to a power-convergent holomorphic mapping ip:T> -> P with Fix(. But a deficiency of Theorem 5.22 is that we do not know how to choose a convergent subsequence of {Cn} in order to approximatefixedpoints. Theorem 5.23 (which also covers the finite-dimensional case) would improve this situation if we could determine the minimal number p for which the mapping Cp defined by (5.119) is power convergent. We give below, inter alia, the simplest possible answer to this question, namely, p = 2. A partial generalization of Theorem 5.23 to unbounded domains was obtained by Do [Do (1992)] who has proved that if P is a convex hyperbolic domain in X, and F e Hol(P) is such that F(V) is contained in some compact convex subset of X with T = Fix(F) ^ 0, then this set T is a holomorphic retract of T>. (We recall, in passing, that a domain T> in a complex Banach space X is said to be hyperbolic if the Kobayashi pseudometric Kf) generates the relative topology of V in X (see Chapter 3.)) This result is only a partial generalization of Theorem 5.23 because of its compactness hypothesis. Indeed, according to a theorem of Krasnoselskii [Krasnoselskii and Zabreiko (1984)], F'(x) is compact for all x £ V and hence each point in Fix(F) is quasi-regular. However, we will show below that this compactness hypothesis is unnecessary. Moreover, we present a simple method for constructing a retraction onto Fix(F). This will also provide a proof of Theorem 5.23 [Reich and Shoikhet (1998c)]. Theorem 5.24 Let T> be a hyperbolic convex domain in X, and let F € Hol(P) be bounded on each subset which is strictly inside T>. (i) If T — Fix(F) contains at least one quasi-regular point a e V, then for

150

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

each A S (0,1), the averaged mapping FX = XI + {I- X)F

(5.120)

is power convergent. (ii) If for at least one A € (0,1) the mapping F\ defined by (5.120) is power convergent, then Fix(i r ) consists of quasi-regular points. First we observe that it is sufficient to establish this theorem for bounded domains. Indeed, if a is a fixed point of F (hence of F\), the hyperbolicity of T> implies the existence of a ball B(a,R) (with respect to the Kv pseudometric) centered at a which is bounded and strictly inside T>. Since F\ is nonexpansive with respect to Kx>, this ball is invariant under F\. Suppose now that (i) is established for that ball and let B(c,r) be any ball (with respect to the norm) which is strictly inside V. Then the family {F£} is bounded on the convex hull of B(a, R) UB(c, r) (which is also strictly inside T>), and therefore FA is power convergent on B{c,r) by the Vitali property. Proof of Theorem 5.24 0) Let a e Fix(F) be a quasi-regular point, and let A\ = XI + (1 - X)F'(a) for A € [0,1). It is clear that a £ Fix(F A ), where F\ is denned by (5.120) and Ax = (Fx)'{a),

Ae[0,l),

(5.121)

with Ao = A = F'(a). We intend to show that for each A G (0,1) the mapping F\ satisfies the conditions of Vesentini's Theorem. First we note that condition (*) (see formula (5.112) in Definition 5.8) is obvious because I-AX

= (1-X)A.

(5.122)

Now we must show that the set a(A\)\{l} lies inside the open unit disk A for each A G (0,1). Once again, the Cauchy integral formula shows that the operator A\ is power bounded. Suppose now that there exists C £ <9A fl o-{A\) and £ ^ 1. Then we have for such C, CI-Ax = (l-X)(tI-A),

(5.123)

where t=YZje

a{A).

(5.124)

The Denjoy-Wolf Fixed Point Theory

151

It is clear that \t\ > 1. But on the other hand, \t\ < 1, since a(A) C A (see (5.118)). So, |i| = 1 and we have, by (5.124), £ = A + (l-A)te8A.

(5.125)

But this is possible only if £ — t = 1. This contradiction proves our assertion. (ii) Now, if for some A 6 (0,1) the mapping F\ is power convergent, then it follows from the Cauchy inequalities that for each o € Fix(F) the operator A\ = (F\)'(a) is power convergent too. This fact in turn implies condition (*) for A\. Hence, it follows from (5.122) that a is quasi-regular. The proof is complete. D So, setting A = ^ in Theorem 5.24, we have that the Cesaro average (see (5.119)) C2 = \{I + F)

(5.126)

is power convergent, i.e., it is sufficient to takep = 2 in Theorem 5.23. As a matter of fact, it turns out that the Cesaro averages defined by (5.126) are power convergent for all p = 2,3 Moreover, we are able to show that all proper convex combinations of the iterates of F are also power convergent. We will describe below a general scheme for constructing a retraction onto Fix(F). A simple but very important consequence of Theorem 5.24 is the socalled Cartan's uniqueness theorem, which is a generalization of the second part of the classical Schwarz Lemma. Corollary 5.3 Let V be a hyperbolic convex domain in X, and let F € Hol(P) be a bounded on each subset which is strictly inside T>. If a € T> is a fixed point of F such that F'{a) = I, the identity operator on X, then F(x) = x for all x € V. Now we return to a general approach to find a retraction onto Fix(F). It follows by Theorem 5.21 that to construct a retraction onto Fix(i r ) by using a power convergent mapping, we must find a mapping $ £ Hoi (D) such that T = Fix(F) = Fix($) and for some point a € T the operator <3>'(a) is power convergent. Lemma 5.3 be such that

Let A be a bounded linear operator in X, and let Ao £ &(A) Ker(A07 - A) © Im(A0/ - A) = X.

(5.127)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Suppose that there exist a domain fi C C and a holomorphic function f defined in a neighborhood ClofQ, with the following properties: (a) U D a(A) and Ao € dCl; (b) f(Q) C A; (c) Ao is a simple root of the equation /(Ao) = 1;

(5.128)

(d) |/(A)| ^ 1 for all A S 90, A ^ Ao. Then the linear operator B — f{A) : X —» X defined by the formula I /(A)(A/ - A)~ld\,

B=TT-.

(5.129)

where T C 0 is a closed path around cr(A), is power convergent. Proof.

It follows from Dunford's Spectral Mapping Theorem that a{B) = f(a(A)).

(5.130)

Hence, conditions (a), (b) and (d) imply that a(B)\{l] c A.

(5.131)

Consider now the function 3(A) = [ l - / ( A ) ] ( A 0 - A ) - 1 .

(5.132)

Condition (c) implies that <7(A) is holomorphic in a neighborhood of the point Ao e dfl, and /(A o ) ^ 0. In addition, g(X) ^ 0 for all A £ a(A), by condition (d), and therefore the operator C = g(A), defined by the formula

C=^~. [ 0(A)(AJ - Ar'dX,

(5.133)

is invertible in X. Furthermore, it follows from the multiplicative property of the calculus of L(X)-valued functions denned by (5.129) and (5.133) (see [Rudin (1973)]) and by formula (5.132) that I - B = C(\0I - A) = (\0I - A)C.

(5.134)

This implies Ker(J - B) = Ker(A0/ - A) and Im(J - B) = Im(A 0 / - A). Now (5.127) and (5.128) imply that B is power convergent. The lemma is proved.

•

153

The Denjoy-Wolf Fixed Point Theory

Theorem 5.25 Let V be a hyperbolic convex domain in X, and let F e Hol(P) with F = Fix(F) ^ 0. If' T contains a quasi-regular point a € V, p

P

then each mapping $ of the form $ = JT akF , where ^ a* = 1 and fc=O fc=O

0 < afc y£ 1 for all k, is power convergent.

Proof. As in the proof of Theorem 5.24, we may assume that V is bounded. Let $ G Hol(P) be defined by p

(5.135)

$ = y£iakFk, fc=o V

where J2 ak = 1 and 0 < at ^ 1 for all A; = 0,1,.. .p. k=o Consider the holomorphic function (polynomial) / : A —» A defined by the following formula: p

f(\) = j2akXk>

AGA-

( 5 - 136 )

fc=0

It is clear that / satisfies conditions (a)-(d) of Lemma 5.3, with Q. = A, and therefore the operator B defined by (5.129) is power convergent to a projection onto Ker(7— A), where A = F'(a). But it follows from the chain rule that $'(a) = B = f(A). Hence, $ is power convergent onto Fix($) (see Remark 5.10). Furthermore, (5.135) implies that Fix(F) C Fix($). At the same time, these sets are connected submanifolds in T> tangent to the same subspace Ker(J — A). Therefore they coincide in V. This completes • the proof. Remark 5.10 combination

IfT> 3 0, then it can be shown that each infinite convex oo

$ = 5> f c F f c ,

(5.137)

fc=o oo

where Yl sk = 1 and 0 < afc ^ 1 for all k, belongs to Hol(2>) and is also fc=o

power convergent to a retraction onto Fix(.F). We now describe an implicit method for approximating fixed points of holomorphic mappings which has been used many times in the theory of nonexpansive mappings (see, for example, [Goebel and Reich (1984)]).

154

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Let V be a bounded convex domain in X, and let F belong to Hol(X>). For t G [0,1) and a fixed y £ V, consider the mapping $(x) = tF(x) + (1 — t)y, x G V. Since $ maps V strictly inside itself, the Earle-Hamilton theorem [Earle and Hamilton (1970)] implies that there exists a unique fixed point z = zt(y) G V of the mapping $. Moreover, z = T-

lim $ n .

(5.138)

n—>oo

The X-valued function zt{y), 0 < t < 1 is called an approximating curve. It was shown in [Kuczumow and Stachura (1990)] that if X = C™ and Fix(F) 7^ 0, then there is a sequence tn G (0,1), tn —> 1 such that for each y G V, the sequence ztn (y) converges to a fixed point of F in V. At the same time, changing our point of view, for each t G [0,1), zt = zt(y) holomorphically depends on y G T>, by (5.138). We denote this mapping by Tt. In other words, Tt is the unique solution of the nonlinear operator equation Tt=tFoTt + (l-t)I.

(5.139)

The mapping % belongs to Hol(2>) and it is easy to check that Fix(7i) = Fix(F)

t G (0,1).

(5.140)

For the Hilbert ball B (as we have already mentioned) the situation is as follows: The net {Tt} is convergent in B as t —» oo to a mapping ^ : B —> B. / / Fix(F) ^ 0, then \& is a holomorphic retraction onto Fix(F) (otherwise, $ is a constant a G SB which is the sink point of F). Also, since for each t > 0 the mapping {Tt} is a firmly holomorphic self-mapping ofM, the iterates {^n}^Li converge weakly to a holomorphic mapping ip : B —» B, which is a retraction onto Fix(F) if it is not empty. A similar situation occurs in general Banach spaces. Theorem 5.26 Let D be a bounded convex domain in a complex Banach space X, and let F G Hol(X>) with T = Fix(F) ^ 0. If F contains a quasi-regular point in V, then the mapping Tt defined by (5.139) is power convergent to a retraction onto J-'.

The Denjoy-Wolf Fixed Point Theory

Proof. Let aeT, operator equation

155

and let B = (Tt)'(a), t G (0,1). Then B satisfies the B = tAoB

+ (l-

t)I,

(5.141)

where A = F'(a). Setting r = ^ , we have B = [I + r(I - A)}'1.

(5.142)

Therefore, B can be defined by formula (5.129) with ^

l

+ ra-A)'

ASA.

(^43)

Since r > 0, this function maps Cl = A into itself and satisfies all the condition of Lemma 5.3. Thus, by this lemma, B is power convergent to a projection onto Ker(7 — A) and so Tt is power convergent to a retraction onto Fix(F). The theorem is proved. • Remark 5.11 Note that it follows from the Neumann series representation of the operator B in (5.129) that the mapping $ t defined by (5.137), oo

$t = J > f e F \

(5.144)

k=o

with ak = t(l-t)k, t e (0,1), has B = [ J + r ( / - A ) ] ~ 1 as its Frechet derivative at the point a. However, generally speaking, $ t and % are different in the nonlinear case. Finally, we observe that P. Mazet and J.-P. Vigue ([Mazet and Vigue (1991)]) have shown that under certain conditions on the Banach space X (for example, reflexivity), condition (*) of Definition 5.8 is not necessary. In particular, they proved the following result: Theorem 5.27 ([Mazet and Vigue (1991)]) Let V be a bounded convex domain in a reflexive complex Banach space X, and let F S Hol(X>) with T — Fix(F) ^ 0. Then T is a holomorphic retract ofT>. The scheme of the proof is as follows. Since T> is a bounded convex domain in X, the Cesaro averages 1

n"1

Cn = -YJFk n

fc=o

(5-145)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

are well defined and bounded on V. Then the Montel property implies that {Cn} contains a subsequence {Cnk} which weakly converges to a holomorphic mapping * 6 Hol(P). It is clear that if a £ Fix(F) C T>, then Cn(a) = a and *(o) = a. At the same time, since X is reflexive, it follows n-l

by the mean ergodic theorem that (Cn)'{a) = £ £ (Fk)'(a) strongly conk=o

verges to P = ('t)'(a) ) which is a projection onto Ker(J — F'(a)). Hence the point a £ Fix(F) C Fix(^) is a quasi-regular fixed point of *&. Moreover, by Theorem 5.22 the mapping ^ is power convergent to a retraction \P G Hol(X>) onto Fix(4r). Without loss of generality, assume now that V is a neighborhood of the origin and a = 0. The crucial point of the proof is the following technical auxiliary lemma: Lemma 5.4 ([Mazet and Vigue (1991)]) Let Sn(h) denote the Taylor polynomial in the power series expansion of a holomorphic mapping h at a neighborhood of the origin. Then for the mappings F and $ defined above, the following equality holds:

Sn{F o *") = S n (* n ) = S n (tf n+1 ).

(5.146)

Now this lemma implies that F o $ = $, hence Fix(<3>) c Fix(F). On the other hand, it follows by the constructions of the mappings * and $ that Fix(F) C Fix(^) = Fix($). Thus we see that Fix(F) = Fix($) = Fix($) is a retract of V, hence a connected submanifold of V tangent to Ker(J - F'(a)) = Ker(J - P).

Chapter 6

Generation Theory for One-Parameter Semigroups 6.1

6.1.1

Continuous and Discrete One-Parameter Semigroups on Metric Spaces Discrete and continuous flows on a domain

Let D be a topological space. A family M(D) of self-mappings of D forms a semigroup with respect to the composition operation on D if FoGGM(D)

(6.1)

whenever F and G belong to M(D). If, in particular, D is a metric space with a metric p on D, then the set Np(D) of p-nonexpansive mappings on D forms a semigroup with respect to the composition operation. If D is a complex domain in a Banach space X, then the set Hol(D) of all holomorphic self-mappings of D also forms a semigroup with respect to the composition operation. The set Aut(D) of all the automorphisms of D forms a group with respect to the composition operation because of the inclusion F-1 £ Hol(£>)

(6.2)

which holds whenever F € Aut(D). Hence F~l G Aut(£>). Let now A be a topological (additive) semigroup with zero, and let there exist a natural ordering of A, i.e., r >t if and only if there i s s e i such that r — t + s. We will mostly consider two cases: (a) A = N U {0} = { 0 , 1 , 2 , . . . } (respectively, A = Z = { 3,-2,-1,0,1,2,3...}; (b) A=[0,T), 0 < T < oo (respectively, A = (-T,T), 0 < T < oo). 157

158

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Definition 6.1 Let A be as above and let M(D) be the semigroup of selfmappings with respect to the composition operation of a topological space D. A family 5 = {Ft} C M(D) is called a one-parameter semigroup on D with respect to A if the following conditions hold: (i) Ft+s = FfFt, whenever s, t and s + t belong to A; (ii) Fo = / , the identity operator on D. In other words, a one-parameter semigroup S c M(D) is defined by a mapping A i-> M(D) which preserves the additive structure of A with respect to the composition operation on M{D). If A = N U {0} = { 0 , 1 , 2 , . . . } , then S = {F0,F1,F2,...,Fn,...}, Fn e M(£>), is called a one-parameter discrete semigroup. Actually, such a semigroup consists of the iterates of a self-mapping F = F\, because of conditions (i) and (ii), i.e., Fo = I, Fn = Fn, n = 1,2, If, in particular, (D,p) is a metric space and M(D) = NP{D), then for each F G NP(D) the set S = {Fn} U {/} forms a one-parameter semigroup of /9-nonexpansive mappings on D. If for each n > 0, Fn is invertible and F~n e NP(D), then S can be extended to a one-parameter group GT = {Fn} U {F~n}, n > 0. In this case condition (i) holds for all t and s in A = Z = {... — 2, —1,0,1,2...}. This set G is called a one-parameter discrete group of isometries on D because of the relation x,y € D.

p(F(x),F(y))=p(x,y),

(6.3)

We have already mentioned that for F G Hol(D) the family of iterates S = {Fn}%L0, Fo = I, can be considered a one-parameter discrete-time semigroup. In this case the vector field / = / — F is referred to as the generator of 5 (see [Kato (1966)]). It is clear that S can be extended to a one-parameter discrete group on D if and only if F € Aut(D). Definition 6.2 Let D be a topological space. A family 5 = {Ft : t £ (0,T)}, T > 0, of self-mappings Ft of D is called a (one-parameter) continuous semigroup if (i) F9+t = FtoFs,

0<s

+ t
(6.4)

Generation Theory for One-Parameter Semigroups

159

(ii) for each x € D, lim Ft(x) = x,

(6.5)

t->o+

where the limit is taken with respect to the topology of D. 6.1.2

Examples

Example 6.1 Let X be a Banach space and let L(X) be the space of all bounded linear operators on X with the norm \\A\\= sup ||Ac||, AGL(X),

(6.6)

xeX.

11*11=1 For a linear operator A e L(X) we can define the family {Bt}tzM. of bounded linear operators by using the following exponential expansion:

Bt = e~tA := J2 ^ p *kAk-

(6-7)

It is clear that the family {Bt} satisfies the semigroup property (i) and (ii), i.e., BtoBs=

e-tA o e~sA = e~^t+s)A - Bt+S,

(6.8)

and Bo = I, the identity operator on X. So if D = X this family forms a one-parameter group of self-mappings of D because etA = [e-*- 4 ]- 1 : X -» X

exists for each

* > 0.

(6.9)

If D is the unit ball of X and A is an accretive operator on X, then the family {Bt}t>o = {e~tA}t>o forms a one-parameter semigroup of so-called proper contractions of D by the inequality ||A||<1,

«>0.

(6.10)

Obviously, this family satisfies the limit property lim Bt = I

t-*o+

(6.11)

with respect to the operator topology on AT, i.e., ^lim \\Bt - I\\L{X) = 0.

