OF SEMIFLOWS OF HOLOMORPHIC MAPS
EDOARDO VESENTINI Received 16 September 2001
Letφbe a semiflow of holomorphic maps of a bounded domainDin a complex Banach space. The general question arises under which conditions the existence of a periodic orbit ofφimplies thatφitself is periodic. An answer is provided, in the first part of this paper, in the case in whichDis the open unit ball of a J∗-algebra andφacts isometrically. More precise results are provided when the J∗-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflowφgenerated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.
1. Introduction
LetDbe a bounded domain in a complex Banach spaceᏱand letφ:R+×D→D be a continuous semiflow of holomorphic maps acting onD.
Under which conditions does the existence of a periodic point ofφ(with a positive period) imply that the semiflowφitself is periodic?
An answer to this question was provided in [22] in the case in whichᏱis a complex Hilbert space andDis the open unit ball ofᏱ, showing that, if the orbit of the periodic point spans a dense linear subspace ofᏱ, thenφis the restriction toR+of a continuous periodic flow of holomorphic automorphisms ofD.
In the first part of this paper, a somewhat similar result will be established in the more general case in whichᏱis aJ∗-algebra andDis the open unit ball BofᏱ. The main result in this direction can be stated more easily in the case in which the periodic point is the center 0 of B. It will be shown that, if the points of the orbit of 0 which are collinear to extreme points of the closureB ofBspan a dense linear subspace ofᏱ, then the same conclusion of [22] holds,
Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:4 (2003) 217–260 2000 Mathematics Subject Classification: 17C65, 32M15, 46G20 URL:http://dx.doi.org/10.1155/S1085337503203109
that is,φis the restriction toR+of a continuous periodic flow of holomorphic automorphisms ofB.
If theJ∗-algebraᏱis a Cartan factor of type one—that is, it is the Banach spaceᏸ(Ᏼ,) of all bounded linear operators acting on a complex Hilbert space Ᏼwith values in a complex Hilbert space—it was shown by Franzoni in [4]
that any holomorphic automorphism ofB is essentially associated to a linear continuous operator preserving a Kre˘ın space structure defined on the Hilbert space direct sum⊕Ᏼ; a situation that has been further explored in [19,20] in the case in which⊕Ᏼcarries the structure of a Pontryagin space.
Starting from a strongly continuous groupT:R→ᏸ(⊕Ᏼ), inducing a continuous flowφof holomorphic automorphisms ofB, it will be shown that, if φhas a periodic pointx0, and if the orbit ofx0is “sufficiently ample,” a rescaled version ofTis periodic. A theorem of Bart [1] yields a complete description of the spectral structure of the infinitesimal generatorXofT.
The particular case in whichᏴCandBis the open unit ball of, which was initially explored in [22], will be revisited, showing that the periodic flowφ fixes some point ofBand that, ifφis eventually differentiable, the dimension of
is finite.
As was shown in [17,19], in the case in which⊕Ᏼcarries the structure of Pontryagin space, a Riccati equation defined onBis canonically associated toX.
The periodicity ofφimplies then the periodicity of the integrals of this Riccati equation.
A similar investigation to the one carried out in Sections3and4for a Cartan factor of type one is developed inSection 5in the case in whichᏱis a spin factor.
In this case, the norm inᏱ is equivalent to a Hilbert space norm. Assuming again, for the sake of simplicity, that the periodic point is the center 0 ofD, a hypothesis leading to the periodicity ofφ, consists in supposing that the points of the orbit of 0 which are collinear to scalar multiples of selfadjoint unitary operators acting onᏱspan a dense linear submanifold of this latter space.
The case of fixed points of the semiflowφacting on the bounded domainD is considered in the second part of this paper, where, among other things, some results which were announced in [16] for discrete semiflows generated iterating a holomorphic map f :D→Dare established in the general case. (One of the basic tools in this investigation was the Earle-Hamilton theorem (see [2] or, e.g., [5,6,9]). This theorem, coupled with the theory of complex geodesics for the Carath´eodory distance, was also used by several authors (see, e.g., [10,11,15, 16,23,24,25,26,27]) to investigate the geometry of the set of fixed points of f. Further references to fixed points of holomorphic maps can be found in [13].) Our main purpose is to obtain some information on the asymptotic behaviour ofφin terms of “local” properties.
In this direction, extending to the continuous case a result announced in [16]
for the iteration of a holomorphic map, it is shown that, if there is a sequence{tν} inR+diverging to infinity and such that{φtν}converges, for the topology of local
uniform convergence, to a function mappingDinto a set completely interior to D, then there exists a unique point x0∈D which is fixed by the semiflow φ;
moreover,φs(x) tends tox0ass→+∞, for allx∈D.