(6.12)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Moreover, it is easy to see that, in fact, ||Bt-J||iW
(6.13)

Now it follows by the semigroup properties that (lim

||Bt+,-JB,|Uw = O

(6.14)

for all real t and s. Therefore such a semigroup is called a uniformly continuous semigroup of linear operators. Remark 6.1 As a matter of fact, we will see below that each uniformly continuous semigroup (group) of linear bounded operators on X can be represented by the exponential form (6.15)

Bt = e~tA for some A G L{X). In addition, it follows from the exponential series for e~tA that hm^(I-Bt) = A,

(6.16)

where the limit again is taken with respect to the norm topology of L(X). In view of the last property, the operator A is usually called the infinitesimal generator of the semigroup Bt (see a more general definition below). Example 6.2 Let X be the space of all bounded uniformly continuous functions defined on R = (—oo, oo) with the topology of uniform convergence. Define the family {Bt}t£R of linear operators on X by the formula [Bt(x)](y) = x(t + y), t G R, y G R,

(6.17)

where x e X is a bounded uniformly continuous function on R = (—oo, oo). It is clear that [Bo(x)](y) — x(y), i.e., Bo = I - the identity operator on X, and

[Bt+.(x)] (y) = x(t + s + y) = x(t + (s + y)) = [Bt(B3(x))} (y), (6.18) i.e., Bt+s = BtoBs,

t,s£R.

(6.19)

Generation Theory for One-Parameter Semigroups

161

So the family {Bt} forms a semigroup of bounded linear operators on X. Actually this semigroup is a group because of [5 t ]- 1 = B_ t

for all

t > 0.

(6.20)

However, regarding the continuity, one can assert only the pointwise convergence of this semigroup to the identity, i.e., lim (7 - Bt){x) = lim {sup|x(i + y) - x{y) I) = 0

t—»0+

*—*0+ t - y g n

J

(6.21)

for each x € X. Moreover, the limit (6.22)

\imUl-Bt)(x)

does not exist for all a; € X. Thus such a semigroup is not uniformly continuous on X. Other examples of linear semigroups can be found, for instance, in [Hille and Phillips (1957)], [Daletskii and Krein (1970)] and [Krein (1971)]. Example 6.3 Let X = C be the complex plane, and let D = A be the open unit disk in C. Let {Ft}t€m be the family of holomorphic self-mappings of A defined by _, . . z + tanh t . Ft(z) = , z&A. ztanh t + 1

, (6.23

It is easy to see that this family forms a one-parameter group of automorphisms of A which has the locally uniform continuity property (6.24)

]\m+Ft(z) = z,

uniformly on each compact subset of A. Moreover, there exists the locally uniform limit lim \{z- Ft(z)) = l - z \

(6.25)

which defines a holomorphic mapping on A. Example 6.4 Another example of a semigroup on A which does not consist of automorphisms may be defined as follows: ft(2) =

e'-^-l)'

**°-

{6"26)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Despite the fact that each Ft is a fractional linear transformation mapping A into itself, its inverse is not a self-mapping on A. So this semigroup cannot be extended to a group of self-mappings of A. At the same time we again have that

(6.27) uniformly on each compact subset of A, and (6.28)

}™=\(z-Ft(z)) = z-z2

exists for each z € A. Note that both the semigroups in Example 6.3 and 6.4 consist of mappings which are nonexpansive with respect to the Poincare hyperbolic metric of A, i.e., (6.29)

p(Ft(z),Ft(w))
for all pairs z and w in A and for each t > 0. Moreover, by the Schwarz-Pick Lemma, in Example 6.4 the strict inequality (6.30)

p(Ft(z), Ft(w))
holds for all z and w in A, z ^ UJ, and t > 0, while in Example 6.3 we have equality p(Ft(z),

Ft(w))

=p{z,w)

z,weA,

t>0.

(6.31)

In this case one says that the semigroup (group) {Ft}t>o consists of pisometries of the metric space (D,p). As a matter of fact, all one-parameter semigroups (groups) of automorphisms of A can be described as follows. Example 6.5

(see [Berkson et al. (1974)]).

(i) Each one-parameter group which consists of elliptic automorphisms of A is of the form F*W

-

(e^-H2)z + a(l-e^) _ l)z + x _ H 2 e ^ t - * > 0 ,

Q(e^t

(6-32)

where ip S R, a G A are given. (ii) Each one-parameter group of hyperbolic automorphisms of A is of the form Ft{z)

= (e*-l)z + /3(l-ae*)'

(6"33)

Generation Theory for One-Parameter Semigroups

163

where

(6.34)

where c G M, c 7^ 0, a £ dA are given. Note that in (i) the point a G A is the unique common interior fixed point of the semigroup, i.e., F t (a) = a,

t > 0.

(6.35)

However, this fixed point is not attractive, i.e., Ft(z) does not converge to a as t —> 00 for all z G A, z 7^ a. In cases (ii) and (iii), a G dA is a boundary locally uniformly attractive fixed point for {Ft}t>o, i.e., Ft(a)=a,

t>0,

and

lim Ft(z) = a,

(6.36)

t—»oo

uniformly on each subset strictly inside A. Observe, however, that in case (ii) there is another (boundary) fixed point (5 G dA of the semigroup {Ft}t>o, while in (iii), a is the unique fixed point in the whole plane. These phenomena will be considered later in the context of the asymptotic behavior of more general semigroups. It turns out that many important classes of linear semigroups are closely connected to corresponding classes of nonlinear semigroups. Moreover, if these connections are one-to-one, one may use different techniques to study both types. An example of such a semigroup is the linear semigroup of isometries with respect to the so-called indefinite metric on Krein spaces. For simplicity we consider this construction just for the Cartesian product of complex planes although this scheme works for any product of Hilbert spaces (and sometimes even Banach spaces) (see, for example, [Vesentini (1987a-b); (1991); (1992); (1994a-b)] and [Khatskevich et al. (1995a)]). Example 6.6 Let H be a complex Hilbert space with the inner product (-, •) and let X — H © C be the Hilbert space of the direct sum of H and C

164

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

with the inner product (•, •) denned by (6.37)

(x, y) = (u, v) + zw,

where u, v e H, z, w € C, and x = u + z, y = v + w are elements of X. Let a : X —> C be a continuous hermitian sesquilinear form on X defined by (6.38)

a(x, y) = {u, v) - zw.

This form is called the indefinite metric on X. The space X endowed with this metric is variously called either a Krein or a Pontryagin space. If J denotes a self-adjoint operator on X defined by the block matrix J =(o-°i)'

< 6 - 39 )

where / is the identity operator on H, then the indefinite metric a : X —> C can be written as (6.40)

a(x,y) = (Jx,y). A linear operator A : X —> X is said to be an a-isometry if

(6.41)

a(Ax, Ay) = a(x, y). In operator form this condition becomes

(6.42)

A'JA = J,

where A* is the adjoint operator of A. It turns out that a one-parameter semigroup of linear block operators on X, which are a-isometries, can be completely described by a (nonlinear) semigroup of fractional linear transformations on H, which preserve the infinitesimal Poincare metric on the open unit ball B of H (see [Vesentini (1987a); Vesentini (1991)]). For simplicity, we consider the case where H — C is the complex plane and B = A - the open unit disk in C. First observe that all groups of automorphisms of A of types (i)-(iii) described in Example 6.5 can be written in the general form Ft{z) = ei6<^^-, 1 - atz

H<1,

MR,

t€R,

(6.43)

Generation Theory for One-Parameter Semigroups

165

where the semigroup properties can be verified by the relations e i(fl, + .-»t-8.)( 1

at+°=l

+ a3ateif>°) = l +

ate~i6'+a3 + asate-«s>

a°

asate-i9°

= *° = °'

(6.44)

(See, for example, [Berkson et al. (1974)].) Let us define the families {a{t)}t>o and {b(t)}t>o by a{t) =

.

l

V 1 - l«t|2

• e'*

(6.45)

and b(t) = -ata(t).

(6.46)

M = ei9'

(6.47)

Then a(t) and

(6.48) Hence,

, _ a(t) ^ + S J

F^-W)^w-Z

a(t)z + b(t)

= W^WY

(6>49)

If we now define the family {At}t > 0 of linear operators on C © C (= C2) by the matrix At=\

fa(t) b(t)\ \b(t) a(t)J

,

6.50

then it is easy to verify that this family forms a one-parameter linear semigroup on C 2 . Moreover, clearly Al JAt = J,

(6.51)

i.e., the family {At}t>o is a semigroup of a-isometries on C 2 , where a : C2 —> C is defined by a(z,w) = ziiBi — z-iw-i, and z = (z\, z2) and w =

166

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

(wi,W2) are elements of C2. In addition, for each t > 0, we have deb At = \a(t)\2 - \b(t)\2 = ^—L-p - M 2 \a(t)\2 1 ~ l"tl 2 - I3^|2 = 1-

(6-52)

Now let A : C2 -» C2 be any matrix operator on I = C2, i4= (cd)«

(6-53)

preserving the a-metric on X and normalized by the condition det A = 1. It then follows by the condition A*JA = J

(6.54)

that c = b and d = a, i.e., A has the form

(i!)

^

with |a|2 - |6|2 = 1. If we define the fractional linear transformation fW then we have

= ]rr^ bz + a

™-it?h-

(6-56)

(6-57)

Since | | | = 1 and |£| < 1, we obtain that F is an automorphism of A. Another useful class of one-parameter semigroups of linear operators is the so-called one-parameter semigroup of composition operators defined on Hardy spaces. Example 6.7 Let F : A —> A be a holomorphic self-mapping on A. If 0 < p < oo, then the operator C : HP(A) -> HP(A) defined by CFf = foF

(6.58)

is a well defined linear operator on HP(A). It is called the composition operator on HP(A) (see, for example, [Shapiro (1993)] and [Cowen and

Generation Theory for One-Parameter Semigroups

167

MacCluer (1995)]). If S = {Ft}t>o is a semigroup of holomorphic selfmappings on A, then the family {At}t>o defined by Atf = foFt,

t>0,

(6.59)

defines a semigroup of linear operators on HP(A). In addition, if 5 = {Ft}t>o is a continuous semigroup, i.e., lim FAz) = z,

t-»o+

z G A,

(6.60)

then {At}t>o is a continuous semigroup on HP(A) in the sense of the strong topology in this space, i.e., lim Atf = /

t->o+

(6.61)

for all / G HP(A). Note that this convergence is not uniform on HP(A). The converse is also true: i.e., for each strongly continuous semigroup {At}t>o of composition operators the corresponding semigroup {Ft} is continuous on the unit disk A in the topology of compact convergence on A (see [Berkson and Porta (1978)]). Of course, a subject of interest is the question when At is an isometry with respect to the norm topology on HP(A). For p ^ 2 the answer was given by F. Forelli [Forelli (1964)]. Namely, each one-parameter semigroup, (actually, group) {At} of composition operators on HP(A), p ^ 2, p > 1, is given by

(Atf)(z) = eimt[Fl(z)]h(Ft(z)), teR, weR,

(6.62)

where Ft is an automorphism of A. On the other hand, for each 1 < p < oo, such an operator At is an isometry on HP(A).

6.2

Linear semigroups

Let D be a domain in a Banach space X. A semigroup S = {Ft}te[o,T) is linear if for each t G [0, T) the mapping Ft : D —> D is the restriction of a linear operator At acting on X. We will mostly be interested in the cases where D is either all of X or the open unit ball in X. So, let us first assume that D — X, and let S = {Bt}t>o be a continuous semigroup of linear bounded operators on X, i.e., (i) Bt+s = BtoBa,

M>0,

(ii) Bo = lim Bt = I ( lim Btx = x, x € X),

(6.63) (6.64)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

where the limit in (ii) is taken with respect to the strong topology on X. Usually, such a semigroup is known as a Co-semigroup. It is easy to see by the semigroup property (i) that condition (ii) implies continuity at each point to > 0, i.e., (6.65)

lim Bt = Bt0.

t—•to

Again we understand the last equality as lim Btx = Btax

(6.66)

t—*to

for each x e X. For each t > 0, define At = (I - Bt)/t. Clearly, At belongs to the space of bounded linear operators on X, for each t > 0. Theorem 6.1 Bt)xj

The set fi =

[x I

G X

:

lim Atx (= V

t -.o+

L(X),

lim \ (I *

t _o+

exists \ in dense X.

Proof. Let X* denote (as usual) the dual of X, and let f(t) be the continuous scalar-valued function on E + = [0, oo) defined by (6.67)

f{t) = t(Btx),

where a; is a fixed element of X and t G X*. Now it follows by the mean value theorem that 1 ft+s f(t) = lim - / f(6)(W. s _0+ S Jt

(6.68)

Since I € X* is arbitrary, we see that for each t > 0 and each x G X, Btx=

1 ft+s lim - / B9xd0. s _o+ s Jt

(6.69)

Define now y=

[ B9xd6, Jo

xeX,

(6.70)

t>0.

Then 1

-[Is

1 fs

1

s Jo

s Jt

B3]y = - / Bexd9 - - /

ft+s

B0xd9.

(6.71)

Generation Theory for One-Parameter Semigroups

169

Letting here s —> 0 + we obtain from (6.69) that (6.72)

lim - (I - Bs)y = x - Btx

s—>0 •

S

exists. In other words, y € SI. Let now a; be an arbitrary element of X. Define xn=n

Jo

As we have seen, xn s fi for all n = 1,2,3 represent xn in the form xn=

(6.73)

Bgxd6.

On the other hand, we can

f Bitxde. Jo

(6.74)

Letting n —> oo, we see that xn —> x strongly. Thus the closure of Q equals X, as claimed. • Definition 6.3

The mapping A : il —> X defined by Ax = lim -(x-

(6.75)

Btx)

is called the infinitesimal generator of the (strongly) continuous semigroup S = {-Bt}t>o- Obviously, the mapping A : Q —> X is linear, but not necessarily bounded unless Q, is all of X. The main property of the infinitesimal generator (which actually is an explanation of the name) is that a given semigroup and its generator are related by the Cauchy problem with an arbitrary initial data in fi. Theorem 6.2 let

Let S = {Bt}t>0 be a strongly continuous semigroup and

il= {x€X

: Ax:=

lim+-(I-Bt)x

existsl.

(6.76)

Then for each i e ! l , Btx G Cl and — (Btx) + ABtx = 0.

(6.77)

Proof. First we note that by the semigroup property, for t and s > 0 the operators Bt and Bs commute. By definition, the right derivative of Bt at

170

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

t > 0 is given by d+ 1 1 — (Btx) = lim - (Bt+3x - Btx) = lim Bt- (B3x - x) at s-»o+ s s-»o+ s = lim - (Bs - I)Btx = -ABtx, s—+0~'~ S

xeQ.

(6.78)

Also d+ — Btx = Bt Ax,

x e Cl.

(6.79)

Thus ABtx = BtAx,

x£il,

t>0,

(6.80)

which means that fi is J5t-invariant for each t > 0. If t > 0, then to obtain the left derivative it suffices to verify that for 0 < s < t, lim - (Btx - Btsx) + BtAx\\ = 0. s—»0+

5

II

(6.81)

To do this, we first observe that it can be shown by using the uniform boundedness principle and the semigroup property that for each t > 0, there is a number 1 < M < oo and w e (-oo, oo)

(6.82)

||-Bt|| < Me"1.

(6.83)

such that

Now we have, for x G fi, \\- (Bt% - Bt-.sx) + BtAx\\ \\s II =

Bt-s(-(Bsx-x))+BtAx

= \\Bt-.([^ (Bsx -x) + (Bs - I)Ax])\\ < Meu(t-J» | | - {Bsx -x) + Ac| + ||(B,-/)i4i|||-»0,

s-^0 + .

(6.84)

171

Generation Theory for One-Parameter Semigroups

Hence, for t > 0 we obtain ^ (Btx) = -ABtx. dt Corollary 6.1

(6.85) n

Each semigroup generator generates a unique semigroup.

Examples of strongly continuous semigroups are given in examples 6.1, 6.2 and 6.6 of the previous section. Note, however, that the semigroup 5 = {Bt = e~tA}t>o defined in Example 6.1 satisfies the so-called property of uniform continuity. Definition 6.4 if

A semigroup S = {Bt}t>o is called uniformly continuous

(6.86) lim | | / - 5 t | | = 0 , t->o+ where the limit is taken with respect to the operator topology on X. As we have already seen, for each bounded linear operator A : X —> X the semigroup S = {-Bt}, defined by Bt = e~tA, is a uniformly continuous semigroup. In this case A=

lim+-(I-Bt)

(6.87)

is its infinitesimal generator. Moreover, we have

1 (e~tA) = -Ae~tA

(6.88)

at uniformly on X. As a matter of fact, each uniformly continuous semigroup S can be represented in the form S = {e~*A}t>o for some A £ L(X). To show this we just need to prove the following assertion. Theorem 6.3 Let S = {Bt}t>o be a uniformly continuous semigroup, and let A : £1 -* X be its infinitesimal generator. Then A £ L(X). Proof.

It follows by the continuity in the uniform operator topology that

lim \\l-\ t->o+ll

f BTdA =0. II

t Jo

(6.89) v

'

Hence there exists to > 0 such that

IIJ - j - I BTdr\\ < 1. II

to Jo

M

(6.90)

172

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Now if we define AQ S L(X) by 1 Ao = -

fto

H> Jo

(6.91)

Brdr,

then AQ is invertible and AQ1 = (/ - (/ - Ao))~ . In addition, 1 1 f fto fto 1 - (I - Bt)B0 = — { BTdr - / Bt+Tdr \ t tot [Jo Jo J 1 (1 /"* 1 /' t+to 1 = -{BTdr-/ B T drL

(6.92)

Letting t —» 0 + , we see that the expression on the right-hand side converges to the limit ^ ( / - Bto) which belongs to L(X). Hence, .4£ 0 = ± (I- Bto) and 4 = i (/ - B^BQ1 E L(X), and we are done. • Despite the fact that the class of uniformly continuous semigroups is quite narrow, this class has a large variety of applications to various problems. In particular, each strongly continuous semigroup in a finite dimensional space is uniformly continuous and is generated by a matrix operator. Another useful example of a uniformly continuous semigroup can be given by the integro-differential equation % + f k(x,y)f(t,y)dy = O, (6.93) ot JD where D is a domain in R" and k(x, y) is a L<2.(D x D) kernel. If we define a linear operator A : L<2(D) —> L?,{D) by (Ag)(x) = f k(x,y)g(y)dy, JD

x G R",

(6.94)

we have that the solution of the Cauchy problem

| U + |^ ( I , vmm.o 1/(0, *) = «(*)•

(695)

is given by the exponent f(t,x) = e~tAx.

(6.96)

Hence we obtan a uniformly continuous semigroup on X = L2(D). An important question in the general semigroup theory is when a given semigroup leaves a given domain D invariant.