If some point x0∈D is fixed by the continuous semiflowφ, the map t→ dφt(x0), wheredφt(x0)∈ᏸ(Ᏹ) is the Fr´echet differential ofφt(x) atx=x0, de- fines a strongly continuous semigroup of bounded linear operators acting onᏱ.
Some situations are explored in which the behaviour of this semigroup deter- mines the asymptotic behaviour of the semiflowφ.
It is shown in Sections7and8that, if the spectral radiusρ(dφt(x0)) ofdφt(x0) isρ(dφt(x0))<1 for somet >0, then, ass→+∞,φsconverges to the constant mapx→x0for the topology of local uniform convergence.
The case in whichρ(dφt(x0))=1 at somet >0 is considered in Sections9and 10, under the additional hypothesis thatdφt(x0) is an idempotent ofᏸ(Ᏹ). As is well known, the spectrumσ(dφt(x0)) ofdφt(x0) consists of two eigenvalues in 0 and in 1 at most.
If
σdφt
x0
= {0}, (1.1)
thendφs(x0)= {0} for alls≥t. As a consequence of Sections7 and8, ifs→ +∞,φsconverges to the constant mapx→x0for the topology of local uniform convergence.
If
σdφt x0
= {1}, (1.2)
thenφis the restriction toR+of a periodic flow of holomorphic automorphisms ofD.
Finally, if
1∈σdφt x0
, (1.3)
and if there is somet >0, witht/t∈Q, such that alsodφt(x0) is an idempotent ofᏸ(Ᏹ), then the semiflowφis constant, that is,φt=id (the identity map) for allt≥0.
2. The general case of aJ∗-algebra
LetᏱbe a complex Banach space, letDbe a domain inᏱ, and let
φ:R+×D−→D (2.1)
be a semiflow of holomorphic maps ofDintoD, that is, a map such that
φ0=id, (2.2)
φt1+t2=φt1φt2, (2.3)
φt∈Hol(D), (2.4)
for allt, t1, t2∈R+, where Hol(D) is the semigroup of all holomorphic maps D→D.
A pointx∈Dis said to be a periodic point ofφwith periodτ >0 ifφτ(x)=x andφt(x)=xfor allt∈(0, τ). The semiflowφwill be said to be periodic with periodτifφτ=id and, whenever 0< t < τ,φtis not the identity map.
We begin by establishing the following elementary lemma, which is a con- sequence of Cartan’s uniqueness theorem (see, e.g., [5]) and which might have some interest in itself.
LetDbe a hyperbolic domain in the Banach spaceᏱ(or, more in general, a domain inᏱon which either the Carath´eodory or the Kobayashi distances define equivalent topologies to the relative topology) and letx0∈Dbe a fixed point of the semiflowφ, that is,φt(x0)=x0for allt∈R+.
Lemma2.1. If there is a vectorξ∈Ᏹ\{0}, for which the mapt→dφt(x0)ξofR+
intoᏸ(Ᏹ)is periodic with periodτ >0, and there is a setK ⊂(0, τ)such that {dφt(x0)ξ:t∈K}spans a dense affine subspaceK˜ofᏱ, thenφτ=id.
Proof. Letx0=0. Since
dφτ(0)dφt(0)ξ=dφτ+t(0)ξ=dφt(0)ξ ∀t≥0, (2.5) thendφτ(0)=id on ˜Kand therefore onᏱ. Cartan’s identity theorem yields the
conclusion.
LetᏴandbe complex Hilbert spaces and letᏸ(Ᏼ,) be the complex Ba- nach space of all continuous linear operators Ᏼ→, endowed with the op- erator norm. ForA∈ᏸ(Ᏼ,),A∗∈ᏸ(,Ᏼ) will denote the adjoint ofA. A J∗-algebra [7] is a closed linear subspaceᏭofᏸ(Ᏼ,) such that
A∈Ꮽ=⇒AA∗A∈Ꮽ. (2.6)
The roles ofᏱandDwill now be played by aJ∗-algebraᏭand by the open unit ballBofᏭ.
LetSbe the set of all extreme points of the closureBofB. As was noted by Harris in [7], ifᏭis weakly closed inᏸ(Ᏼ,), thenS= ∅.
Lemma2.2. LetS= ∅. If0is a periodic point of the semiflowφ:R+×B→B, with periodτ >0, and if there is a setK⊂(0, τ)such that, for everyt∈K,φt(0) is collinear to some point of S, and the set{φt(0) :t∈K} spans a dense linear subspace ofᏭ, then the semiflowφis periodic with periodτ.