Generation Theory for One-Parameter Semigroups

173

For linear semigroups this question mostly arises when D is the open unit ball in X. In this case the semigroup is said to be a semigroup of contractions on X. In other words, the problem is to find the properties of the infinitesimal generator A of a semigroup {Bt}t>o with ||Bt|| < 1

(6.97)

for all t > 0. Basic results in this connection are given in the famous Hille-Yosida and Lumer-Phillips theorems. Although these theorems are known for strongly continuous semigroup in locally convex spaces, it is enough for our further purposes to formulate them for uniformly continuous semigroups in Banach spaces. Definition 6.5 Let X be a Banach space and let A G L{X). The set p(A) = {X e C : (XI - A)'1 G L(X)} is called the resolvent set of the operator A. If p(A) is nonempty, then the operator-valued function R = (XI — A)"1 is called the resolvent of the operator A. Now we formulate a version of the Hille-Yosida theorem (see [Yosida (1974)]) for a uniformly continuous semigroup of contractions on X. Theorem 6.4 Let A € L(X). Then A is an (infinitesimal) generator of a semigroup of proper contractions S = {Bt}, \\Bt\\ < 1, if and only if (i) p(A)D(-oo,0);

(ii) XR(X, —A) = (I + j Aj

is a contraction for all X > 0.

Moreover, the following exponential formula holds:

Bt = e~tA= lim | 7 / + - A V 1 1 .

(6.98)

Remark 6.2 It can be shown that, actually, the spectrum a(A) of the operator A lies in the closed right-half plane. This, of course, implies condition (i). Definition 6.6 A linear operator A c L(X) is said to be accretive (respectively, dissipative) if for each x € X, there is x* £ J(x) such that Re(Ax,x*)>0

(6.99)

Re«x,a:')<0).

(6.100)

(respectively,

174

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

A very useful characterization of accretive (respectively, dissipative) operators can be given as follows. Theorem 6.5

A linear operator A G L(X) is accretive if and only if ||(A/ + A)a:||>A||i||

(6.101)

\\{XI-A)x\\>X\\x\\)

(6.102)

(respectively,

for all x £ X and A > 0. This result and the Hille-Yoside theorem imply the following important assertion known as the Lumer-Phillips theorem (see [Yosida (1974)]). Theorem 6.6 Let A G L(X). If A is accretive and for some Ao G R j = (0,oo), (AoJ + A)'1 G L(X), then A is the generator of a one-parameter semigroup of proper contractions on X, i.e, \\e-tA\\ < 1,

t>0.

(6.103)

t > 0,

(6.104)

Conversely, if for some A G L(X) \\e-tA\\ < 1,

then A is accretive (or - which is the same - A is dissipative). Definition 6.7 hermitian) if

An operator A G L(X) is said to be conservative (or Re{Ax,x*) = 0

(6.105)

for all x G X and x* G X*. A conservative operator A is simultaneously accretive and dissipative. The following assertion holds. Theorem 6.7 A linear operator A G L(X) is conservative if and only if its spectrum lies on the imaginary axis of the complex plane and it is a generator of a group {Bt = e~tA : t G (—00,00)} of linear isometrics, i.e., \\e~tAx\\ = \\x\\ for alltGR

andx£X.

(6.106)

175

Generation Theory for One-Parameter Semigroups

6.3

Generated Semigroups of Nonexpansive and Holomorphic Mappings

Now let D be a domain (open, connected subset) in a Banach space X with the topology induced by the norm of X. Definition 6.8 A semigroup S = {Ft : t e (0,T)}, T > 0, on D is said to be generated if for each i € D , there exists the strong limit f(x)=\im+j{x-Ft(x)).

(6.107)

In this case the mapping / : D —> X is called the (infinitesimal) generator of S. One of the most important problems in the context of semigroup theory is to find out when a continuous semigroup of nonexpansive mappings in the norm or metric sense (in particular, />-nonexpansive or holomorphic mappings) has a generator. To trace an analogy with the classical linear case, we note that each semigroup of bounded linear operators which is continuous in the operator topology is differentiable at zero, and its generator is also a bounded linear operator. And conversely, if / is a linear holomorphic mapping, then it is bounded by definition, and we obtain the simplest case: the semigroup 5 generated by / is a uniformly continuous linear semigroup Ft = e~ff. In addition, we have the exponential formula e~tf = lim (/ + tf/n)~n. n—>oo

(6.108)

For the nonlinear case the analogous facts are not trivial. For nonlinear semigroups of norm nonexpansive mappings this problem has been considered by many mathematicians, mostly in the framework of the study of the asymptotic behavior of semigroups as well as the structure of their stationary point sets. For semigroups of holomorphic mappings it was also studied in connection with the theory of branching processes. In the onedimensional case, the differentiability with respect to the parameter of nonlinear semigroups of holomorphic mappings was proved by E. Berkson and H. Porta [Berkson and Porta (1978)] in their study of linear semigroups of composition operators on Hardy spaces. This nice result was extended by M. Abate [Abate (1992)] to the case of C n . However, it is no longer true in the infinite dimensional case. E. Vesentini has investigated semigroups of those fractional linear transformations which are isometries with respect to

176

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

the infinitesimal hyperbolic metric on the unit ball of a Banach space. He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Observe that in general, such semigroups are not everywhere differentiable. As a matter of fact, it turns out that a continuous one-parameter semigroup of holomorphic self-mappings of a domain in a Banach space is differentiable with respect to the parameter if and only if it is right locally uniformly continuous with respect to the parameter at zero. Let us recall that a net {fj}jeA C Hol(D,X) is said to converge to a mapping f 6 Ho\(D,X) in the topology of locally uniform convergence over D (or, briefly, T-converge) if for every ball B CC D (B is strictly inside D), limsup \\fj(x) - f(x)\\ = 0.

(6.109)

jeAxGB

We write in this case / = T— lim /,•. For the finite dimensional case jeA

this topology coincides with the compact open topology on D. Definition 6.9 A family S = {Ft}t>o C HOI(JD) is said to be a locally uniformly continuous one-parameter semigroup (or, briefly, a T-continuous semigroup) if it satisfies the semigroup property s,t>0, (i) F.+t = FtoFt, and (ii) T - lim Ft = I\Dt-»o+

Theorem 6.8 ([Reich and Shoikhet (1998b)]) Let D be a bounded domain in X, and let S = {Ft}t>o C Hol(D) be a strongly continuous semigroup. The following conditions are equivalent: (a) S is a T-continuous semigroup; (b) The differences ft = -AI- Ft) c are uniformly bounded on each subset strictly inside D; (c) For each x G D, there exists the strong limit hm+-t(I-Ft)(x)

= f(x),

which is bounded on each subset strictly inside D.

(6.110)

(6.111)

Generation Theory for One-Parameter Semigroups

Remark 6.3

177

It is remarkable that actually f = T-hm\{I-Ft),

(6.112)

i.e., the convergence in (c) is actually locally uniform convergence over D (see [Reich and Shoikhet (1998a)]). It is clear that if (c) holds, then the infinitesimal generator f belongs to Ho\(D,X). Since for the finite dimensional case a continuous semigroup of holomorphic self-mappings is T-continuous, it follows that such a semigroup always has a holomorphic infinitesimal generator (see also [Abate (1992)]). We now prove the above assertions in a more general setting. Definition 6.10 Let X be an arbitrary Banach space and let D be a in X. We say that a family [G3 : s e (0, T)}, T > 0, of self-mappings of domain D satisfies the approximate semigroup property if for each subset D strictly inside D the following conditions hold: (i) for each e > 0, there is a positive S = S(D,s) < T such that sup\\Gs(x) - Gp3/p\\ < es

x€D

(6.113)

for all positive integers p and all s 6 (0,6); (ii) for each pair s,t £ (0,T), s + t
(6.114)

Theorem 6.9 Let D be a domain in a complex Banach space X, and let {G s : s £ (0, T)} be a family of holomorphic self-mapping of D which satisfies the approximate semigroup property. Suppose that Gs converges to the identity as s —> 0 + , uniformly on each subset D strictly inside D, i.e., limsup ||G,(a;) - x\\ = 0.

(6.115)

s->0+ x€D

Then the strong limit lim - (/ - Gs) = f

(6.116)

s—»0+ S

exists and is a holomorphic mapping from D into X, which is bounded on each subset strictly inside D.

178

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

To prove our theorem we need two lemmas. Lemma 6.1 Let Dfeea domain in a complex Banach space and let <j) € Hol(.D). Suppose that for some subset Di c D withdist(Di,dD) > 0 there are two numbers /i and d, 0 < n < d, an integer p > 1, and a domain D2, DxCD2cD with dist(Di,dD2) > d such that sup \\x-<j>k(x)\\ < / i

(6.117)

for all k = 0,1,2,..., p— 1. Then for x S D\ the following inequality holds: \\x - 4?{x) - p{x - 4>(x))\\ < -/— (p - l)\\x - # c ) | | .

(6.118)

Proof. Let x € D\ and z G D2 be such that \\z — x\\ < fi. Then the ball Bd-n(z) with its center at z and radius d — fi lies in D2. Hence it follows from (6.117) and the Cauchy inequality that ||(J-0*)'(*)||<-Ji-.

(6.119)

Qi — fJi

Therefore, for x G D\ and y G D such that ||a; — y|| < \i we have, by (6.119), ||z - 4>k{x) ~{y- $k(y))\\ < - / - \\x - 2/H-

(6.120)

CL iX

Now setting y = (f>(x) and using (6.117) and (6.120) we obtain by the triangle inequality NP-1

||z - r(x) - P{x - 0(a:))|| = Y, [ ^ t o - {k+1)(x) ~x + {x)} " fc=0

<^2\\4>k{x)-x-[4>k((x))-4>(x)]\\ fc=l

<-^-(p-l)\\x-(x)l

(6.121) •

and we are done.

Lemma 6.2 Let D be a domain in a complex Banach space and let a family {Gs : 0 < s < T}, Ga € Hol(D), satisfy the approximate semigroup property. Suppose that G3 converges to the identity as s —> 0 + , uniformly on each subset strictly inside D. Then for s > 0 small enough the net fB = -AI-G.) 5

(6.122)

179

Generation Theory for One-Parameter Semigroups

is uniformly bounded on each subset strictly inside D. Proof. Let D\ be a subset strictly inside D and let 0 < d < dist(dD, Di). Take any domain D2 CC D such that D\ CC D2 and dist(Di,dD2) > d. Choose fi, 0 < (i < d, such that (i(d — /z)" 1 < ^, and choose a, 0 < a < T, such that for all T € (0,
(6.123)

l

In addition, it follows by the approximate semigroup property (i) that there exists 0 < 8 < | such that \\Gk3(x)-Gsk(x)\\<%

(6.124)

Now set n = [^]. For s £ (0, S), we for all x £ £>2 and each k = 1,2, have n > 2, ns > | , and ks < a for all k = 1,2,..., h. Hence it follows from (6.123) and (6.124) that sup \\x - Gks(x)\\ <»,

x€D2

se {0,6).

(6.125)

Now for all x € D\ we get, by Lemma 6.1, n\\x - G.(x)\\ - ||z - Gna(x)\\ < \\n(x - Gs(x)) - (x - G?(z))|| <^n||a:-G.(x)||,

(6.126)

or

\\x-G3(x)\\
(6.127)

Therefore, by (6.122)-(6.127), we obtain

||/x(*)|| < ^ (IN - G ns (x)|| + \\Gns(x) - GJ(x)||)

(6.128) whenever s G (0,6). The lemma is proved.

•

Proof of Theorem 6.9. Let D be a domain in a complex Banach space X, and let {Gs : s £ (0,T)} be a family of holomorphic self-mappings of D which satisfies the approximate semigroup property. Suppose that G3

180

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

converges to the identity uniformly on each subset strictly inside D. To show that the net / . = - ( / - G.)

(6.129)

is a Cauchy net, as s —> 0 + , on each subset D\ strictly inside D, assume that e > 0 has been given. Choose 0 < d < dist(Di,dD), 0 < // < d, such that -7^— < e,

(6.130)

d — ^jj

and choose 0 < w < T such that sup ||i-G T (a;)|| <%

(6.131)

for all r e (0,w). Let 0 < S = (5(£>!,e) < w b e such that condition (i) of the definition is satisfied, i.e., \\GT(x) ii

GPT(X)\\ ii

p

(6.132)

< ET

whenever x e D\ and r e (0,6), p= 1,2, Now choose an integer N > 0 such that AT"1 < 6 and e • AT"1 < 5 /i. Then, for all integers m,n> N and allfc= 0,l,2,...,p = max{m,n}, we have, by (6.131) and (6.132), sup \\x - Gk_L. (x) I < sup \\G\_ (x) - G_k_ (x)\\

X6-D1

x€Di

m n

m

"

mn

+ J S " X - G ^ ( i B ) l l < e ^ + 2<'Z-

(6133)

Therefore, by (i) and Lemma 6.1, setting in this lemma (j> = GJL. and nm

p = m we get for all x G D\,

\\x - Gi (x) - mix - G_i_ (x)) II < |x - G^_ (a:) - mix - G j _ (x)) II + ||G!!L(a;)-Gj.(x)|| < T ^ - m l | x - G ^ ( x ) | | + e - - . 11

nra

n

"

d — (1

'

n m

(6.134)

Tl

Multiplying this inequality by n and using (6.129) and (6.130) we obtain, for x £ D\, ||/i(x)-/s!_(a:)||<e(||/_j_(»)|| + l ) .

(6.135)

Generation Theory for One-Parameter Semigroups

181

Now it follows, by Lemma 6.2, that there is C — C(Di) such that ||/^(x)||<£

(6.136)

for all x € D\, whenever N (and therefore n • m) is big enough. by (6.131) we have \\fi(x)-f^.(x)\\<e(C + l).

So,

(6.137)

In a similar way we can get \\fi(x)-f^.(x)\\<e(C + l)

(6.138)

for all x in Dx and n,m> N, and hence \\f±(x)-fx(x)\\<2e(C+l)

(6.139)

for all x € D\ whenever n,m> N. This inequality means that the sequence {/A }n-jv c o n v e r ges as n —> oo uniformly on each subset D\ strictly inside D. In particular, it converges uniformly on each ball strictly inside D and is uniformly bounded on such a ball. Therefore, its limit /=

lim fx

n—»oo «

(6.140)

is a holomorphic mapping from D into X. Now we show that the net {/s}s€(o,T) converges to / uniformly on each subset D\ strictly inside D. This will conclude the proof of our theorem. For given e > 0 and x € D\, setting n = [^-], we can choose s so small that ||/jL(aO-/(z)||<e.

(6.141)

In addition, for such s and n we have fs-fi.

= -s(I-Gs)-n(l-Gx)

+ H( j - G l r 1 )- n *( j - G i)]-

<6-142)

Observe that in our setting n = [^r], so that we have ns —» oo and J2|l -> 1 as s -» 0. Thus we can find 8 > 0 such that 1 - &£ < e and Gjsni (x) e £>2 CC D whenever s G (0,6) and a; € Di. Using the n

182

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

approximate semigroup property (ii) (see the Definition) we get for such s and all x G £>i, - \\G^ (x) - G.(x)|| < - llGiad (x) - G on

S

+ p^oG

o G,_^ (x)|| n

i£n1{x)-G3(x)\\

n

n

<-\M\\x-G S [

H n

TI

"

S~

^11

n

M

(6.143) where M =

sup

xeZ?2,s6(0,i)

||(Gs)'(x)||. Once again, using Lemma 6.2, we have

\\x-G.-M\\
(6-144)

and therefore (6.143) implies that - ||
n

(6.145)

II

Now condition (i) of the definition implies i \\G[?n](x)\\<e[-^<e. (6.146) s II n II sn Finally, by Lemmas 6.1 and 6.2, we obtain for x G Di and s G (0,5),

- ||x - G[tn] (x) -ns(xS II

n

V

Gx (x)) II "

/ II

< - \\x - G[inl(a:) - [ns](x - Gi(x)) II + - \[ns] - ns\ \\x - Gi (x)||

< fej H + 1 | H - ns\) \\x - Gx (x)\\

< ^ H + H _ 1 \ c < 2£e.

(6.147)

Thus for \ xns6 £>i and ns s € (0, / J) we get from (6.141)-(6.147) ||/a(x) - /(x)|| <

||/A(X)

- /(x)|| + ||/a(x) - /i(x)||

<e{2 + MC + L + 2C),

(6.148)

Generation Theory for One-Parameter Semigroups

and we are done.

183

•

Proof of Theorem 6.8. It follows by Theorem 6.2 that condition (a) implies condition (c). By Lemma 6.2, conditions (a) and (b) are equivalent. The implication (c)=^(b) is obvious. • 6.4

The Cauchy Problem and the Product Formula

We know already that a T-continuous semigroup of holomorphic mappings is right-differentiable with respect to the parameter t at zero. As a matter of fact, this implies, in turn, that Ft is differentiable at each point t € M+ and that the function u(t, x) = Ft(x) is the solution of the Cauchy problem u't(t,x) = -f(u(t,x)), u(0,x) = x [Cartan (1967)]. In the context of the Hille-Yosida theory the following question is also of interest. If in the infinite dimensional case we have a family of holomorphic mappings which satisfies the approximate semigroup property and converges to the identity uniformly on each subset strictly inside D, is this family differentiable with respect to the parameter and does its derivative generate a semigroup which may be represented by the so-called product or exponential formula? To answer these questions we first formulate the following general assertion. Theorem 6.10 Let (D,p) be a complete metric space, and let {Gs : 0 < s < T} be a family of p-nonexpnsive mappings on (D,p) with the following properties: (i) For each p-ball B C (D, p) and each e > 0, there is a positive 6 = 6{B, e)
G|(:r)) < e - s

(6.149)

for all x € B and for all integers p whenever s G (0, <5); (ii) For each p-ball B C (D,p) there exist /i = /x(B) > 0 and C = C(B) such that p(Gs{x),x)
(6.150)

for all x e B, whenever s G (0, /i). Then for each pair s € (0, T) and t > 0 and each sequence of integers {£„}

184

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

such that - ^ - > 1 , as n-»oo,

(6.151)

Tit

there exists the limit lim G*£ = Ft

n—*oo

(6.152)

n

uniformly on each p-ball in (D,p). This limit does not depend on the sequence {tn} and it is a locally uniformly continuous one-parameter semigroup with respect to t > 0. Proof. First we establish two simple inequalities. For each T 6 (0,T) and each integer £, we have e-i

p(GeT(x),x) < J2p{Gl+1(x),

Gl(x)) < tp(GT(x),x).

(6.153)

3=0

Now if £i and £2 are two arbitrary integers, (6.153) implies p{&?{x), G^{x)) = pi^a^M{G^-t2\x)),

G™in^2)(z))

< p(G^-^{x),x) < \h-e2\p(GT(x),x) (6.154) for each r G (0,T). Take a p-ball B C (D, p) and choose /i > 0 so that condition (ii) holds. Then, for each r G (0,/x) we have , by (6.153) and (6.154), p(Gir(x),x)<£-CT

(6.155)

p(GeTHx), G?(x)) < \h-l2\CT

(6.156)

and

for all x G B and for all integers l,l\,ti. For a given s G (0, T) and t > 0, consider the sequence of mappings NOO Gj 1 > on 5 , where {i n } is a sequence of integers which satisfies (6.151).

{

n J 1

Taking an integer N so that s/N < n we get, by (6.155), p (G'jUx), X) < ^

L < 00

(6.157)

for all n > iV and all x & B. In addition, for each j = 1,2,... ,tn and m = 1,2,..., pfG'7(a;),xN) <mj-~

C < 00

(6.158)

Generation Theory for One-Parameter Semigroups

185

whenever n> N This means that there exists a p-ball B\ c (D, p) such that the sequences (G*r(z)| In

and \G"J!_(Z)\

J jV

I

nm

are in Bi for all x e B, j = J JV

1,2,..., tn, m = 1,2, Now for a given e > 0 we can choose, by (i), 6 = 8(e, Bi)
(6.159)

for all z £ Bi and all m = 1,2,..., whenever 0 < r < 8. Taking N so large that s/N < min{/i, 8} and setting z = GJ_7. (a:), x £ B, m = 1,2,..., j = 1,2,..., tn, n > N and T = ^ , we obtain, by the triangle inequality, the nonexpansiveness of Gs and inequality (6.159), that

p((ft{x), G^ix)) < ^(C'r^fef^)), G^-UG^^ix)))

GT^HG^^ix)))

< t n • e • - < a • e, (6.160) n where a = sup {^f2} < oo because of (6.151). In the same way and for the same e > 0 we obtain the inequality P ( G V {X), G V " {X)) <e-a \

(6.161)

/

nm.

m

for all x € B wheneverTO> AT, n = 1,2, by (6.156) that

In addition, it follows

p ( G V m (X), Gtjr (x)) < ^2- - ^=2- £ < e • a \

nm

nm

'/

n

nTO

(6.162)

and ,o fG*1?" (a;), G V (a)) < ^ 2 . - ^ ^ L < e • a

(6.163)

186

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

for n,m> N, where N is large enough. Thus, by the triangle inequality, (6.160)-(6.163) imply that for a given e there is N > 0 such that p (G*? (X), G*r (x)) < 4ae

(6.164)

whenever n,m > N. This means that < Glr i is a Cauchy sequence uniformly on each p-ball B C (D,p), and since {D,p) is complete, its limit exists and is a /^-nonexpansive mapping on (D,p). Once again it follows from (6.156) that it {r n } is another sequence of integers such that — -»t

(6.165)

then, for a given e > 0 and x 6 B, p (G*? (X), GrP (x)) < — - — £ < e \

"

n

/

n

Tl

(6.166)

whenever n is large enough. This means that Ft = lim G'l1

(6.167)

does not depend on the sequence {tn} satisfying (6.151). Now let s e (0,T), t > 0 and r > 0 be given numbers, let {tn}i° and {rn}T be two sequences of integers such that ^ p —» t and ^ ^ - » r a s n —> oo. Then, for a given e > 0 and x € B, p(Ft(*V(x)), F t + r (x)) < p(F t (F r (x)), Gi" +r "(x)) +p(Fi" +I -"(x), T t + r (x))

<^(Ft(Fr(x)), ^ - ( G t ^ J + e < p(Ft(Fr{x)),

G'HFrix)))

+p(Gt£(Fr(x)),

G*j(Gt(x)))+e

< p(F r (x), G r /(x)) + 2e < 3e

(6.168)

whenever n is big enough. Since e > 0 is arbitrary, we have F t + r = Ft • F r .

(6.169)

187

Generation Theory for One-Parameter Semigroups

Thus, Fr : R+ —> (D, p) is a one-parameter semigroup which is uniformly continuous on each p-ball in {D,p) with respect to t > 0. The theorem is proved. • A consequence of this theorem is the following important assertion. Theorem 6.11 Let D be a metric domain in X with a metric p G (SPS) and let {Gs : s G (0,T)} be a family of holomorphic self-mapping ofD which satisfies the approximate semigroup property. Suppose that Gs converges to the identity as s —> 0 + , uniformly on each subset D strictly inside D, i.e., limsup ||Gs(x) - z|| = 0.

(6.170)

s-»0+ x€D

Then, for each pair s and t, s G (0, T), t > 0, and each sequence of integers {£„} such that — ->1 as n->oo, nt there exists the strong limit

(6.171)

lim G'a" = Ft, n—»oo

(6.172)

n

uniformly on each subset strictly inside D. This limit does not depend on oo} is a one-parameter {tn} and s in (6.171), and the family {Ft :0
f=

lim I (J _
(6.173)

s-»0+ S

For x G D the mapping u(t,x) — Ft(x) defined by (6.172) is the solution of the Cauchy problem

I

9t

;

(6 . 174)

lim u(t, x) = x

K. t->o+

where f is defined by (6.173).

Proof. Let D be a metric domain in X with some metric p G (SPS). Then it follows that conditions (i) and (ii) of the above theorem are satisfied. Therefore, by this theorem, for each pair s and t, s G (0,T), t > 0, and each sequence of integers {£„} such that ^ —> 1, there exists the strong limit Ft — limn-Kx, G*? uniformly on each subset strictly inside D. This

188

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

limit, F t : R+ —> Hol(D), is a one-parameter semigroup of holomorphic self-mappings of D. Now we already know that the limit

(6.175) exists and is a holomorphic mapping on D. We want to show that this mapping / : D —» X generates the semigroup {Ft}, i.e., that it also satisfies the condition / = thm

T-^

,

(6.176)

where the convergence in (6.176) is uniform on each Di CC D. Indeed, for given e > 0 and D\ CC D, we can find a small enough 5 > 0 such that

|/(s)_£z£(E)|<e

(6.177)

for all x € D\ and all t £ (0,6). In addition, setting s = t and tn = n we can find <5i < <5 such that

j||GT(x)-Ft(x)||
(6.178)

(see (6.160)) for all t £ (0,5). Once again, using (6.175), we take S2 < S such that - \\Gt(x) - G\ (x)|| < s for 0 < t < S2.

(6.179)

Thus we get for t e (0,J 2 ),

|| / W _£^M|<| / ( I ) _£Z|*)| + i ||C, - 0 1 (x)|| + -t ||G1 (i) - F,(x)|| < 3e,

(6.180)

and we are done. Finally, it follows by the semigroup property and (6.176) that for each x e D the mapping u(t,x) = Ft(x) is a solution of the Cauchy prob• lem (6.174). This concludes the proof of the theorem. Corollary 6.2 Let D be a metric domain in X with a metric p G (SPS) and let {Ft : t € (0,T)}, T > 0, be a one-parameter semigroup of holomor-

Generation Theory for One-Parameter Semigroups

189

phic self-mappings of D such that lim Ft = / ,

(6.181)

t-»o+

uniformly on each subset strictly inside D. Then this semigroup can be continuously extended to a flow {Ft : 0 < t < oo} on all of M + . Remark 6.4 Formula (6.172) is called the product formula. The crucial point in establishing this formula is that the family {Gs : s € (0, T)} of holomorphic mappings in D satisfies the approximate semigroup property and has a right-hand derivative at s = 0 which is equal to f. The question is what happens when we have an arbitrary continuous family {G s } s >o C Hol(D) which is differentiate at s = 0 + . Actually, we will see below that for a bounded convex domain, the above theorem and the so-called resolvent method and range condition imply a somewhat more general assertion. Theorem 6.12 Let D be a bounded convex domain in a complex Banach be an arbitrary family of holomorphic selfspace X, and let {G3}3e^,T) mappings of D such that lim

s->0+

X~G^X)

(6.182)

= f{x)

S

exists uniformly on each subset strictly inside D and is bounded on such subsets. Then (1) the Cauchy problem (6.174) has a global solution u(-, •) defined on R+ x D; (2) this solution can be obtained by the following product formula: u(t,-)=

lim Gl,

n-»oo

n

(6.183)

where the limit is uniform on each subset strictly inside D. An immediate but very important consequence of this theorem is the following assertion. Corollary 6.3 Let D be as in Theorem 6.10, and let f and g be two holomorphic generators of one-parameter semigroups on D, i.e., {Ft}t>o and {Gt}t>o, respectively. Then the mapping h = / + g is also a generator and the semigroup Ht generated by it can be obtained by the formula Ht=

lim [Ft. -G±]n,

(6.184)

190

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

where the limit is uniform on each subset strictly inside D. Corollary 6.4 The set of holomorphic generators on a bounded convex domain is a real cone. 6.5

Nonlinear Resolvents, the Range Condition and Exponential Formulas

Definition 6.11 We will say that a mapping / : D —> X satisfies the range condition if there exists a positive T > 0 such that for each s € (0,T), (I + sf)(D) D D

(6.185)

and Ts = (I + s / ) " 1 is a well-defined self-mapping of D. This mapping Ts is called the (nonlinear) resolvent of / . It turns out that similarly as in the linear Hille-Yosida theory, the resolvent method plays an important role in the study of nonlinear generators of semigroups, as well as in the study of the structure of the fixed point sets of self-mappings and null point sets of vector fields. In this context it is natural to look for geometrical conditions which will ensure that any semigroup of holomorphic mappings can be represented by exponential formulas or, in other words, to find out when the range condition holds for each holomorphic generator. To answer this query we need the following formula, which is usually called the resolvent identity. Lemma 6.3 Let D be a convex domain in a Banach space X, and let f : D —» X be a mapping which satisfies the range condition. Then for 0 < s
(6.186)

Proof. For each x 6 D the element y = f z - f (l- j)Tt(x) belongs to D, by the convexity of D. It follows by the definition of the resolvent that I-Tt = tf(Tt) for t 6 [0,T). Thus,

y = Tt(x) + -t(x- Tt{x)) = Tt(x) + sf(Tt(x)) = (I + sf)Tt(x)l6.187) Hence Ts{y) = (/ + sf)~l{y) = Tt(x), and we are done.

•

191

Generation Theory for One-Parameter Semigroups

Lemma 6.4 Let D be a convex metric domain with a metric p G (SPS). If f G Hol(D, X) is bounded on each subset strictly inside D and satisfies the range condition, then the family {Ts = (I+sf)~l : s £ (0,T)} converges to the identity, uniformly on each subset strictly inside D and satisfies the approximate semigroup property. Proof. Indeed, let D be a metric domain with a metric p £ (SPS), and let D\ be a subset of D such that dist(£>i, dD) > 0. Denote Mi = sup{||/(j/)|| : y 6 £>i} and 6 = min { ^ , T } , where 0 < d < dist(D\,dD). Setting y = x + sf(x) for x £ D and s £ (0, J), we have Ts(y) = cc, ||a;-1/11
< LMxs. (6.188)

Since for each T G (0, T) and each integer £,

p{T?{x),x) < "£P(TJ+I(x),

(6.189)

Tj{x)) < £p(TT(x),x),

3=0

we have that for x € D, n = 1,2,..., andfc= 1,2,..., n, | | 7 | (a:) - x \ \ < ± p ( T t ( x ) , x ) < C - s < ± ,

(6.190)

whenever 0 < s < 5^ = min{<5, ^ } . Firstly, (6.188) and (6.190) mean that the subset D2 = Di (J

Bd(x),

x€Di

which lies strictly inside D, where Bd(x) is the closed ball with its center at x and radius d, contains the sets {Ts(x)}^(o,^) and {Tt}, s G {0,6), n = 1,2,..., and k — l,...,n, where x € D\. Secondly, it follows from (6.188) that the net {7^}s€(o,<$) converges to the identity uniformly on D\. Now denote Mi = sup {||/(x)||}. It follows from the Cauchy inequalities that for each x € D\ and y £ D such that \\x ~ v\\ < f ) ll/'(y)ll < 2M/d, and hence, for such i and y we have

ll/W-/(2/)ll<^alk-2/||-

(6.191)

Now, because of the identity x - T.(x) = s/(T s (x)), x G D,

(6.192)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

we have that /(*) = lim X~T^X\ s—>0+

(6.193)

S

uniformly on D\. In addition, (6.190)-(6.192) imply that

\\sf(x)-x + T2(x)\\ < ElJ/W + r, (rt\x)) - (rl-\x))\\

<£.^-s2,i6D.

(6.194)

Thus we obtain, by (6.188), (6.192) and (6.194), \\%(x) - Tf || < \\x - Tf (i) - s/(x)|| + ||s/(i) -x + T.{x)\\ < C2-^ s> + s\\f(x) - f(Ts(x))\\


f o r a l l x e D i , s € (0,<Ji), n = 1,2,{6.195)

This inequality shows that condition (i) of the definition of the approximate semigroup property is satisfied. Now take positive s,t such that s + t < Si. Then it follows by the resolvent identity and (6.188) that

\\Ts+t(x)-Tt(x)\\ < ip(r. + t (x), Tt(x))

^hp(Tt(^-tx

+

7T-tTt+s{x))'Tt{x))

^ip{jhx+7rtTt+°{x)>x) <^\\X-Ts+t{x)\\^-t<^.C.s.

(6.196)

Generation Theory for One-Parameter Semigroups

193

Thus we have

\\Ts+t(x)-Z(Tt(x))\\

< ^p{Ts+t(x), Tit


T,Tt(x))

\C+-

~ m s+t L

m

c]
2

(6.197)

This proves condition (ii) of the definition of the approximate semigroup property, and we are done. • Thus we have proved the main theorem of this section. Theorem 6.13 Let D be a convex metric domain in a complex Banach space X with a metric p € (SPS), and let f £ Ho\(D,X) be bounded on each subset strictly inside D and satisfy the range condition. Then f is the infinitesimal generator of the one-parameter semigroup of holomorphic self-mappings of D, which can be defined by the following analogs of the exponential formula:

Ft= lim ( / + - / ) " " n—>oo \

n

(6.198)

/

or /

1

\ [-**»]

Ft = lim ( / + - / )

,

(6.199)

where the convergence in (6.198) and (6.199) is uniform on each subset strictly inside D. As a matter of fact, for bounded convex domains the converse also holds. We prove this assertion in a more general setting. Let D be a metric convex domain in a Banach space X with a corresponding metric p on D. We recall that the metric p is compatible with the convex structure of D if the following conditions hold:

(i)

p(sx + (1 - s)y, sw+ (1 - s)z) < max[p(x, to), p(y,z)];

(6.200)

194

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

(ii) there is a real function

(6.201)

and for each three elements x,y,z £ D and s G [0,1], the following inequality holds: p(sx + (1 - s)z, sy + (l-

s)z) <
(6.202)

Of course, each convex bounded domain in a complex Banach space is a compatible metric domain with the hyperbolic metric p on D. Theorem 6.14 Let D be a metric convex domain in a Banach space X with a corresponding metric p on D, and let the metric p be compatible with the convex structure of D. Suppose that {Ft : 0 < t < T} is a family of p-nonexpansive self-mappings of D, i.e., (6.203)

p(Ft(x),Ft(y))
L=A

=:

/

(6.204)

exists uniformly on each p-ball in D and / : £ ) — » X is a continuous mapping, then f satisfies the range condition. Moreover, the resolvent Ts — ( / + sf)~x : D —> D is defined for each s > 0 and is a p-nonexpansive selfmappings of D.

Proof. Suppose that the conditions of the theorem are satisfied. For a fixed z € D and s, t > 0, we consider the equation x = - ^ - Ft(x) + -^-z S ~y L

S "|

t

(6.205)

= G.,t(x).

Since p is compatible with the convex structure of D we have, by definition, that

p(G.,t{x), G.,t{y)) < K l T i ) ^ ^ '

-(p(7Ti)p(x'y)'

Ft(j/))

(6"206)

Generation Theory for One-Parameter Semigroups

where 0 < f{^+i)

195

< 1> a n d limsup

T

r

< oo.

(6.207)

So, by the Banach fixed point theorem, it follows that equation (6.205) has a unique solution in D, x = xS}t for each s, t > 0. Since the equation (6.208)

x + sft(x) = z, z€D,

where ft = \ {I-Ft) is equivalent to (6.205), this implies that the mapping TStt = (I + sft)*1 is a well-defined self-mapping of D and T3
(6.209) where n = 1,2,..., and x° t (z) = y is an arbitrary element of D. Thus it follows by induction that


(6.210)

because Ft is a p-nonexpansive mapping on D. Hence Ts,t : D —> D is also /)-nonexpansive on D. Now we want to show that the net {TStt)s>o converges to a mapping 7^ : £> —> D, as t —» 0 + . If this holds it is clear that Ts = (I + s/)" 1 is a /9-nonexpansive mapping of D, and this will conclude our proof. First we show that for each z G D and each s > 0, the net {TStt(z)}(= {xs,t}) is strictly inside D for £ small enough. Indeed,

p{T.A*),z)=p(7±rtFt{x.,t) +

-KlTi)^:**

7L-z,z)

i) + piFt(z), z)].

(6.211)

Thus it follows by the approximate semigroup property of the resolvent and

196

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

by (6.207) that limsupp(:rs t, z) < limsup

-.

r- —

tJ^J±

< C < oo. (6.212)

So, there is a p-ball in D which contains the set {xS]t(z)} whenever t is small enough. Now we want to show that for a fixed s > 0 and a given z G D this net is a Cauchy net as t —> 0 + . Indeed, if we denote, as before, xa,t = Ts,t(z) a n d zt,r = (I + sfr)(xSit), we have z = (I + s/ t )(x,,t), and {zt,r} converges to z as t,r -» 0 + , because {ft} is a Cauchy net. But Ts,r(zt,r) = xs,t and we get p(Z,t(z),

Zir(z))

< p(Tt,t(z), Ts,r(zt,r))+p{Ts,r(zt,r), 0,

Z,r(z)) (6.213)

as t, r —> 0 + . The proof is complete.