Proof. Let∆be the open unit disc ofC. Fort∈K,
∆ζ−→ ζ
φt(0)φt(0) (2.7)
is, up to parametrization, the unique complex geodesic whose support contains both 0 andφt(0). (For the Kobayashi or Carath´eodory metrics onB, for the basic notions concerning complex geodesics, see, e.g., [14,15].)
Sinceφτ(0)=0 and
φτφt(0)=φtφτ(0)=φt(0), (2.8) thenφτis the identity on the support of the complex geodesic (2.7). Hence
dφτ(0)φt(0)=φt(0) ∀t∈K, (2.9) and thereforedφτ(0)=IᏭ. Thusdφτ(0) maps the setSonto itself. By Harris’
Schwarz lemma [7, Theorem 10],φτ=dφτ(0)=id.
Let nowx0∈Bbe a periodic point ofφwith periodτ >0.
As was shown in [7], the Moebius transformationMx0is a holomorphic au- tomorphism ofBwhich maps anyx∈Bto the point
Mx0(x)=
I−x0x0∗
−1/2 x+x0
I+x0∗x−1I−x0∗x0
1/2
=x0+I−x0x0∗
1/2
xI+x0∗x−1I−x0∗x0
1/2
. (2.10)
Furthermore,
Mx0(0)=x0, Mx0−1=M−x0, (2.11) andMx0is the restriction toBof a holomorphic function on an open neighbour- hood ofBinᏭ, mapping∂Bonto itself.
ApplyingLemma 2.2to the semiflowt→ψt=M−x0φtMx0, we obtain the fol- lowing theorem.
Theorem2.3. Ifx0∈Bis a periodic point ofφwith periodτ >0and if there is a setK⊂(0, τ)such that
(i)for anyt∈K,M−x0(φt(x0))is collinear to some point inS;
(ii)the set{φt(x0) :t∈K}spans a dense affine subspace ofᏭ(as was shown by Harris in[7, Corollary 8],Bis the closed convex hull ofS),
then the semiflowφis periodic with periodτ.
Remark 2.4. Under the hypotheses ofTheorem 2.3, setting ψt=φt when t≥ 0, andψt=φ−t whent≤0, one defines a flowψ:R×B→Bof holomorphic automorphisms ofB, whose restriction toR+isφ.
The flowψis continuous if and only if the semiflowφis continuous, that is, the mapφ:R+×B→Bis continuous.
In the case in whichn=dimCᏭ<∞, a similar statement toTheorem 2.3 holds for a discrete semiflow, that is to say, for the semiflow generated by the iterates fm= f◦f◦ ··· ◦f (m=1,2, . . .) of a holomorphic map f :B→B.
Theorem2.5. If f has a periodic pointx0∈B, with periodp > n(i.e., fp(x0)= x0,fq(x0)=x0ifq=1, . . . , p−1), ifM−x0(fq(x0))is collinear to some point in the Shilov boundary ofBforq=1, . . . , p−1, and if the orbit{fq(x0) :q=1, . . . , p−1} ofx0spansᏭ, thenf is periodic with periodp.
For example, let f1:z→e2πi/3zand let f2be another holomorphic function
∆→∆such that f2(0)=0 but f2≡0. Let f :∆×∆→∆×∆be the holomorphic map defined by
fz1, z2
= f1
z1
, f2
z2
, z1, z2∈∆. (2.12)
If f2 has a periodic point in∆\{0}, and therefore is periodic, f is periodic with period≥3. If f2 is not periodic, f is not periodic. However, every point (z1,0) withz1∈∆\{0}is a periodic point of f with period 3.
3. Cartan domains of type one
Let theJ∗-algebraᏭbe a Cartan factor of type one,Ꮽ=ᏸ(Ᏼ,). Let J=
I 0 0 −IᏴ
, (3.1)
and letΓ(J) be the group of all linear continuous operatorsAon⊕Ᏼwhich are invertible inᏸ(⊕Ᏼ) and such that
A∗JA=J. (3.2)
It was shown by Franzoni in [4] that the group of all holomorphic automor- phisms of the unit ballBofᏭ, which is called a Cartan domain of type one, is isomorphic to a quotient ofΓ(J), up to conjugation when dimCᏴ=dimC.
To avoid conjugation, we will consider now the case in which∞ ≥dimCᏴ= dimC≤ ∞.