•

We denote by RNp(D) the class of all mappings / : D —» X for which the resolvent (J + rf)~1 is a well-defined /9-nonexpansive self-mapping of D for each positive r. We say that a mapping / : D —> X is an infinitesimal p-generator if for some T > 0, there exists a one parameter semigroup S = {St}te(o,T) of self-mappings St • D ^ D (St+r(x) = St{Sr{x)), x £ D, 0 < t + r < T) which are /O-nonexpansive for each t € (0,T): p(St(x), St(y)) < p(x,y),

x,y G D,

(6.214)

and such that lim \ p(x - tf(x), St(x)) = 0

(6.215)

exists for all x e D uniformly on each p-ball in D. In this case we will also say that / is a p-generator on (0, T). if T = oo we will write / € GNP(D). So if, in particular, D is a bounded convex domain in a complex Banach space and p is its hyperbolic metric, then a continuous p-generator on some interval (0,T) belongs to the class RNP(D). Thus we have the following analogs of the Hille-Yosida theorem. Theorem 6.15 Let D be a bounded convex domain in a complex Banach space X, and let p be its hyperbolic metric. Suppose that f : D —+ X is

Generation Theory for One-Parameter Semigroups

197

bounded and uniformly continuous on each p-ball in D. Then f 6 GNP(D) if and only if f e RNP{D). Theorem 6.16 Let D be a bounded convex domain in a complex Banach space X, and let f : D —> X be a holomorphic mapping. Then f generates a one-parameter semigroup on M.+ of holomorphic self-mappings of D if and only if for some T > 0 and for each r £ (0, T], the mapping Jr = ( 7 + r / ) " 1 is a well-defined holomorphic self-mapping of D. Moreover, in this case, Jr = (7 + rf)~l is a well-defined holomorphic self-mapping of D for all r >0.

Chapter 7

Flow-Invariance Conditions

7.1

Boundary Flow Invariance Conditions

Let V be a convex subset of a Banach space X and let / : V —» X be a continuous mapping on V, the closure of V. Then the following tangency condition of flow invariance lim dist(z - hf(x), V)/h = 0,

i£»,

(7.1)

is a necessary condition for the solvability of the Cauchy problem

(t + '<">-

«,2,

{ u(o) = i e D

oni?+ = [0,oo). It was shown in [Reich (1975); Reich (1976)] that condition (7.1) can be rewritten in another form which sometimes is more convenient. Namely, let X be a Banach space, and let V be a convex domain in X with 0 G V. We denote by p (— p-o) the Minkowski functional of V, that is, p(x) := inf{A > 0 : x G XD},

x € X.

(7.3)

(Note that since V is open, it is an absorbing set, i.e., for each x G X there is A > 0 such that x & \iD for all /x with |/x| > A.) It is known that p : X —* [0, oo) is continuous on X and that V = {x € X : px>(ar) < l } , ^ = { i £ X : p D (x) < l } .

(7.4)

Note also that p is a gauge function: p(x + y)< p(x) +p(y), p(Xx) = Xp(x), A > 0, 199

(7.5)

200

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

whence each one of the sets Ve = {x e X : p(x) < £}

(7.6)

is an open convex absorbing subset of X, the closure of which is Vt = {x S X : p(x) <£}. For 0 < £ < 1 we will call {Ve} the level sets of p. Now it follows by the Hahn-Banach theorem that for each x G X, there is a linear functional x* £ X such that

{

Re(x,x*)

=pv{x)

and

(7.7)

Re(y,x*)
(7.8)

and this means that j x is a support functional of T> at x. If T> is the open unit ball in X, then x* = j x for x G dV is a selection of the duality map J : X —* 2X' defined by J{x) = {x* G X* : (x,x*) = ||x||2 = ||z*|| 2 }.

(7.9)

Note that we can assume that j \ x — j x for all X > 0. If now / : T> —> X generates a strongly continuous semigroup S = {Ft}t>o, Ff.V^V, t>0, i.e., thm

ft(x) = f(x),

x£V,

(7.10)

where ft(%) = a ~ t ' x ' and the limit is taken with respect to the norm of X, then we have Re{ft(x),jx) = \ Re(z - Ft(x),jx) = i (l - Re(F t (x), j x )) > 0, a; G dV. Thus, it is evident that / satisfies the following boundary condition: Re{f(x),jx)

> 0, xe&D,

j x is a support functional of V at x. (7.11)

As a matter of fact, it follows by a result in [Reich (1975)] that (7.1) is equivalent to condition (7.11) if we require it to hold for all support

201

Flow-Invariance Conditions

functional j x . To see this, we define the set /(£>,x) = {z £ X : z = x + a(y - x) for some y € V and a > 0}. (7.12) The closure I(V, x) of this subset is sometimes called the support cone to V at x. Lemma 7.1 Let V be a convex subset of X. For z in X and x in V, the following are equivalent: (1) z G I(V,x).

(7.13)

(2) lim dM,({l-h)x

+ hz,V)/h

=

ti.

(7.14)

h—»0+

(3) Ifx* € X* supports V at x, then Re(z,x*) < Re(x,x*). Proof. (1)=>(3) is immediate. If z does not belong to I(D,x), then there exists x* in X* such that Re(z,x*) > sup{Re(y,x*) : y £ I(D,x)}. The functional x* must support I{V,x) at x. Consequently, (3)=^(1). The implication (2)=>(1) is also immediate. To prove (l)^=s>(2), let e > 0 be given. There are a > 0 and y € T> such that \z — (x + a(y — x))\ < e. LetO is the open unit ball in X and / maps T> into X, then (7.11) can be written as ^inf

Re(f(x),x*) > 0,

a; € dV.

(7.15)

For the classes of monotone and accretive mappings, theflowinvariance condition (7.1) (or equivalently, (7.11)) was systematically used to study the null point set of / and the Cauchy problem (7.2) (see, for example, [Brezis (1973)], [Reich (1975); Reich (1976)], [Goebel and Kirk (1990)], [Brezis (1970)], [Martin (1973)] and [Webb (1996)]. A result of Martin [Martin (1973)] shows that if V is a convex subset of X and / : T> —* X is a continuous accretive mapping on T>, then (7.1)

202

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

(respectively, (7.11) or (7.15)) is also sufficient for the existence of solutions to the Cauchy problem (7.2). This yields a continuous semigroup of nonexpansive mappings on T>. Theorem 7.1 Let T> be a convex subset of a Banach space X and let f :T> —> X be a continuous mapping on V. Assume that f is an accretive mapping on T>. Then f is a generator of a continuous semigroup of norm nonexpansive mappings on T> if and only if it satisfies the boundary flow invariance (7.1): lim dist(x-hf(x),V)/h

= Q, x € V.

(7.16)

For holomorphic mappings an analog of Martin's theorem can be formulated as follows. Theorem 7.2 ([Aizenberg et al. (1996)]) Let V be a bounded convex domain in a complex Banach space X and let f : T> —> X be a uniformly continuous mapping on T>, which is holomorphic in T>. Then f is a generator of a continuous semigroup of holomorphic self-mappings of T> if and only if it satisfies the flow invariance (7.1). Note that in contrast with Martin's theorem, we require that / should be uniformly continuous on V. The first question one can now raise is: What happens when f is not necessarily uniformly continuous on the closure VofV? We will answer this question as well as prove the above theorem (in a somewhat more general setting) by using the notion of the so-called numerical range. 7.2

Numerical Range of Holomorphic Mappings

Let V be a convex domain in a complex Banach space X, and suppose that V contains the origin. Let h : V —> X be a holomorphic mapping. Define hs(x) = h(sx),

0 < s < 1,

(7.17)

and note that hs is holomorphic in the enveloping domain £ V. Clearly, this domain contains T>, the closure of T>. For x e dT>, let J(x) be the set of all continuous linear functionals on X which are tangent to (support) V at x, i.e., J(x) = {I G X* : l(x) = 1, ReZ(y) < 1 for all y€V}.

(7.18)

Flow-Invariance Conditions

203

Let Q(x) be a non-empty subset of J(x). Definition 7.1 If h has a continuous extension to V, the numerical range of h (taken with respect to Q) is the set W(h)={l(h{x)):leQ(x),

xGdV}.

(7.19)

We write V in place of W when the numerical range is taken with respect to J. Note that in the case where V is the open unit ball B = {x£X: \\x\\ < 1}.

(7.20)

This definition agrees with that given by L. A. Harris in [Harris (1971b)]. We now define the upper and lower bounds L+(h) and L~{h) for the numerical range of holomorphic mappings as follows: L+(h) := lim supReV(/is)>

(7-21)

L~(h) := lim inf ReV(hs).

(7.22)

8-*l~

respectively, s—>1~

Using the maximum principle one can show that the limits in the above definition exist when V is balance. Definition 7.2 A holomorphic mapping h : T> —> X is said to be holomorphically dissipative (respectively, holomorphically accretive) if L+{h) < 0,

(7.23)

L'{h)>0.

(7.24)

respectively,

It is clear that h is holomorphically dissipative if and only if the mapping —h is holomorphically accretive. Definition 7.3 A holomorphic mapping h : V —» X is said to be holomorphically conservative if both h and — h are holomorphically dissipative (respectively, holomorphically accretive) i.e., L-(h) = L+(h) = 0.

(7.25)

Let p ( = px>) be the Minkowski functional of V and h : V —> X be a holomorphic mapping. Put ||ft||2> = sup{||M*)||:a:€P},

(7.26)

204

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

when this is finite. Suppose h has a uniformly continuous extension to T>. Then h is holomorphicaly dissipative (respectively, holomorphically accretive) if and only if supReV(/i) < 0 (respectively, inf ReV(/i) > 0), since ||/i — /i s || —> 0 as s —» 1~. Moreover, exactly as it was shown in [Harris (1971b), Theorem 2], one can prove that lun+ l | J + ^ l | p ~ 1 = sup ReW(h),

(7.27)

lim I I ^ - ^ H P - 1

(7.28)

respectively, =

infRe^(fc).

Thus, even though the numerical range may differ when taken with respect to different choices of Q, the number supReW(/i) (respectively, inf ReW(h)) remains fixed. Hence we can replace it by supRef(/i) (respectively, inf ReV(h)). In fact, this remains true for mappings which do not necessarily have a continuous extension to T>. (See the end of this section.) For the case of the open unit ball, the only properties of J{x) our arguments use are that J{x) ^ 0 and (7.29)

XJiXx) C J(x)

whenever |A| = 1 and ||a;|| = 1. For the case of bounded linear operators the notion of dissipativeness coincides with the classical one (see, for example, [Yosida (1974)]). In the theory of linear operators and its applications (to evolution equations, probability and ergodic theory), the classical Lumer-Phillips theorem (see, for example, [Lumer and Philips (1961)] and Theorem 6.6 in Chapter 6) plays an important role. For the case of bounded linear operators the Lumer-Phillips theorem can be reformulated as follows: A bounded linear operator A : X —> X (where X is a Banach space) is dissipative if and only if its resolvent (I — A-A)"1 is well defined on X for all X > 0 and satisfies the condition

||(7-AA)- 1 1| < 1,

A>0.

In other words, for all t > 0, the operator (I — tA)"1 contraction on X.

(7.30) : X —» X is a

205

Flow-Jnvariance Conditions

It turns out that for holomorphic mappings the notion of dissipativeness is equivalent to the flow invariance condition. First, we formulate the following extension of the Lumer-Phillips theorem. Theorem 7.3 Let B be the open unit ball of a complex Banach space X and let h : B —> X be holomorphic. Then h is holomorphically dissipative (respectively, holomorphically accretive) if and only if (I — th)(B) D B and (I — th)-1 is a well-defined holomorphic mapping of B into itself for each t > 0 (respectively, (I + th)(B) D B and (I + th)~1 is a well-defined holomorphic mapping of B into itself for each t > 0). In other words, h £ Hol(B, X) is holomorphically accretive if and only if it satisfies the range condition on B. We will, in fact, prove (the necessity part of) this theorem in a more general setting. Theorem 7.4 Let V be a bounded convex domain in X and leth :T> —> X be a holomorphically dissipative mapping bounded on each subset strictly inside V. Then for each t > 0, we have (I - th){V) D T> and (I -th)'1 is a well-defined holomorphic mapping ofV into itself. Proof. Fix y £ V and t > 0 and define (suspending our subscript convention) (7.31)

g.{x) = -(y + th{8x)) s

for x e V and 0 < s < 1. Given x £ dV and / £ J(x), we have

where

Rel(gs(x)) < - (ReJ(y) + s as = supReV(/i s ).

tas),

(7.32)

(7.33)

Assume without loss of generality that V contains the origin and let p be the Minkowski functional of V. Since p(y) < 1, it follows from our assumption that there exist a constant k < 1 and a number 6 > 0 such that - {p{y) + taa) < k s whenever 1 - S < s < 1. Then f := I — rga satisfies P(f(x)) > Re/(x - rgs(x)) > 1 - rk

(7.34)

(7.35)

206

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

whenever r > 0. The space X is an F-space with respect to p since V is open and bounded. Moreover, by hypothesis and the mean value theorem, there exists a number Ms satisfying P(9s(x) - 9s{y)) < Msp(x -y),

x,yeV

(7.36)

for each s with 0 < s < 1. Fix s with 1 — 6 < s < 1. Then by the inverse function theorem, for all small r > 0 the function / maps T> homeomorphically onto a subset of X containing 0. Hence f{V) 2 ( 1 - rk)V,

(7.37)

by (7.35). Therefore the function F:=f-\(l-r)I)

(7.38)

is a holomorphic mapping of cD into V, where c=(l-rk)/(l-r).

(7.39)

Clearly, c > 1. Since V lies strictly inside cD, the Earle-Hamilton theorem implies that F has a unique fixed point x* in T>. This is also a fixed point for g3 and hence z = sx* satisfies (I~th)(z) = y.

(7.40)

Suppose now that there is another solution z\ of (7.40). Choose s\ < 1 so that s\ > s and si > p{z{). Clearly, x\ = zi/si is in T) and satisfies gSl{xi) = x\. One can show as above that x\ is the unique fixed point of gSl in T>. On the other hand, if we set xi = z/si, then X2 £ "D and 9si(x2) = %2 by (7.40). Hence #2 = xi, so z\ = z. It can be shown (as, for example, in [Harris (1977), Theorem5]) that the solution z of (7.40) depends holomorphically on y £ V. This completes the proof of our asser• tion. Thus we have proved the necessity part of our generalization of the Lumer-Phillips theorem. The sufficiency part is proved in [Harris et al. (2000)]. Another immediate consequence of this theorem and the holomorphic analog of the Hille-Yosida theorem is the following boundary flow invariance condition.

Flow-Invariance Conditions

207

Corollary 7.1 Let T> be a bounded convex domain in a complex Banach space X, and let f :V —> X be a uniformly continuous mapping on V which is holomorphic in T>. If f satisfies the boundary condition inf

x*GJ(x)

Re0,

x £ dV,

(7.41)

then it generates a semigroup of holomorphic self-mappings of T>. This corollary proves, in fact, the last theorem of the previous section. On the other hand, regarding the sufficiency part of the generalized Lummer-Phillips theorem, it does not seem natural to consider a boundary condition to characterize generators. This is because a bounded convex domain itself is a complete metric space with respect to its hyperbolic metric. In addition, there are many examples of holomorphic generators which have no continuous extension to V (see the examples below). In particular, if F £ Hol('D), then / = I — F is a generator because it satisfies the range condition. Thus, the following question arises: 7s there an interior flow invariance condition which characterizes the class of generators ? 7.3

Interior Flow Invariance Conditions

It would be desirable to find such an interior flow invariance condition from which (7.41) could be derived in the case where / £ Hol(P, X) has a continuous extension to V. For the Euclidean ball T> in X = C n , a certain condition in this direction was established by M. Abate [Abate (1992)]. Namely, he proved that / G Hol(I', C n ) is a generator if and only if it satisfies the estimate 2[\\g(x)\\2

-\(g(x),x)\2]Re(g(x),x)

+ (1 - ||a;||2)2Re{/'(x)/(a;), g(x)) > 0,

(7.42)

where g{x) = (1 - ||z|| 2 )/(z) + {f{x), x)x.

(7.43)

For n = 1 this condition becomes Ref(z)z>-~Ref'(z)(l-\z\2),

(7.44)

where z £ A, the open unit disk in the complex plane C and / € Hol(A, C).

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Despite the simplicity of condition (7.44) it is not clear how (7.41) can be derived from (7.44) when / has a continuous extension to A. On the other hand, this condition may be useful in studying the behavior of the derivative of a semi-complete vector field in A. Therefore it is natural to ask the following question: Can this condition be extended to a general Banach space in a form similar to (7.44) (instead of condition (7A2))? Note also that in the one-dimensional case it follows from the maximum principle for harmonic functions that if / E Hol(A,C) has a continuous extension on A, then the boundary flow invariance condition Ref(z)z > 0,

(7.45)

z e A

implies the following interior condition: Ref(z)z>Ref(0)z(l-\z\2),

z e A.

(7.46)

Conversely, it is clear that (7.45) does result from (7.46) if / has a continuous extension to all of A. Thus another question arises: Are (7.46) and its Banach space analog necessary and sufficient for f to be a generator? The following result provides affirmative answers, in any Banach space, to all the questions raised above. Theorem 7.5 Let V be the open unit ball in a complex Banach space X. Then f G Hol(P, X) is a generator on T> if and only if it is bounded on each subset strictly inside T> and one of the following conditions holds: (a) For each x € V there exists x* G J(x) such that (7.47)

Re(f(x)-{l-\\x\\2)f(0),x*)>0; (b) inf

x*& J(x)

R e ( 2 | | z | | 2 / ( z ) + (1 - \\xf)f'(x)x,

x*)>0

x e P ; (7.48)

(c) for each x g T> and for each x* £ J{x),

Re(^-^f'(0)x+(l-\\xe)f(0),x*)
I


I

(7.49)

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 a generator of a group of automorphisms ofV.

209

Flow-Invariance Conditions

Note that in the one-dimensional case condition (a) reduces to (7.46) and condition (b) becomes Abate's condition (7.44). Proof. First we observe that condition (a) (as well as condition (c)) implies that / must be a holomorphically accretive mapping, hence satisfy the range condition on V. Thus, to prove our theorem we just have to show the equivalence of conditions (a), (b) and (c), and that the property of / to be a generator implies condition (a). So let / be a generator of the one-parameter semigroup {Ft}t>o of holomorphic self-mappings of V. We intend to prove that condition (a) of the theorem is satisfied. Indeed, fix any x e V and x* e J(x), and set u = a;/||a;||, u* = a;*/||a;||. Consider the holomorphic function / on the unit disk A C C defined as follows: /(A) = (/(Au),u'>,

AeA.

(7.50)

Similarly we define a family {Ft}t>o of holomorphic self-mappings of A: F t (A) = (Ft (Au), u*),

A e A , t > 0.

(7.51)

It is clear that Fo(A) = A and that there exists the limit tlim+

\ (A - Ft (A)) = /(A).

(7.52)

Now let Mb denote the Mobius transformation on A defined by Mb(X) = ^ % , 1 — Ao

(7.53)

be A.

Consider the family {Ht}t>o of holomorphic self-mappings of A denned by Ht(\)

= Mh{0)(Ft(X)),

(7.54)

\£A,t>0.

Note t h a t since W«(0) = 0 for all t, it follows by the Schwarz L e m m a t h a t |Wt(A)| < |A| for all A e A and t > 0.

(7.55)

Now, by simple calculations, one can conclude t h a t for each A e A t h e curve Wt(A) : M + —> A is right-differentiable at zero a n d t Um

A

~ ^ t ( A = / ( A ) - / ( 0 ) + ?(0)A 2 : = h(X),

AeA.

(7.56)

It follows now by (7.55) a n d (7.56) t h a t for all A e A ,

Reh(\)X > 0

(7.57)

210

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and ft(0) = 0.

(7.58)

Re/(A)A > Re/(0)A(l - |A|2).

(7.59)

But (7.57) means that

Setting in this inequality A = ||a;|| we have by (7.50), Re{f(x),x*} > Re
(7.60)

Since x and x* £ J(x) are arbitrary, this proves the following implication: If / is a generator, then / satisfies (a). Now we will show that (a) is equivalent to (b). To do this, we return again to the function /(A) = /(0)-7(0)A 2 + MA)

(7-61)

denned by (7.50), where h(X) satisfies (7.56) and (7.57). But these conditions are equivalent to the conditions h(X) = X-p(X),

A€ A

(7.62)

with Rep(A) > 0.

(7.63)

So, (7.59) is equivalent to (7.62) and (7.63) for / of the form (7.61). Now, in the same terms, we translate condition (b). If we define / as above by (7.50), then (b) implies Re[2/(A)A + /'(A)(l - |A|2)] > 0.

(7.64)

If we substitute here / in the form (7.61) with h(X) = Xp)X) we see that (7.64) is equivalent to the condition

Re [Ap'(A) + i ± | ^ p ( A ) ] > 0, A e A.

(7.65)

We intend to show that (7.65) is equivalent to (7.63). Then setting again A = ||z|| in (7.50) and (7.64) and noting that /'(A) = (f'(Xu)u,u') = (f'(x)x,x*)Tr^ for x ^ 0, we will get by continuity the equivalence of conditions (a) and (b) of the theorem.

211

Flow-Invariance Conditions

So, let p e Hol(A, C) satisfy (7.63). Define F = (p - l)(p + I)" 1 . Since the mapping w = |^y maps the right half-plane into A, F is a self-mapping of A. Applying the Schwarz-Pick Lemma to F, we obtain (p-l\ \p+l)

_

2\p'\ |p + l | 2 - | p - l | 2 | l + p P - | P+ 1|2(1_|A|2)

V™>

or

b ' W < ^ .

(7.67)

This implies

Re(-Ap'(A)) < |Ap'(A)| < ' ' ^ f f l < \ ^ Rep(A), (7.68) which is equivalent to (7.65). In the opposite direction ((7.65) implies (7.63)) we prove the following somewhat more general fact: Let p S Hol(A,C) and suppose that there is a positive function ip : [0,1) —> R+ such that the following condition holds: Re(Ap'(A) + V(|A|)p(A)) > 0,

A e A.

(7.69)

Then Rep(A) > 0 everywhere on A. Indeed, setting A = re%e we have V(A)=r^

(7.70)

and (7.69) becomes Re (r^j

+ V>(r)Rep(A) > 0,

A = reie e A.

(7.71)

Assume now that there exists Ao = roei9° in A such that Rep(Ao) < 0.

(7.72)

Since (7.69) implies that Rep(0) > 0, there exist 0 < J~I < ro such that Rep(7-iei9°) = 0 and Rep(roeie°) < 0. Thus one can find r 2 e (ri,r0) such that Rep(r2ei9°) < 0

(7.73)

Re^(rac<*")<0.

(7.74)

and

212

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

This implies Re (r-2^)

+ rp(r2)Rep(r2eieo) < 0,

(7.75)

a contradiction. Thus Rep(A) > 0 for all A e A and we are done. In other words, conditions (a) and (b) are equivalent. To obtain (c) we now represent /(A) = (f(Xu),u*) in the form

f(X) = a + ibX-aX2 + X f I±££dji(O. JdA 1 - C*

(7 . 76)

and we calculate: 1 + C A = 1-|A[2 1 - C"A |1 - CA|2

=

1 - 1 * 1 (1 + |A|) 2 1 + |A| |1 - CA|2

=

1 + 1A| ( l - j A | ) 2 1 - |A| ' |1 - CAP • (7.77)

Since |C| = 1, this equality shows t h a t

U * l < Rel±CA 1+JA[

(7.78)

Noting that p(0) = /'(0) (see (7.61) and (7.62)), we obtain from (7.76) and (7.78)

Re/(0)A(l - |A|2) + | A | 2 ^ j /'(0) > Re/(A)A > Re/(0)A(l - |A|2) + | A | 2 [ ^ J /'(0).

(7.79)

Once again, setting A = ||x|| we get from the last inequality condition (c). Since by (b), Re(f'(0)x,x*) > 0, it is clear that (c) is stronger than (a).

•

The following fact is a direct consequence of Theorem 7.5. Corollary 7.2 Let f 6 Hol(P, X) be a holomorphic generator on T>. Then the linear operator A = f'(0) is (totally) accretive, i.e., inf

y*€J(y)

Re(Ay,y*) > 0,

y G X.

(7.80)

Proof. Substitute x = ty, x* = ty* in (b), where \\y\\ = \\y*\\ = 1, y* £ J(y) and t 6 (0,1). Letting t tend to zero we get (7.80). In its turn, (7.80) implies that the left-hand inequality in (c) is sharper than (a). Moreover, it implies that (a) holds for all x* £ J(x). Thus we

Flow-Invariance Conditions

213

have that if / has a continuous extension to V, then (a) yields the flow • invariance boundary condition (7.41) (equivalently, (7.1)).

7.4

Semi-Complete and Complete Vector Fields

In connection with flow invariance conditions we give the following definition. Definition 7.4 A holomorphic vector field

(7.81) on a domain V is determined by a holomorphic mapping / £ Hol(£>, X) and can be regarded as a linear operator mapping Hol(X>, X) into itself, where Tfg £ Hol(£>, X) is defined by (Tfg)(x)=g'(x)f(x),

xGV.

(7.82)

The set of all holomorphic vector fields on V is a Lie algebra under the commutator bracket

[Tg,Th] = \g(x)^

, h ( x ) ^ ] := {g'(x)h(x) - h'(x)g{x))^.

(7 . 8 3)

(see, for example, [Dineen (1989)]). Furthermore, each vector field (7.81) is locally integrable in the following sense: for each x £ T> there exist a neighborhood ft C T> of x and 8 > 0 such that the Cauchy problem (7.2) has a unique solution {u(t, x)} C T> defined on the set {\t\ <S}xQ,cRxV. Definition 7.5 A holomorphic vector field Tf defined by (7.81) and (7.82) is said to be (right) semi-complete (respectively, complete) on T> if the solution of the Cauchy problem (7.2) is well-defined on all of R+ x V (respectively, R x P), where R + = [0, oo) (respectively, R = (-00,00)). Thus, if V is hyperbolic, then Tg is semi-complete (respectively, complete) if and only if g is the generator of a one-parameter continuous semigroup (respectively, group). On the other hand, if T> is bounded and Hol(P, X) is the subspace of Hol(P, X) consisting of all those g £ Hol(r>, X) which are bounded on each ball strictly inside V, then a semigroup (group) {Ft}, t G R +

214

Nonlinear Semigroups, Fixed Points, and Geometry of Domainsh

(respectively, t G R), induces ^Jinear semigroup (group) {L(t)} of linear mappings L(t) : Kol(D,X) i-> Hol(£>,X), denned by (L(t)g)(x) := g(Ft(x)),

(7.84)

where i € l + ( t 6 K) and x e V. This semigroup is called the semigroup of composition operators on Hol(P,X). If {F t }, t € R+(t € R), is T-continuous (that is, differentiate), then {L(t)}, t e R + (i £ R), is also differentiable and

(7.85)

I £(0)<7 = 5

for all g 6 Hoi(P, X), where f = ^ _ . In other words, a holomorphic vector field Tf, denned by (7.81) and (7.82) and considered a linear operator on Hol(£>, X), is the infinitesimal generator of the semigroup {L(t)}. It is sometimes called the Lie generator. Thus a holomorphic vector field Tf is semi-complete (respectively, complete) if and only if it is the Lie generator of a linear semigroup (respectively, group) of composition operators on 1101(1?, X). This follows from the observation that (7.86)

L(t)Iv = Ft and Tflv = /,

(7-87)

where I-D is the restriction of the identity operator to V. Moreover, using the exponential formula representation for the linear semigroup: L®9

= E ~ljr-

Tf9

= eM-tTf]g

(7.88)

= <*PHT/]J O .

(7.89)

fc=O

(see Chapter 6), we also have 5W

= E -TT~ fc=o

K-

Tffv

Flow-Invariance Conditions

215

So, a locally uniformly continuous semigroup of holomorphic self-mappings can be represented in exponential form by the holomorphic vector field induced by its generator. In particular, Theorem 7.5 (see the previous section) asserts that if / £ Hol(D, X) then T/ (= f{x)-§^) is a semi-complete vector field if and only if / satisfies the interior flow invariance condition (a) (respectively, (b) or (c)). Furthermore, condition (c) also leads to a characterization of a semicomplete vector field to be complete. Following S. G. Krein [Krein (1971)] (see also [Vesentini (1996a]), we say that a linear operator A : X —> X is conservative if for all x € X and x* £ J{x) there holds Re(Ax,x*) = 0.

(7.90)

In order to simplify our terminology we will sometimes identify a semicomplete (complete) vector field Tf with its determining holomorphic mapping / , which is, in fact, the infinitesimal generator of a semigroup (group) of holomorphic self-mappings on T>. The following corollaries are consequences of Theorem 7.5 in the previous section. Corollary 7.3 Let f e Hol(X>, X) be a semi-complete vector field. Then f is actually complete if and only if its derivative at zero, /'(0), is a conservative linear operator. It is well known that a complete vector field g on the open unit ball V in a Banach space X is a polynomial of degree at most 2 (see, for example, [Arazy (1987)], [Dineen (1989)] and [Upmeier (1986)]). More precisely, g has the form g{x) = a + Ax + Pa{x),

(7.91)

where a is an element of X, A is a conservative operator on X, and Pa is a homogeneous form of the second degree such that Pia = iPa. One of the consequences of this representation is an infinitesimal analog of Cartan's uniqueness theorem ([Arazy (1987)], [Dineen (1989)] and [isidro and Stacho (1984)]): If g 6 Hol(Z>,X) is a complete vector field such that 5(0) = 0 and g'(0) = 0, then g = 0. Applying Corollary 7.3, we obtain at once the following extension of this theorem.

216

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Corollary 7.4 If f £ Hol(D, X) is a semi-complete vector field on V such that /(0) = 0 and /'(0) = 0, then f = 0. We know already that if / e HolCD, X) has the form (7.92)

f = I-F,

where F is a self-mapping of V, then / is semi-complete. Thus Corollary 7.4 is also a generalization of Cartan's uniqueness theorem (sometimes this theorem is also called the Generalized Schwarz Lemma (see Chapter 3)). Suppose now that a complex Banach space X is a so-called JB* triple system. This is equivalent to saying that its open unit ball I? is a homogeneous domain, i.e., for each pair x,y £ T> there exists a holomorphic automorphism of V such that F(x) = y (see, for example, [Upmeier (1986); Dineen (1989); Isidro and Stacho (1984)]). Then it is well-known that for each a £ X there exists a homogeneous polynomial Pa{x) such that Pia = iPa and the mapping g :T> —> X defined by g{x)=a-Pa{x)

(7.93)

is a complete vector field on T>, which is called a transvection of T>. Using this fact and Corollary 4 in [Aharonov et al. (1999b)] we get the following representation theorem. Corollary 7.5 Let X be a JB* triple system and let V be its open unit ball. Then the cone Q of semi-complete vector fields on V admits the decomposition (7.94)

g = go@g+,

where QQ is the real Banach subspace of HitOO(V,X) consisting of transvections and Q+ is the subcone of Q such that for each h £ Q+, XD£ Re(h(x),x*)>0

for all z £ P .

(7.95)

In other words, / G Q admits a unique representation f = 9+ h

(7.96)

where g = /(0) - Pf(0)(x) is complete, h € G+ and h(0) — 0. The natural examples of JB* triple systems are a complex Hilbert space H, the space of bounded linear operators L(H) on H, and its subspaces J such that A £ J if and only if AA* A £ J (such subspaces are called

217

Flow-Invariance Conditions

J*-algebras); see [Harris (1974a); Dineen (1989)]. In the latter case the general form of transvections on V is (7.97)

g(x) = a - xa*x,

where a £ J and a* is its conjugate. Thus each semi-complete vector field on the open unit ball of a J* -algebra has the form (7.98)

f(x) = f(0)-xf*(0)x + h(x),

where he 6+ and h(0) = 0. In particular, when X = C is the complex plane and V = A the open unit disk in C, (7.98) becomes /(*) = / ( 0 ) - 7 ( 0 ) * 2 + *P(*),

(7-99)

where p{z) e Hol(A, C) and Rep(z) > 0,

(7.100)

zGA.

That is, p(z) is a function in the class of Caratheodory. Using the RieszHerglotz integral characterization of this class (see, for example, [Duren (1983)] and [Aleksandrov (1994)]) we deduce the following conclusion: / € Hol(A, C) is a semi-complete vector field if and only if it admits the representation

f{z) = a + ibz-az2 + z [

1+iZ

JdA 1 - C*

d/x(C),

(7.101)

where a e C, b G M and fi is a positive measure on dA. As a matter of fact, this representation is the key to proving the results in the previous section because we have partially used a reduction to the one-dimensional case. We have also obtained (7.99) with (7.100) by an independent method.

Chapter 8

Stationary Points of Continuous Semigroups 8.1

Generalities

Let V be a topological space, and let a family S = {Ft € (0, T)}, T> 0, of self-mappings Ft of V form a (one-parameter) continuous semigroup, i.e., (i) F3+t = Ft-Fs,

0<s + t
(8.1)

(ii) hmFt(x) = x,

(8.2)

where the limit is taken with respect to the topology of V. Definition 8.1 A subset W of V is said to be the stationary point set of S if it consists of all the points a £ T> such that Ft(a) = a for all te (0,T).

(8.3)

In other words,

W=

P) FixF t .

(8.4)

0
The first problem in this context is the existence problem of a stationary point of S, in other words, to determine when W ^ 0. The second problem is the uniqueness problem, that is, to find out when W consists of a unique point a € T>. This problem is connected with a more general, rather important in applications, problem - to determine the structure of the set W in relation to the topological structure of T>. Another important problem is to find constructive methods for the approximation of W. In 219

220

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

the case when T = oo this problem, in particular, is closely related to the study of the asymptotic behavior of semigroups. Definition 8.2 A point a 6 P i s said to be a periodic point of a semigroup S = {Ft : t £ (0, T)} if for some t0 > 0 this point is a fixed point of Fto, i.e., a = Fto(a), hence a = -Ft"(a) for all n = 0,1,2,... . The simplest situation is, of course, when for some t0 > 0 the mapping Ff0 : T> —> T> has a unique fixed point a = Ft0 (a) in T>. Then directly from the semigroup property (i) it follows that W = {a} ^ 0. But even in this case the approximation problem is still open. Sometimes one can study the existence and the structure of W as the common fixed point set of the commuting family {Ft}. Indeed, if, for example, V = B™, where B is the open Hilbert ball, then the following assertion, due to I. Shafrir ([Shafrir (1992a)]) is useful for the study of the common fixed point sets of one-parameter semigroups, which consist of nonexpansive mappings with respect to the hyperbolic metric p on V (= B n ). Lemma 8.1 Let {Fa : a £ J} be a commuting family of pnonexpansive (holomorphic mappings) self-mappings o/B n and assume that there is a p-bounded subset M C B™ which is Fa-invariant for all a £ J. Then there exists a common fixed point a £ B n of this family. Remark 8.1 Observe that if in Lemma 8.1 we omit the assumption that there exists a p-bounded subset M c B n which is Fa-invariant for all a £ J, then one can construct a counterexample even for n = 1. Example 8.1 ([Kuczumow et al. (2001a)]) Let B be the open unit ball in a separable Hilbert space (H, (•, •)) with the orthonormal basis {en}n6pjWith each affine subset Y = (x + Y) nB of B, where Y is a closed subspace of X and x £ B is the unique point of least norm in Y, we associate the nearest point holomorphic retraction of © onto Y given by the formula R = Mxo Projy o M_ x ,

(8.5)

where Mx is a Mobius transformation. We construct the family {-Fn}n6N in the following way: we choose a sequence of positive real numbers {an}nen such that this sequence is strictly decreasing, 0 < an < h n and lim an = n—*oo

0. Let us set Xo = X, Co = B, a0 = 0, r0 = ||ci - ao||.

(8.6)

221

Stationary Points of Continuous Semigroups

For n = 1, the mapping R\ is the nearest point holomorphic retraction of B onto the affine set C\ in B, where

(8.7) gi = e 2 - a0 - j .

j72 (e 2 - a0, ei - o 0 )(ei - a 0 ) ,

ra*-

*i =

(8.8)

(8-9)

ai = a o + ro ( - + - cosai)/ x + ( - s i n a ^ M ,

(8.10)

ri = ||ei - a i | | ,

(8.11)

Xi = {xeX0:x

= a1+y

and (ai,y) = 0}

(8.12)

and Ci=Xinl.

(8.13)

£ X, fi, f2,..., / „ £ If we already have r 0 , n,..., rn £ M, ai, a2,...,an X, ffi,ff2, • • • i3n € X, hi,h,2,...,hne X, X0,Xi,... ,Xn, CQ,C\, . . . , C n and Ri,R2,...,Rn, where each Xi is an affine set in X, e i , a , 6 d c Xuek € ^ for /c > i + 2(i = 0 , . . . ,n),Xi+1 C Xi for i = 0 , . . . , n - 1, and each Rt is the nearest point holomorphic retraction of B onto d for i = 1 , . . . , n, then the next holomorphic retraction Rn+i is the nearest point holomorphic retraction onto Cn+1=Xn+1nB,

(8.14)

where fn+i = ij jT (ei - an), \\ei - an\\ 9n+l = e-n+2 - On — T, ftn+i = M

r,9n+\,

(8.15) TTo ( e "+2 "" an el ~ an){ei

||ei — an\\

~ an),

(8.16) (8-17)

an+i = a\\9n+l\\ n + rn U - + - c o s a n + i j / n + i + ( - sina n + ij/i n + i ,(8.18) r-n+i = llei-On+iH,

(8.19)

222

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and Xn+1 = {x e Xn : x = an+i + y and ( a n + 1 , y ) = O } .

(8.20)

Now it is sufficient to take Fn = RnORn^

o.-.oRx

(8.21)

= Fmo Fn

(8.22)

for n = 1,2,... to get FnoFm for every m, n £ N, FixF n = Cn

(8.23)

lim diamCn = 0,

(8.24)

for each n s N, n—too

and oo

f ) FixF n = 0.