LetT:R→ᏸ(⊕Ᏼ) be a strongly continuous group such that
T(t)∗JT(t)=J, (3.3)
or equivalently
T(t)JT(t)∗=J, (3.4)
for allt∈R. If
T(t)=
T11(t) T12(t) T21(t) T22(t)
(3.5) is the representation ofT(t) in⊕Ᏼ, withT11(t)∈ᏸ(),T12(t)∈ᏸ(Ᏼ,), T21(t)∈ᏸ(,Ᏼ), andT22(t)∈ᏸ(Ᏼ), then (3.3) and (3.4) are equivalent to
T11(t)∗T11(t)−T21(t)∗T21(t)=I, T22(t)∗T22(t)−T12(t)∗T12(t)=IᏴ, T12(t)∗T11(t)−T22(t)∗T21(t)=0,
(3.6)
T11(t)T11(t)∗−T12(t)T12(t)∗=I, T22(t)T22(t)∗−T21(t)T21(t)∗=IᏴ, T21(t)T11(t)∗−T22(t)T21(t)∗=0.
(3.7)
HereT11(t)∗∈ᏸ(),T12(t)∗∈ᏸ(,Ᏼ),T21(t)∗∈ᏸ(Ᏼ,), andT22(t)∗∈ ᏸ(Ᏼ) are the adjoint operators ofT11(t),T12(t),T21(t), andT22(t).
From now on, in this section, latin letters x and y indicate elements of ᏸ(Ᏼ,) and greek lettersξandηindicate vectors inᏴand.
It was shown in [4], that, ifx∈B,T21(t)x+T22(t)∈ᏸ(Ᏼ) is invertible in ᏸ(Ᏼ), and the functionT(t), defined onBby
T(t) :x−→
T11(t)x+T12(t)T21(t)x+T22(t)−1, (3.8) is, for allt∈R, a holomorphic automorphism ofB.
Setting
φt=T(t) (3.9)
fort∈R, we define a continuous flowφof holomorphic automorphisms ofB.
Ifx0∈B is a periodic point ofφwith period τ >0, and if the hypotheses of Theorem 2.3are satisfied,φis periodic with periodτ.
SinceT(τ)=id, then
T11(τ)x+T12(τ)=xT21(τ)x+xT22(τ) ∀x∈ᏸ(Ᏼ,), (3.10) whence
T12(τ)=0, T21(τ)=0, (3.11)
and therefore, by (3.6),
T11(τ)∗T11(τ)=T11(τ)T11(τ)∗=I,
T22(τ)∗T22(τ)=T22(τ)T22(τ)∗=IᏴ, (3.12) that is,T11(τ) andT22(τ) are unitary operators in the Hilbert spacesandᏴ.
Furthermore, (3.10) becomes
T11(τ)x=xT22(τ) ∀x∈ᏸ(Ᏼ,). (3.13) SinceT22(τ) is unitary, every pointeiθτ(θ∈R) in the spectrumσ(T22(τ)) of T22(τ) is contained either in the point spectrum or in the continuous spectrum.
In both cases, there exists a sequence{ξν}in Ᏼ(which may be assumed to be constant ifeiθτis an eigenvalue), withξν =1, such that
νlim→+∞
T22(τ)ξν−eiθτξν=0. (3.14)
Since, by the Schwarz inequality,
T22(τ)ξν|ξν−eiθτ=T22(τ)ξν−eiθτξν|ξν
≤T22(τ)ξν−eiθτξν, (3.15) then
νlim→+∞
T22(τ)ξν|ξν=eiθτ. (3.16)
Hence, letting, for anyη∈,xν=η⊗ξν∈ᏸ(Ᏼ,), thenxν(ξν)=ηand
νlim→+∞xνT22(τ)ξν= lim
ν→+∞
T22(τ)ξν|ξνη=eiθτη. (3.17)
Thus, by (3.13), T11(τ)η= lim
ν→+∞T11(τ)xνξν= lim
ν→+∞xνT22(τ)ξν=eiθτη (3.18) for allη∈. Therefore,
T11(τ)=eiθτI, (3.19)
and (3.13) yields
T22(τ)=eiθτIᏴ. (3.20)
In conclusion,
T(τ)=eiθτI⊕Ᏼ. (3.21)
Thus, the rescaled groupL:R→ᏸ(⊕Ᏼ), defined by
L(t)=e−iθtT(t), (3.22)
is periodic with periodτ.