(8.25)

n=l

Nevertheless, for a one-parameter continuous semigroup of pnonexpansive mappings the following assertion holds. Theorem 8.1 Let B be the open unit ball in a complex Hilbert space H and let p be the hyperbolic metric on B n , n > 1. If S = {Ft : t > 0} is a continuous semigroup of p-nonexpansive self-mapping ofW1, then the following statements are equivalent: (i) S has a periodic point in W1, i.e., there is to > 0 such that Ft0 has a fixed point in B n ; (ii) Ft has a fixed point in B n for each t > 0; (in) there is a stationary point a S B n of the semigroup S. We prove this theorem in a somewhat more general setting. In this connection we give the following definition. Definition 8.3 Let V be a topological space. We say that a class G of self-mappings of V has the compact commuting property (CCP) if for each commuting family {Fa : a £ J} the set f] Fix.Fa is not empty a£j

whenever there is a compact subset M of V which is Fa-invariant for all aE J.

Stationary Points of Continuous Semigroups

223

Proposition 8.1 Let T> be a topological space, and let G be a class of self-mappings of V which has the compact commuting property (CCP). If S = {F t : t > 0} C G is a continuous semigroup of self-mappings of V which has a periodic point in V, i.e., there is SQ > 0 such that FSo has a fixed point in T>, then there is a stationary point of the semigroup S, i.e., (1 FixF t ^ 0. t>o

Proof. Let a £ V be a periodic point of S, that is, a is a fixed point of FSo for some s 0 > 0. Consider the compact subset M of V defined by M = {Ft(a) : 0 < t < s0}. We claim that FS{M) C M for all s > 0. Indeed, if x G M, then there is 0 < p < so such that x = Fp(a), and we have, by the semigroup property (i), that Fs(x) = Fs(Fp(a)) = Fs+P(a).

(8.26)

On the other hand, the number s+p can be represented as s + p = USQ +1 for some 0 < t < SQ. Once again, by the semigroup property (i), we see that Fs(x) = Fs+P(a) = Ft+n30(a) = Ft(Fnso(a)) = Ft(F?0(a)) = Ft(a)eM.

(8.27)

This proves our claim which, in turn, implies the existence of a common fixed point of S. O Now Theorem 8.1 is seen to be a consequence of Proposition 8.1 and Lemma 8.1. Corollary 8.1 Let S be a continuous semigroup of holomorphic mappings ofM such that for some to € (0,oo), Fto — I- Then S is a group of automorphisms ofM of elliptic type, i.e., Ft = M~l o etA o M, where A is a linear conservative operator (Re{Ax, x) — 0 for all x in H) and M is a Mobius transformation o/B. Proof. Since Fto = /, Theorem 8.1 shows that S has a common fixed point o £ l . Let M = M-a denote a Mobius transformation of B such that M(a) — 0, and consider the semigroup $(£) = M o Fto M" 1 defined on B. It is clear that $(t)(0) = 0 for all t > 0, and that $(t 0 ) = /. Since $(s) o $(t 0 - s) = $(*o) = /

(8.28)

224

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

for all 0 < s < t0, we see that for all 0 < s < t0, <J>~J(s) = $(£0 — s) and $(s) is an automorphism of B. Hence $(s) is the restriction to B of a linear isometry of H onto itself for each s > 0. This implies our assertion. • By using the notion of the duality mapping one can easily reformulate this result for B n too. Occasionally, the mappings Ft, t £ M.+ , will also be denoted by F(t). The above approach implies also the following observation. If V is a strongly convex bounded C 2 domain in C n and {F(t) : t > 0} is a semigroup of holomorphic self-mappings of V such that for each t > 0, F(t) has a fixed point in V, then it follows by a result in [Abate (1989a)] (see also [Abate and Vigue (1991)]) that this semigroup has a common fixed point (stationary point) in I? as a commutative family of holomorphic mappings. Moreover, as in the case of the Hilbert ball, it is enough to require the existence of an interior fixed point only for one to > 0 to provide the existence of such a point for the whole semigroup. Theorem 8.2 (cf. [Abate (1988b)]) Let V be a strongly convex bounded C2 domain in Cn and let {Ft : t > 0} be a semigroup of holomorphic self-mappings ofV. Then the following assertions are equivalent: (a) The semigroup {Ft} has a stationary point in T>. (b) The semigroup {Ft} has a periodic point in V, i.e., there exists to > 0 such that Ft0 has a fixed point in T>. (c) There exists x £ T> and a sequence tn —> oo such that {Ftn(x)} is strictly inside T>. (d) For each x e V there is a sequence tn —» oo such that {Ftn(x)} is strictly inside V. We will give a proof of this theorem which is different from the one in [Abate (1988b)] by using the infinitesimal generator of a semigroup in a somewhat more general situation. Unfortunately, we cannot assert the full analog of this theorem for the infinite dimensional case (even for the Hilbert ball), but for some classes of p-nonexpansive or holomorphic mappings one can formulate a similar statement. We need the following notion. Definition 8.4 Let V be a domain in a Banach space X. We will say that a class G of self-mappings of V has the Denjoy-Wolff iteration property (DWIP) if whenever F G G has no fixed point in V, the sequence of iterates {Fn} strongly converges to a point on the boundary of V, uniformly on each compact subset of V.

225

Stationary Points of Continuous Semigroups

Proposition 8.2 Let G be a class of p-nonexpansive which satisfies the Denjoy-Wolff iteration property. Let a one-parameter continuous semigroup of p-nonexpansive which has no common fixed point inM. If Fto € G for at then there is a point e € <9B such that S converges toeast uniformly on each compact subset o/B.

mappings on B S = {Ft}t>o be mappings on B least one to > 0, tends to infinity,

Proof. First we note that by Theorem 8.1, Fto has no fixed point in B. Therefore there is a point e £ dM such that Fnt0 converges to e uniformly on each compact subset of B. Let C be a compact subset of B. Since the semigroup S = {Ft}t>o is continuous, the set C : = {zs = Fs(z) :z&C,

0 < s < t0}

(8.29)

is also a compact subset of B". Therefore, for each e > 0 one can find n 0 € N = {1,2,... } such that s u p | | F t " W - e | | = sup sup | | F n t o ( z s ) - e | | z€C

i€c

0<s
= sup sup ||F nto+s (5) - e|| < e sec o<s
(8.30)

for all n> UQ. If follows now that ||Ft(z)-e||<e for each t > no^o a n d z € C.

(8.31) •

Remark 8.2 For the Hilbert ballM the following classes of self-mappings o/B are known (see [Kuczumow et al. (2001a)]) to satisfy the Denjoy-Wolff iteration property: (1) the class Q\ consisting of the condensing holomorphic mappings; (2) the class Qi consisting of the firmly p-nonexpansive mappings of the first kind; (3) the class Qz consisting of the firmly p-nonexpansive mappings of the second kind; (4) the class Gi consisting of the averaged mappings of the first kind, i.e., F = (1 — c)I © cT, where T is p-nonexpansive and c G (0,1); (5) the class Q5 consisting of the averaged mappings of the second kind, i.e., F = (1 — c)I + cT, where T is p-nonexpansive and c £ (0,1). Thus if a semigroup S = {Ft}t>o contains at least one element Ft0, to > 0, of the class Gi,l
226

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

In a somewhat more general situation one can say the following. Proposition 8.3 Let D be a domain in a Banach space X, and let G be a class of self-mappings of V which has both the (CCP) and the (DWIP) properties. Suppose that {Ft : t > 0} C G is a semigroup of self-mappings of T>. Then the following assertions are equivalentfa) The semigroup {Ft} has a stationary point in T>. (b) The semigroup {Ft} has a periodic point in V, i.e., there exists to > 0 such that Ft0 has a fixed point in V. (c) There exists x G T> and a sequence tn —•> oo such that {Ftn(x)} is strictly inside V. (d) For each x G V there is a sequence tn —* oo such that {Ftn{x)} is strictly inside T>.

8.2

Generated Semigroups

As a matter of fact, we already know that if P is a bounded domain in C n , then each continuous semigroup of holomorphic self-mappings of V is locally uniformly continuous and hence differentiable at t = 0. We consider now such a situation in general. That is, we assume that T> is a domain in a Banach space X and a semigroup S — {Ft : t > 0} is generated on T>, i.e., for each x € T> there exists the strong limit f(x)=lim+-t(x-Ft(x)).

(8.32)

If in this case the mapping / : V —> X (the infinitesimal generator of S) is locally Lipshitzian on V, then, by using the uniqueness of the solution to the Cauchy problem

lim u(t, x) = x, \ t-»o+

(8.33)

where u(t, x) = Ft(x) and / is defined by (8.32), it follows that the stationary point set W is the null point set (Nullp/) of / in V, i.e., W = p ) Fixx,Ft = Null©/.

(8.34)

l>0

Once again, we continue with a somewhat more general situation.

227

Stationary Points of Continuous Semigroups

Definition 8.5 We say that a class M of self-mappings of V has the common fixed point property (CFPP) if the following condition holds: If {F3}S£A i s a ne * of commuting mappings in Ai such that for each s € A, Fs has a fixed point in V, then f] Fixpi^ ^ 0. s€A

Proposition 8.4 Let X be a real or complex Banach space, and let T> be a domain in X. Suppose that M is a class of self-mappings ofT> with both the (CFPP) and the (DWIP). Let f : V -> X be the infinitesimal generator of a one-parameter continuous semigroup {F(t) : t > 0} C M such that {F(t)} is locally uniformly Lipschitzian on V. If f has no null point in V, then for each x 6 V the net {F(t)(x)} strongly converges to a point b on the boundary ofV. Proof. First we note that there is an interval (0, /i) such that for each t G (0, /J,), F(t) has no null point in V. Indeed, if we suppose that there is a sequence tn —> 0 such that for each n the mapping F(tn) has a fixed point in V, then by the (CFPP) there is a point x GV such that F(tn)(x) = x for all n and hence f[x) = 0. This is a contradiction. So, for each m large enough (m > l//x) all the mappings F ( ^ ) have no fixed point in V. This means that for each x e V the sequence F ( ^ ) (x) (= Fn (~^)) converges strongly as n —> oo to a point bm on the boundary of T>. But it follows from the semigroup property that Fnm ( ^ ) (x) = Fn(l)(x) -> bm, a s n - » o o , and hence bm = b does not depend on m. We are now able to show that the semigroup {F(t)} strongly converges to 6 as t tends to infinity. In fact, for any given e > 0 and x G V, we can choose S > 0 such that ||.F(t)(a:) - F(t)(y)\\ < e/2 for all t > 0 whenever y € V and ||y-a;|| < 5. For such 8 we take m e N so large that m" 1 € (0, /x) and \\F(h){x) - x\\ < 6 for all h € [0,1/m). Finally, for such m and t > 0, setting n = [tm], we have \\F(£)(X)

~b\\ = \\Fn(±)(x)-b\\ < e/2

(8.35)

for t > 0 big enough. Since h = t- % € [0, i ) , we get for such t > 0, \\F(t)(x)-b\\ = \\F(~)(F(h)(x))-b\\<

IK)w)w)-^)w||+HS)w-*i


(8.36)

228

iVon.Kn.ear Semigroups, Fixed Points, and Geometry of Domains

and we are done.

•

Corollary 8.2 Under the conditions of Proposition 8.4, assume in addition that f is also locally Lipschitzian. Then the semigroup {F(t)} has a stationary point in V if and only if for some t 0 > 0 the mapping F(to) has a fixed point in V. Note now that by results in [Abate (1988a)] (see also [Kuczumow and Stachura (1990)]) for each strongly convex bounded C 2 domain V in C n the class of holomorphic self-mappings of T> has the Denjoy-Wolff iteration and the common fixed point properties. In addition, it is known [Abate (1992)] that each one-parameter semigroup of holomorphic self-mappings on V is differentiable with respect to the parameter. In this way we are led to a proof of Theorem 8.2. Remark 8.3 The following example shows that formula (8.34) is no longer true for the closure of V even in the case when f is continuous onV. Example 8.2 Let T> be the open unit disk in the complex plane C, i.e., V = {x e C : |x| < 1}. Consider f(x) — x - 1 + y/l - x. It is clear that / € Ho^X*, C) and that it is continuous on T>. In addition, Nul%/ = {0,1}. Ho"wever, the Cauchy problem (8.33) has the solution Ft : V ^ V, t>0

(8.37)

Ft(x) = l - [ l - e - ^ + e - ^ v / T 1 ^ ] 2 ,

(8.38)

defined by the formula

and for all £ > 0 we have Ft(l) = l - [ l - e - 5 f ] 2 < l .

(8.39)

Thus % ^ Nul%/. We will study boundary null points of generators later.

8.3

The Resolvent Method

Since in most situations the generator of a one-parameter semigroup satisfies the range condition, one can study the common fixed point set of the

Stationary Points of Continuous Semigroups

229

semigroup as the fixed point set of a single self-mapping defined by the resolvent of the generator. Lemma 8.2 Let T> be a domain in X, and let f : T> —> X satisfy the range condition on T>, i.e., the resolvent JT = (I + rf)~1 is well-defined for allr>0. Then for each r > 0, Fix c Jr = Nullp/. Proof. identity

(8.40)

Indeed, by definition, for each x £V and each r > 0 we have the Jr(x)+rf(Jr(x)) = x.

(8.41)

If a £ FixpJr, then we have by (8.41) f(a) = f{Jr(a)) = 0, i.e., a £ NUIID/. Conversely, if /(a) = 0, then (/ + rf)(a) = a for all r > 0. Thus J r (a) = (/ + r / ) - 1 ( a ) = a. D We now describe the resolvent method for the study of the null point set of generators. This method can be used to determine its structure, to solve the existence problem, as well as to find methods for approximating null points of generators. First we will trace a parallel with an implicit method for the study of the fixed point set of a single self-mapping of a domain. Let P be a bounded convex metric domain in X endowed with a metric p compatible with the convex structure of V (in particular, p may be the hyperbolic metric on V). Suppose that F : V —> V is a /)-nonexpansive self-mapping of V. For t e [0,1) and a fixed y &T>, consider the mapping $(x) = tF(x) + (1 — t)y, x e V. Since $ is a strict contraction on (V, p) (in particular, $ maps T> strictly inside itself), the Banach principle implies that there exists a unique fixed point z = zt(y) £ V of the mapping $. Moreover, zt(y) = lim $ n (y), n—>oo

(8.42)

where the limit is uniform on each p-ball in (T>, p). (For holomorphic mappings the same conclusion follows from the Earle-Hamilton theorem.) The X-valued function zt{y), 0 < t < 1, is called an approximating

curve. At the same time, changing our point of view, formula (8.42) implies that for each t £ [0,1), zt = zt{y) can be considered a /9-nonexpansive selfmapping of V depending on y 6 V (in particular, if F is holomorphic, then

230

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

zt depends holoraorphically on y G P). We denote this mapping by %. In other words, % is the unique solution of the nonlinear operator equation Tt=tFo%

+ {\- t)I.

(8.43)

The mapping % belongs to Np(V) (in particular, to Hol(2?)), and as we already know Fix(Ti) = Fix(F),

t e (0,1).

(8.44)

On the other hand, we know that the mapping g = I — F is a generator of a continuous flow of p-nonexpansive self-mappings of D. In addition, g : V —» X satisfies the range condition, i.e., for each r > 0 the resolvent Gr = (I + rg)-1 is a well defined p-nonexpansive (holomorphic) self-mapping of V. Claim 8.1 Gr = % with t = j ^ . Indeed, since Gr = (I + r(I - F))~x is the solution of the equation (/ + r(I - F)) oGr = I, we have (1 + r)Gr - rF o Gr = I or

Gr = -?—FoGr + -±-I.

(8.45)

1+r 1+r Setting t = jq^;, we get our claim by the uniqueness of the solution of this equation. Returning to the general case, let / : T> —> X be the generator of a continuous flow of p-nonexpansive self-mappings of V, and let Js = (I + s/)" 1 , s > 0, be its resolvent. Setting F — Js in our previous considerations, we see that for each r > 0 the mapping Gr = (I + r(I — J s ) ) - 1 is a well-defined p-nonexpansive self-mapping of V. The following relation is the key to our approach in the sequel. Lemma 8.3 Let f : V —> X be the generator of a continuous flow of p-nonexpansive self-mappings ofV, and let Ja = (I + s/)" 1 , s > 0, be its resolvent. Define GT = (/ -I- r(I - Js))-1 : V -> V. Then J(r+l)s = Js(Gr) = J.{I + r(I - Js))-1.

(8.46)

Proof. Fix s > 0 and set F = J3. By the previous claim we know that for each r > 0, the mapping Gr = (I + r(I — Ja))~x is a well-defined pnonexpansive self-mapping of V and for each x G V this mapping can be defined as the unique solution of the equation °r(X)

= 7^1

J'(Gr(x))

+ ^l*-

(8-47)

Stationary Points of Continuous Semigroups

231

On the other hand, since [I + r(I - Ja)]Gr = / and I-Js

— sf(Js) we have

[/ + rsf(Js)]Gr(x) = Gr{x) + rsf(Js(Gr{x)))

= x,

xGV.

(8.48)

Hence it follows from (8.47) that Js(Gr(x)) + (1 + r)sf(Ja(Gr(x))) = x,

xeV.

(8.49)

Since the equation z + (1 + r)sf(z) = x,

x€T>,

(8.50)

has the unique solution z = J(r+i)s(x) this means that

J(r+1)s(x)

= J3(Gr(x)).

(8.51J

Definition 8.6 Let 2> be a bounded convex metric domain in X endowed with a metric p compatible with the convex structure of V. We say that V satisfies the implicit approximation property if for each x € V and for each /J-nonexpansive self-mapping F of V its approximating curve {zt(x) : t 6 [0,1)} denned by the equation zt{x) = tF{zt{x)) + {\-t)x

(8.52)

is convergent to a point of T>, the closure of V. Proposition 8.5 Let V be as above and let f :V —» X satisfy the range condition, i.e., for each r > 0 the resolvent Jr = (I+rf)^1 is a well-defined p-nonexpansive self-mapping ofD. Then (1) For each x 6 T>, there exists a (= a(x)) € T> such that Jr(x) —> a and f(Jr(x)) —> 0 in the topology of X as r —» oo. (2) The following hypotheses are equivalent: (a) Nullx,/ ^ 0; (b) for some x G T>, the net {«7r(x)}, where Jr = (I + r / ) - 1 is strictly inside V; (c) for each x 6 V, the net {Jr(x)} is strictly inside V. Proof. Fix s > 0 and x e V. For r > 0, consider Gr{x) = (/ + r ( / J s ))~ 1 (a;). By our assumptions and the above claim, GT{x) —> a for some a (= a{x)) e 2?. In turn, by (8.47), we have that J,(Gr{x)) - Gr{x) -» 0,

r -> oo.