Note that
L(t)∗JL(t)=J ∀t∈R. (3.23)
If
L(t)=
L11(t) L12(t) L21(t) L22(t)
(3.24) is the representation ofL(t) in⊕Ᏼ, withL11(t)∈ᏸ(),L12(t)∈ᏸ(Ᏼ,), L21(t)∈ᏸ(,Ᏼ), andL22(t)∈ᏸ(Ᏼ), then
Lα,β(t)=e−iθtTα,β(t) (3.25) forα, β=1,2. Therefore, setting, forx∈B,
L(t)(x) : x−→
L11(t)x+L12(t)L21(t)x+L22(t)−1, (3.26) then
L(t) =φt ∀t∈R. (3.27)
IfX:Ᏸ(X)⊂⊕Ᏼ→⊕Ᏼis the infinitesimal generator of the groupT, the operatorX−iθI⊕Ᏼ, with domainᏰ(X), generates the groupL.
The structure of the spectrumσ(X−iθI⊕Ᏼ) is described in [1] by a theorem of Bart, whereby
(i)σ(X−iθI⊕Ᏼ)⊂i(2π/τ)Z;
(ii)σ(X−iθI⊕Ᏼ) consists of simple poles of the resolvent functionζ→ (ζI⊕Ᏼ−(X−iθI⊕Ᏼ))−1;
(iii) the eigenvectors ofX−iθI⊕Ᏼspan a dense linear subspace of⊕Ᏼ. According to [1], ifXis the infinitesimal generator of a strongly continuous groupT, and if conditions (i), (ii), and (iii) hold, the groupLdefined by (3.22) is periodic with periodτ.
Summing up, in view ofTheorem 2.3, the following result has been estab- lished.
Theorem3.1. If there is a periodic pointx0∈Bforφ, with periodτ >0, and if there is a setK⊂(0, τ)such that, for anyt∈K,M−x0(φt(x0))is collinear to some point ofS, and the set{φt(x0) :t∈K}spans a dense affine subspace ofᏸ(Ᏼ,), then there exist a strongly continuous groupT:R→ᏸ(Ᏼ,)and a real numberθ such that the rescaled groupRt→L(t)is a periodic group with periodτ.
IfX:Ᏸ(X)⊂⊕Ᏼ→⊕Ᏼis the infinitesimal generator of the group T, conditions (i), (ii), and (iii) characterize the periodicity ofLwith periodτ.
Thus, ifXgenerates a strongly continuous groupT, and if conditions (i), (ii), and (iii) hold, the groupLdefined by (3.22) is periodic with periodτ. As was proved in [19, Proposition 4.1], the groupT satisfies (3.3) for allt∈Rif and only if the operatoriJXis selfadjoint. If that is the case, setting
⊕0=(⊕0)∩Ᏸ(X), 0⊕Ᏼ =(0⊕Ᏼ)∩Ᏸ(X), (3.28) [19, Lemma 5.3] implies that the linear spaces andᏴ are dense inandᏴ. We consider now the case in which the semigroupT|R+ is eventually differ- entiable (i.e., there ist0≥0 such that the functiont→T(t)xis differentiable in (t0,+∞) for allx∈⊕Ᏼ). By (3.22), alsoL|R+is eventually differentiable.
According to a theorem by Pazy (see, e.g., [12]), there exista∈Randb >0 such that the set
ζ∈C:ζ≥a−blog|ζ|
(3.29) is contained in the resolvent set ofX−iθI⊕Ᏼ. Thus, the intersection ofσ(X− iθI⊕Ᏼ) with the imaginary axis is bounded. Condition (i) implies then that σ(X−iθI⊕Ᏼ) is finite. But then, by [1, Proposition 3.2],X−iθI⊕Ᏼ∈ᏸ(⊕Ᏼ), and thereforeX∈ᏸ(⊕Ᏼ), proving thereby the following proposition.
Proposition3.2. Under the hypotheses ofTheorem 3.1, if moreover the semigroup T|R+is eventually differentiable, the groupTis uniformly continuous.
Remark 3.3. The above argument holds for any strongly continuous semigroup Tof linear operators, which is periodic, showing that, ifTis eventually differen- tiable, thenTis uniformly continuous.
IfTis eventually norm continuous, then (see, e.g., [3]) its infinitesimal gen- eratorXis such that, for everyr∈R, the set
ζ∈σ(X) :ζ≥r (3.30)
is bounded.
At this point, [1, Proposition 3.2] implies that, ifTis also periodic, then the operatorXis bounded, and thereforeTis uniformly continuous.
This conclusion holds, for example, if the periodic semigroupTis eventually compact.
4. The unit ball of a Hilbert space
Theorem 3.1has been established in [22] in the case in whichBis the open unit ball of the Hilbert space(i.e., whenᏴ=C).
In this case,T11(t)∈ᏸ() is invertible inᏸ(),T12(t)∈,T21(t)=(•|T12(t)), andT22(t)∈Care characterized by the equations
T22(t)2−T12(t)2=1, T11(t)∗T11(t)=I+ 1
T22(t)2
• |T11(t)∗T12(t)T11(t)∗T12(t). (4.1)
As was shown in [22], there is a neighbourhoodUofBsuch that
x|T11(t)∗T12(t)+T22(t)=0 ∀x∈U, t∈R. (4.2)
The orbit ofx0∈Bis described by φt
x0
=T(t)x0
= 1
x0|T11(t)∗T12(t)+T22(t)
T11(t)x0+T12(t). (4.3)
The infinitesimal generatorXofTis represented in⊕Cby the matrix X=
X11 X12
• |X12
iX22
, (4.4)
whereX12∈,X22∈R,iX11 is a selfadjoint operator, and the domainsᏰ(X) andᏰ(X11) ofXand ofX11are related by
Ᏸ(X)=ᏰX11
⊕C. (4.5)
Sinceφτis the identity, by [17, Proposition 7.3] and by (3.27), the set Fixφ= x∈B:φt(x)=x∀t∈R (4.6) is nonempty.
The ballBbeing homogeneous, there is no restriction in assuming 0∈Fixφ.
Thus, by (3.8),T12(t)=0 for allt∈R, and thereforeX12=0. Furthermore, as a consequence of (4.1),
T22(t)=eiX22t, (4.7)
and the skew-selfadjoint operatorX11 generates the strongly continuous group T11:t→T11(t) of unitary operators in.
Equation (3.9), which now reads
φt(x)=e−iX22tT11(t), (4.8) yields the following lemma.
Lemma4.1. The setFixφis the intersection ofBwith a closed affine subspace of. Because of (3.21),
X22=θ+2nπ
τ (4.9)
for somen∈Z, and therefore
φt(x)=e−(2nπ/τ)itL11(t)x (4.10)
for allx∈Band somen∈Z.
The strongly continuous periodic group L11:t→L11(t), with periodτ, of unitary operators inis generated by
Y11:=X11−iθI:ᏰX11
⊂−→. (4.11)
By [1],σ(Y11)⊂i(2π/τ)Zconsists entirely of eigenvalues, and the correspond- ing eigenspaces, which are mutually orthogonal, span a dense linear subspace of
.
Form∈Z, letPm be the orthogonal spectral projector associated with (2π/
τ)mi. By [1, (3)],L11is expressed by L11(t)x=
m
e(2mπ/τ)itPmx (4.12)
for allx∈and allt∈R. ThusL11(t) leaves invariant every spacePm(), and acts on it by the rotation
x−→e(2mπ/τ)itx. (4.13)
Hence, the following lemma follows.
Lemma 4.2. If the orbit of x0∈B spans a dense affine subspace of , then dimCPm()≤1for allm∈Z.
Since, by (3.25),
σY11
=σX11−iθI (4.14)
ifσ(X11) is finite, alsoσ(Y11) is finite.
A similar argument to that leading toProposition 3.2yields now the following theorem.
Theorem4.3. If the continuous flowφof holomorphic automorphisms of the open unit ballBofdefined by a strongly continuous groupT:R→ᏸ(⊕C)has a periodic point whose orbit spans a dense affine subspace of, and if moreoverTis eventually differentiable, thendimC<∞.
According to [17, Theorem VII], for anyγ >0 and every choice ofx0∈B∩ Ᏸ(X11), the function
φ•x0
|[0,γ]: [0, γ]−→ᏰX11
, (4.15)
defined by (4.3) for 0≤t≤γ, is the unique continuously differentiable map [0, γ]→withx([0, γ])⊂Ᏸ(X11), which is continuous for the graph norm
x−→ x+X11x (4.16)
onᏰ(X11), and satisfies the Riccati equation d
dtφtx0
=X11φtx0
− φtx0
|X12
+iX22
φtx0
+X12 (4.17)
with the initial conditionφ0(x0)=x0∈B∩Ᏸ(X11).
Hence,Theorem 3.1can be rephrased.
Proposition4.4. If the Riccati equation (4.17) has a periodic integral which spans a dense affine subspace of, (4.17) is periodic (i.e., all integrals of (4.17) satisfying the above regularity conditions are periodic).
We consider now the case in which one of the two spacesandᏴhas a finite dimension, and thereforeJdefines in⊕Ᏼthe structure of a Pontryagin space.