(8.53)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Together with (8.46) we get that for a given x G T>, lim Jr(x) = a (= a(x)) G V.

(8.54)

On the other hand, it follows by the definition of the resolvent that X

~ Jrr{x)

= f(Jr(x)),

r > 0.

(8.55)

Thus f(Jr(x)) —* 0 as r —> oo, and assertion 1 is proved. To prove assertion 2, we just have to show that (b) implies (a) and (a) implies (c) since the implication (c) =>• (b) is trivial. Indeed, if we assume that for some x G V the net {Jr{x)} is strictly inside V, then a = lim Jr(x) G V, and it follows from assertion 1 that a is a null point of r—>oo /•

Conversely, if / has a null point a G V, then for each x G V we have by (8.40), p(Jr(x),a) = p(Jr(x),Jr(a)) < p(x,a). Hence for each x G V, the net {Jr(x)} is /3-bounded in (T>,p), or, which is the same, this net is strictly inside T>. • Remark 8.4 Two standard examples of metric domains which satisfy the implicit approximating property are once again the open Hilbert ball and bounded convex domains in Cn endowed with the hyperbolic metric (see Chapters 4 and 5). In addition, it follows that Nullp/ is a p-nonexpansive retract of T>, and therefore, by Theorem 5 in [Kuczumow and Stachura (1990)] it is a metrically convex subset of (T>,p). We consider these two cases independently in the next section in the situation where a generator f has no interior null points in T>. 8.4

Null Point Free Generators

Let V be a bounded convex metric domain in X endowed with a metric p compatible with the convex structure of V. Assume also that V satisfies the implicit approximating property. Let / be the generator of a one parameter semigroup S = {Ft : t G (0, oo)} of /9-nonexpansive self-mappings of V. Suppose that / is null point free, i.e., the stationary point set W=

f)

FixpFt = Null©/ = 0.

(8.56)

0
In this case, we already know by Proposition 8.5 that for each x & V the net of resolvents {JT(x) (= (/ + rf)~1(x)) : t > 0} strongly converges to a

233

Stationary Points of Continuous Semigroups

boundary point a (= a(x)). We will now show that for the two cases when V is the open Hilbert ball or a bounded strongly convex domain in C n this point a G dT> does not depend on x G T> and is the sink point (Wolff point) for the semigroup {Ft : t G (0,oo)} generated by / . Now let X = H be a complex Hilbert space with the inner product (•,•), and let V = M be the open unit ball in H with the hyperbolic metric p. We recall that a boundary point a G dM is called the sink (or Wolff) point for a self-mapping F G NP(M) if all the ellipsoids E ( a , K ) = {x<= B : \ l - ( x , a ) \ 2 / ( I - \ \ x \ \ 2 )


K > 0,

(8.57)

internally tangent to dM at the point a are invariant under F. Theorem 8.3 Let f be the generator of a one parameter semigroup S = {Ft : t G (0,oo)} of p-nonexpansive self-mappings ofD and let f have no null point in B. Then there is a unique boundary point a G dM such that (i) for each x G P the net of resolvents {Jr(x) (= (/ + rf)~1(x)) : r > 0} strongly converges to a; (ii) for each r > 0 the point a G dM is the sink point for the resolvent mapping Jr : B —> B; (in) for each t > 0 the point a G dM is the sink point for the mapping Ft : M —> B, an element of the semigroup S. Proof. If N U I I B / = 0, then we claim that there exists a point a G dM such that for each x G B, Jr(x) strongly converges to a as r —> oo. Indeed, for each i £ l , the net {Jr(x)} is />-unbounded, i.e., ||J r (a;)|| —> 1 as r —> oo. Now fix ro > 0 and consider the mapping Jro : B —> B. Since Jro has no fixed points in B, it is known (see Chapter 5) that there exists a point a G dM (the sink point) such that all the ellipsoids

E(a,K)={x€B:\l-{x,a}\2/(l-\\x\\2)
K > 0, (8.58)

are invariant under Jro. But it follows from the resolvent identity Jrx - Jro ( ^ a; + ( l - - ) J r (z)) ,

(8.59)

r > ro, that E(a, k) is also invariant under Jr for all r > ro. NOW if for a fixed i £ l w e choose K > 0 such that x G E(a, K), then we get \l-(Jr(x),a)\2
(8.60)

234

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

and consequently, \l - (Jr(x),a)\ -» 0 as r -> oo. Hence Jr(x) converges strongly to a as r —> oo. If we now assume that for some 0 < r < TQ there exists K > 0 such that the ellipsoid E(a, K) is not invariant under Jr we get that there is another boundary sink point b G dM of Jr for such r G (O,ro). That is, all the ellipsoids E(b,K)={x£M:\l-(x,b)\2/(l-\\x\\2)
K > 0, (8.61)

are invariant under Jr. Once again, using the resolvent identity Jro(x) = Jrf—x+(l— ) Jro(x)) , (8.62) \f0 \ To/ J we get that all the ellipsoids E(b,K),K > 0, are invariant under JTQ. Clearly, we can find K > 0 such that the set M = E(a, K) n £(6, if) is not empty and is contained in B. But this set is a p-bounded subset of B which is also invariant under J r o . Thus Jro must have an interior fixed point in M. This is a contradiction. So, for all r > 0, all the ellipsoids E(a,K) are invariant under J r . Finally, by using the exponential formula Ft = lim UtT we see that these ellipsoids E(a,K) t > 0. The theorem is proved.

(8.63)

are also invariant under Ft for each •

Corollary 8.3 Let B be the open unit ball in a complex Hilbert space H, and let f : B —> H satisfy the range condition on B. If f has a continuous extension to M, then it has a null point in B. Proof. If / has no null point in B, then as we saw above there exists a sink point a G dM such that for each x G B, Jr(x) strongly converges to a, as r —• oo. But since f(Jr(x)) converges to zero as r —> oo, it follows by continuity that f(a) = 0 and we are done. D Remark 8.5 We will see below that, in fact, for holomorphic generators the following assertion is true: if f has no null point in B, then there is a point a G dM such that lim f{ra) = 0.

(8.64)

i—>i-

Moreover, we will see that for the finite dimensional case (when M is the open unit ball in Cn), if a £ 9B is the sink point of the semigroup S = {Ft •

Stationary Points of Continuoxis Semigroups

235

t G (0, oo)}, then there exists the so-called angular derivative Zf'(a) = P,

(8.65)

which is a real nonnegative number. It is clear that (8.65) implies (8.64). In addition, in this case, the semigroup S = {Ft : t G (0,oo)} strongly converges to a as t —> oo. Unfortunately, we cannot claim the latter fact in general. More details on the asymptotic behavior of semigroups in general Hilbert spaces can be found in Chapter 9. Here we show that in the particular case when the generator / has the special form / = / — T, where T is a self-mapping of B, then the semigroup 5 = {Ft : t £ (0, oo)} strongly converges to the sink point a G dM of the mapping T as t —> oo. Theorem 8.4

Let T be a p-nonexpansive mapping ofM.

(1) The Cauchy problem

| « « M +„(,,„ -T W l l I )) = ,

(866)

[u{0,x) = x has a unique solution {u(t, x) : t € R + } c B for each i £ l . (2) The semigroup {F(t)}t>o defined by F(t)x := u(t, x)

(8.67)

consists of p-nonexpansive mappings F(t), t > 0. (3) If T has no fixed point in B, then F(t) is also fixed-point-free for all t>0, and F(t)x converges as t tends to infinity to a point a G dM, the boundary ofM, for all x G B. Proof. First we note that the curve Gt = (1 — i)I + tT converges to / as t -> 0+ and -t (I - Gt) = I - T.

(8.68)

p(Gt(x),Gt(y)) < max[p(x,y),p(T(x),T(y))],

(8.69)

Since

236

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

each Gt is /j-nonexpansive. Hence it follows again by the product formula that for each t > 0 the mapping F(t) given by F(t) = lim G l n—>oo

n

(8.70)

is the solution operator for the Cauchy problem (8.66) and is also a pnonexpansive mapping on B. If now T has no fixed point in B, then it follows that

p | FixF(t) = 0.

(8.71)

t>o

By Theorem 8.1 we now see that each Ft has no fixed point in B. At the same time direct calculations show that the following formula holds:

F(t)(x) = e-*x + [ e-<*-«T(F(0(a:))de. Jo

(8.72)

\\T(F(0(x))\\ < 1

(8.73)

Since

for all £ € [0, t] and x € B, we can write F(t) = e-*I + (1 - e-*)A,

(8.74)

A = r J - j j * e-(*-«(T o F(0)d£

(8.75)

where

is a p-nonexpansive mapping on B. Hence Ft is an averaged mapping of the second kind and the result follows by Proposition 8.2 and Remark 8.2. D In the setting of finite dimensional Banach spaces it is known that each convex domain in C™ satisfies the implicit approximation property. If, in particular, V is a strongly convex C2 domain, and / £ Hol(£>, X) has no null point in V, then there exists a unique point a € &D such that for each x £ V, the net {J r (x)} converges to a, as r —> oo. Moreover, in this case the class of holomorphic self-mappings of V satisfies the Denjoy-Wolff iteration property, so the semigroup generated by / strongly converges to a boundary point b £ &D. The question is, of course, whether these two points o and b coincide.

237

Stationary Points of Continuous Semigroups

Theorem 8.5 Let V be a strongly convex bounded C2 domain in C" and let {F(t) : t > 0} be a semigroup of holomorphic self-mappings of T>. If {F{t)} has no stationary point in T>, then there exists a unique point a € dT> such that (a) For all x €T> the net {F(t)(x)} strongly converges, as t —> oo, to a. (b) For all x 6 V the net Jr{x) = I — rF'(Q)~l{x) strongly converges as r —» oo to the same point a. Proof. To establish our assertion it is enough to identify in our situation the points a and b obtained in Propositions 8.4 and 8.5. Setting / = —F'(0), we consider again the resolvents Js = (I + s/)" 1 , s > 0, which are holomorphic self-mappings of T>. Since / has no null points in ~D, J3 has no fixed point in V and as we saw above, the net {G r (0)} defined by formula (8.47) converges to a € &D as r -> oo (Gr(0) = ^ j J,(G r (0))V It follows from the proof of Theorem 2.3 in [Abate (1988a)] that for each n € N and for every x € T> and R > 0, the following inclusion holds: (8.76)

J?{Ex(a,R))cFx{a,R),

where Ex(a, R) and Fx(a, R) are the small and the big horospheres of center a, pole x, and radius R defined by Ex{a,R) = \z GV :limsup[p(z,u;) - p(x,w)] < \

loS#).

(8-77)

Fx(a,R) = [z e V : limmf [p(z,w) - p(x,w)] < ^ logi?},

(8.78)

where p is the hyperbolic metric on V. Therefore it follows again by the exponential formula (8.63) that for all t > 0, F(t)(Ex(a,R))

C Fx(a,R).

(8.79)

Hence, if 6 = lim F(t)x, then b = a and we are done. t—>oo

8.5

n

The Structure of Null Point Sets of Holomorphic Generators. Retractions

By Nullp/ we denote the analytic set defined as the null point set of / € Hol(P, X). Even in the finite dimensional case it is a complicated problem

238

Nonlinear Semigroups, Fixed Points, and Geometry of Domains

to recognize when an analytic set N consists only of irreducible components (see, for example [Aizenberg and Yuzhakov (1983)] and [Chirka (1985)]). It is known that this is the case when N is locally a complex analytic manifold. By using the above approach one can study the null point set of a generator / as the common fixed point set of the commuting family {-Ft}, the semigroup generated by / . Indeed, if D is a bounded convex domain in C n and {Ft} is a semigroup of holomorphic mappings, then it is well known ([Abate and Vigue (1991)]) that the commonfixedpoint set of this commutative family is a holomorphic retract of V. This approach becomes less transparent if X is an infinite dimensional space. Equality (8.40) and the Mazet-Vigue theorem [Mazet and Vigue (1991)] (for example) immediately imply that the null point set of a holomorphic generator on a convex bounded domain in a complex reflexive Banach space is also a holomorphic retract of T>. (This, in turn, implies that Nullp/ is a connected submanifold of V [Cartan (1986)].) Theorem 8.6 Let V be a convex bounded domain in a complex Banach space X and let f € QHol(T>). Suppose that a G Nullp/ and that one of hte following hypotheses holds: (1) X is reflexive, (2) KerA © Im A = X, where A = f'(a). Then Nullp/ is a connected complex analytic submanifold in T>, which is tangent to KerA Proof. It is sufficient to note the a G Null©/ = FixpJr(f) for all r > 0, where FixpJ r (/) is the fixed point set of the resolvent Jr{f) of / in V. In addition, it follows by the chain rule that / — Jr(A) — rJr(A), where Jr{A) = [Jr(f)]'(a) is the resolvent of the linear operator A for r > 0. Thus Ker(/ - [Jr(f)]'(a)) = KerA, and the theorem follows from the MazetVigue theorem. • Corollary 8.4 Let V be a bounded convex domain in a reflexive complex Banach space X, and let {Ft : t £ (0, oo)} be a locally uniformly continuous semigroup of holomorphic self-mappings of V. Then W=

p | F\xT,Ft 0
is a holomorphic retract ofT>.

(8.80)

Stationary Points of Continuous Semigroups

239

Thus W is a connected analytic submanifold o}T>. Moreover, ifD = B is the open unit ball in a Hilbert space H, then W is an affine submanifold ofB. In the context of the above theorem we will also need the following notion. Definition 8.7 Let / be a holomorphic mapping from V into X and let W = Null-p/ ^ 0. A point a € W is said to be quasi-regular if the condition Kerf (a) © Im /'(a) = X

(8.81)

holds. If, in addition, Ker/'(a) = {0}, then we say that a is a regular null point of / . Corollary 8.5 Let T>,X and f be as in Theorem 8.6. If a € V is an isolated point o/Nullx>/, then it is unique. In particular, if a £ Nullp/ is regular, i.e., f'(a) is invertible, then a is unique. Thus from Theorem 8.6 we can obtain the global description of the (interior) stationary point set of a semigroup {Ft}, t > 0, generated by a holomorphic mapping. But even in this case we only know that a retraction exists, but we have no constructive approximation process for finding the points in W. So, the question is how to construct a retraction onto this set. A possible way is to investigate the asymptotic behavior of the semigroup as t —> oo. It will become clear that this may be done only if we know a priori at least one point a EW and the spectrum of the linear operator f'(a) satisfies certain conditions (i.e., it does not intersect the imaginary axis with, perhaps, the exception of zero). Another way would be to apply equality (8.40), and for a fixed r > 0 to construct the sequence of the discrete Cesaro averages

G" = ^ £ ^ ' j=0

(8-82)

so that a subsequence {Gnic} weakly converges to a mapping G : T> —» W which is a holomorphic retraction of V onto W. As a matter of fact, this method turns out to be supefluous because as we will see below, the iterates of the resolvents Jr strongly converge to a holomorphic retraction of V onto W.

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

Theorem 8.7 Let V be a bounded convex domain in X', and let f e Hol(D, X) be the generator of a one-parameter semigroup of holomorphic self-mappings ofD. Suppose that W = Nullp/ ^ 0. Then (i) If W contains a quasi-regular point a G T>, then for each s > 0 the sequence {J™ = (I + sf)~n}f converges to a holomorphic retraction ofD onto W, as n —> oo, uniformly on each ball strictly inside T). (ii) IfW contains a regular point a € V, then W = {a} and the net {Js — (/ + s/)~1}s>o converges to a as s —> oo, uniformly on each ball strictly inside T>. To prove our theorem we need the following lemma. Lemma 8.4 (cf. [Lyubich and Zemanek (1994)]) Let A be a bounded linear operator on a Banach space X such that \\{I-A)n\\<M,

n=l,2,...

(8.83)

for some M < oo. Then the following conditions are equivalent: KerA©ImA = X

(8.84)

I m i = ImA

(8.85)

Proof. The implication (8.84) => (8.85) is obvious. Now let (8.85) hold, and let a functional x* £ X* vanish on the sum KerA © ImA Then x* £ KerA*. Furthermore, it follows from (8.85) and the Banach-Hausdorff theorem that the condition (u, x*) — 0 for all u € KerA implies that x* £ ImA*. Thus x* £ KerA* n ImA*. But because of (8.83), KerA* n ImA* = {0} by the Yosida mean ergodic theorem ([Yosida (1974)]). So x* = 0, and this implies (8.84). We recall that a linear operator A : X —> X is said to be m-accretive if for each r > 0, the operator Xr = (I + rA)^1 is well denned on X and • ||(7 + rA)~l\\ < 1. Lemma 8.5 Let A be a bounded linear operator on X which is maccretive with respect to some norm equivalent to the norm of X. If A satisfies condition (8.84), then for each r > 0, the linear operator Xr = (I + rA)~l satisfies the condition Ker(7 - I r ) © Im(7 - Ir) = A.

(8.86)

Stationary Points of Continuous Semigroups

Proof.

241

Returning to the original norm of X, we have by the definition ||2?||<M
71=1,2,...,

r>0.

(8.87)

By Lemma 8.4 and (8.87), it is sufficient to show that Im(/ — I r ) is closed in X. Indeed, if yn G Im(7 — lr) converge to y G X, we get a sequence {xn} C X such that (/ - lr)xn

= rAIrxn

-> y G ImA.

(8.88)

Note that it follows by the definition of J r that Ker(7 - lr) = KerA.

(8.89)

Therefore, if we represent xn £ X'm the form xn = un+vn, where un G ImA and vn G KerA (see (8.84)), we get from (8.88) and (8.89), (/ - lr)un = r(Alrun + AIrvn) = r(Alrun + Avn) = rAIrun -^>ye ImA Denote zn = lrun

(8.90)

G X. We have by (8.90), Zn — un - vAzn

(8.91)

and hence zn G ImA. Since ImA is closed and invariant under A, and Azn —» p y G ImA, the sequence {z n } converges to some element z € X and hence un = zn + rAzn —> z + y = x. Once again, it follows by (8.90) a that (/ - Ir)x = y. Lemma 8.6 Let the conditions of Lemma 8.5 hold. Then er(Xr), the spectrum of the operator IT, is contained in the open unit disk A, except perhaps, for 1, i.e.,
(8.92)

Proof. Fix r > 0. It follows by (8.87) that
(8.93)

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Nonlinear Semigroups, Fixed Points, and Geometry of Domains

It follows by the spectral mapping theorem (see, for example, [Dunford and Schwarz (1958)]) that a(f(A)) = f(a(A)). Thus, if we assume that ei 0. This • implies that