Assuming
∞>dimCᏴ<dimC≤ ∞, (4.18) the extreme points ofBare all the linear isometriesᏴ→; by [19, Theorem III],Xis represented by the matrix
X=
X11 X12
X12∗ iX22
, (4.19)
where X11:Ᏸ(X11)⊂→ and X22 ∈ᏸ() are skew-selfadjoint, X12 ∈ ᏸ(Ᏼ,), andᏰ(X)=Ᏸ(X11)⊕Ᏼ.
The Riccati equation (4.17) is replaced in [19] by the operator-valued Riccati equation
d
dtx(t)=X11x(t)−x(t)X22−x(t)X22−x(t)X12∗x(t) +X12 (4.20) acting onC1maps of [0, γ] into
Dˇ = x∈ᏸ(Ᏼ,) :xξ∈ᏰX11
∀ξ∈Ᏼ (4.21)
which are continuous for the norm (4.16).
For any γ >0, any choice ofu invertible inᏸ(Ᏼ) and of v∈Dˇ such that x0=vu−1∈B, the functiont→x(t) expressed by (3.8), withx=x0, fort∈[0, γ]
is the unique solution of (4.20) satisfying the conditions stated above, with the initial conditionx(0)=x0.
Theorem 3.1yields then the following proposition.
Proposition4.5. Let the integralt→x(t)be periodic with periodτ >0, and let there be a setK⊂(0, τ)such thatx(K)spans a dense affine subspace ofᏸ(Ᏼ,).
If, for anyt∈K,M−x0(x(t))is collinear to some linear isometry ofᏴinto, the Riccati equation (4.20) is periodic.
5. Spin factors
Similar results to some of those ofSection 3will now be established in the case in which the J∗-algebra Ꮽis a spin factor. In this section, is, as before, a complex Hilbert space, andC∗is the adjoint ofC∈ᏸ(). A Cartan factor of type four, also called a spin factor, is a closed linear subspaceᏭofᏸ() which is∗-invariant and such thatC∈Ꮽimplies thatC2is a scalar multiple ofI.
Since, forC1, C2∈Ꮽ,C1C2∗+C2∗C1is a scalar multiple, 2(C1|C2)I, of the identity, thenC1, C2→(C1|C2) is a positive-definite scalar product, with respect to whichᏭis a complex Hilbert space. (For more details concerning spin factors, see, e.g., [7,18,21].) Denoting by| · |and by · the operator norm and the Hilbert space norm onᏭ, then
|C|2= C2+
C4−C|C∗2 ∀C∈Ꮽ. (5.1) The open unit ballBfor the norm| · |, also expressed by
B=
C∈Ꮽ:C2<1 +C|C∗2 2 <1
, (5.2)
is called a Cartan domain of type four. The setSof all extreme points ofBis the set of all multiples, by a constant factor of modulus one, of all selfadjoint unitary operators acting on the Hilbert space, which are contained inᏭ[7,21].
Changing again notations, we denote byx, yelements of the spin factorᏭ, andx→x stands for the conjugation defined by the adjunction in the Hilbert spaceᏭ. For anyM∈ᏸ(Ꮽ),Mt will indicate the transposed ofM. The same notation will be used to indicate the canonical transposition inC2and the trans- position inᏭ⊕C2.
According to [7,21], any holomorphic automorphism f of B can be de- scribed as follows.
Let
J=
I 0 0 −IC2
, (5.3)
and letΛbe the semigroup consisting of allA∈ᏸ(Ꮽ⊕C2) such that
AtJA=J. (5.4)
EveryA∈Λis represented by a matrix A=
M q1 q2
• |r1
e11 e12
• |r2
e21 e22
, (5.5)
whereM∈ᏸ(Ꮽ) is a real operator,q1,q2,r1, andr2are real vectors inᏭ, and E:=
e11 e12
e21 e22
(5.6) is a real 2×2 matrix such that detE >0, and
MtM−RtR=IᏭ, (5.7)
MtQ−RtE=0, (5.8)
EtE−QtQ=IC2. (5.9)
HereR:Ꮽ→C2andQ:C2→Ꮽare defined by Rx=
x|r1
x|r2
∈C2 ∀x∈Ꮽ, Qz=z1q1+z2q2 ∀z=
z1
z2
∈C2.
(5.10)
It was shown in [18] that the setΛ0= {A∈Λ: detE >0}is a subsemigroup ofΛ.
Forx∈Ꮽ, let δ(A, x)=2x|r1−r2
+e11−e22+ie12+e21
(x|x) +e11+e22+ie21−e12
. (5.11)
One shows (see [18,21]) that, ifA∈Λ0,δ(A, x)=0 for allx in an open neighbourhoodUofB. Hence, the map
Aˆ:Ux−→ 1 δ(A, x)
2Mx+1 + (x|x)q1−i1−(x|x)q2
(5.12)
is holomorphic in U. Its restriction toB, which will be denoted by the same symbol ˆA, is the most general holomorphic isometry for the Carath´eodory- Kobayashi metric ofB[21]. This isometry is a holomorphic automorphism ofB if, and only if,Ais invertible inᏸ(Ꮽ⊕C2).
If ˆA(0)=0, thenq1−iq2=0, and thereforeq1=q2=0 becauseq1andq2are real vectors; (5.9) reads nowE∈SO(2), and (5.8), which now becomesRtE=0, yieldsr1=r2=0. Thus, by (5.7),Mis a real linear isometry ofᏭ. Setting
E=
cosα −sinα sinα cosα
(5.13) for someα∈R, then
A(x)ˆ =eiαMx ∀x∈B. (5.14)
As a consequence,
A(x)ˆ =x∀x∈B⇐⇒A=
e−iαIᏭ 0 0 0 cosα −sinα
0 sinα cosα
. (5.15)
Now, letT:R+→ᏸ(Ꮽ⊕C2) be a strongly continuous semigroup such that T(t)∈Λ0for allt≥0. Setting
φt=T(t) (5.16)
fort≥0, one defines a continuous semiflow φ:R+×B→Bof holomorphic isometricsB→B.
Ifx0∈Bis a periodic point ofφwith periodτ >0, and if the hypotheses of Theorem 2.3are satisfied, then
(i)φis the restriction toR+of a continuous flowR×B→B, which will be denoted by the same symbolφ;
(ii)T is the restriction to R+of a strongly continuous groupR→ᏸ(Ꮽ⊕ C2), which will be denoted by the same symbolT;
(iii) (5.16) holds for allt∈R.
Since,T(τ)(x)=xfor allx∈B, by (5.15), there is someα∈Rsuch that
T(τ)=F(τ), (5.17)
where
F(τ)=
e−iατIᏭ 0 0
0 cos(ατ) −sin(ατ) 0 sin(ατ) cos(ατ)
. (5.18)
Thus,
σT(τ)=σF(τ). (5.19)
Setting
L−= (ζ, iζ) :ζ∈C, L+= (ζ,−iζ) :ζ∈C, (5.20) ifατ∈πZ,σ(T(τ)) consists of the eigenvaluee−iατ, with the eigenspaceᏭ⊕ L−⊂Ꮽ⊕C2, and of the eigenvalueeiατ, with the eigenspace 0⊕L+⊂Ꮽ⊕C2. Ifατ∈πZ,T(τ)=IᏭ⊕C2whenατ/πis even , andT(τ)= −IᏭ⊕C2whenατ/πis odd.
In conclusion, the following theorem has been established.
Theorem 5.1. If there is a periodic point x0∈B forφ, with period τ >0, and if there is a setK⊂(0, τ)such that, for anyt∈K,M−x0(φt(x0))is collinear to a multiple, by a constant factor of modulus one, of a selfadjoint unitary operator which acts on the Hilbert spaceand is contained inᏭ, and the set{φt(x0) :t∈ K}spans a dense affine subspace ofᏭ, then there exist a strongly continuous group T:R→ᏸ(Ꮽ⊕C2)and a real numberαfor which (5.17) and (5.18) hold.
The infinitesimal generator
X:Ᏸ(X)⊂Ꮽ⊕C2−→Ꮽ⊕C2 (5.21) of the groupThas a pure point spectrum, consisting of at least one and at most two distinct eigenvalues.
Ifατ∈πZ,σ(T(τ)) consists of the eigenvaluee−iατ, with the eigenspaceᏭ⊕ L−, and of the eigenvalueeiατwith the one-dimensional eigenspace 0⊕L+.
Ifατ∈πZ, the groupT is periodic with periodτ whenατ/π is even, and period 2τwhenατ/πis odd.
According to [18, Theorem 4.1],Ᏸ(X)=Ᏸ⊕C2, whereᏰis a dense linear subspace ofᏭ, andXis expressed by the matrix
X=
X11 X12 X13
• |X12
0 X23
• |X13
−X23 0
, (5.22)
whereX23∈R,X12andX13are real vectors inᏭ, andX11is a real, skew-selfad- joint operator onᏭwith domainᏰ.
