FOR DOMAINS IN BANACH SPACES
LAWRENCE A. HARRIS Received 29 November 2001
We discuss the Earle-Hamilton fixed-point theorem and show how it can be ap- plied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also ex- tend the Lumer-Phillips theorem estimating resolvents to dissipative holomor- phic functions.
1. Introduction
Our purpose is to provide an introduction to holomorphic fixed-point theorems and to describe some recent joint work with Simeon Reich and David Shoikhet.
The main results are a fixed-point theorem and a Lumer-Phillips theorem, which is a mapping theorem for resolvents. These theorems apply to holomorphic functions whose numerical range is a set of complex numbers that lies to the left of a certain vertical line.
We begin by recalling some basic facts about holomorphic functions with do- main and range in a Banach space. We give two equivalent definitions of holo- morphy (weak and strong) and discuss power series and Taylor’s theorem. The holomorphic functional calculus is included as an example, although this is not needed later.
Next we give a brief overview of fixed-point theorems for holomorphic func- tions beginning with the Earle-Hamilton fixed-point theorem. This theorem may be viewed as a holomorphic version of Banach’s contraction mapping the- orem since a holomorphic function that maps a bounded domain strictly inside itself is a contraction with respect to a natural holomorphically invariant metric on the domain. (As usual, a contraction is a Lipschitzian mapping with Lips- chitz constant less than 1.) Our formulation is slightly more general than the
Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:5 (2003) 261–274 2000 Mathematics Subject Classification: 46T25, 47H10, 47A12 URL:http://dx.doi.org/10.1155/S1085337503205042
original since we require only the given function to be bounded rather than that its domain of definition.
We give a self-contained proof of the Earle-Hamilton theorem and deduce a closely related fixed-point theorem of Khatskevich, Reich, and Shoikhet. The lat- ter theorem obtains a unique fixed point for a holomorphic function of a convex domain into itself provided that the function has a uniformly continuous ex- tension to the closure of the domain and there are no sequences of approximate fixed points on its boundary.
Since the main results refer to the numerical range of holomorphic functions, we introduce this subject by beginning with the classical definition of the numer- ical range for operators on a Hilbert space. We then discuss Lumer’s extension of the numerical range and its properties for linear operators on a Banach space.
This leads to the author’s extension to holomorphic functions on the open unit balls of Banach spaces. Finally, we give the general definition for holomorphic functions on convex domains containing the origin.
Our first main result is an extension of the Earle-Hamilton theorem. This asserts that (under mild restrictions) if the numerical range of a holomorphic function lies strictly to the left of the vertical linex=1, then the function has a unique fixed point in its domain. A consequence is that a uniformly continuous holomorphic functionhon a domain is dissipative (i.e., its numerical range lies on or to the left of the imaginary axis) if and only if the resolvent (I−th)−1is defined on the domain and maps it into itself. In the case wherehis a linear operator on a Banach space, this reduces to a theorem of Lumer and Phillips.
2. Holomorphic functions
Holomorphic functions on abstract spaces were considered by M. Fr´echet around 1910 and subsequently studied by many authors. (See [3,4,14].) We give two definitions of holomorphy. LetXandYbe complex Banach spaces and letᏰbe an open subset ofX.
Definition 2.1(strong definition). A functionh:Ᏸ→Yisholomorphicif for each x∈Ᏸthere exists a continuous complex-linear mappingDh(x) :X→Ysuch that
limy→0
h(x+y)−h(x)−Dh(x)y
y =0. (2.1)
There is an alternative weaker definition which reduces matters to the case of a complex-valued function of a complex variable. (We useλto denote a complex variable.)
Definition 2.2 (weak definition). A functionh:Ᏸ→Y is holomorphicif it is locally bounded and if the mapping
λ−→h(x+λy) (2.2)
is holomorphic atλ=0 for eachx∈Ᏸ,y∈Xand linear functional∈Y∗.
Clearly the strong definition implies the weak definition. It was shown by Dunford in 1938, using the uniform boundedness principle, that both defini- tions are equivalent. (See [14, Theorems 3.10.1 and 3.17.1].) In many cases,Ᏸ will be the open ball of radiusrand centerx, that is,
Br(x)=
z∈X:z−x< r. (2.3) IfSis a set of complex numbers, we define
|S| =sup|λ|:λ∈S. (2.4) Example 2.3. Define a mappingP:X→Y to be a homogeneous polynomial of degreenif
P(x)=F(x, . . . , x
n
), (2.5)
whereF:X× ··· ×X→Y is a continuous (complex) multilinear map of degree n. ThenPis holomorphic onXand
DP(x)y=
n−1 k=0
F(x, . . . , x
k
, y, x, . . . , x
n−k−1
), x, y∈X. (2.6)
If{Pn}∞0 is a sequence of homogeneous polynomials wherePnis of degreenand if there exist positive constantsM andrsuch thatPnrn≤M forn=0,1, . . ., then it follows easily from the WeierstrassM-test and the weak definition that
h(x)=∞
n=0
Pn(x) (2.7)
converges to a holomorphic function inBr(0).
Thenth derivativeDnh(x) ofhatxcan be represented as a multilinear map of degreen. We denote the associated homogeneous polynomial of degreenby Dˆnh(x). It can be shown that
Pn= 1
n!Dˆnh(0), n=0,1, . . . . (2.8) The following theorem shows that every holomorphic function can be writ- ten as the sum of an infinite series of homogeneous polynomials in some neigh- borhood of each of the points of its domain.
Theorem2.4 (Taylor’s theorem [14, Theorem 3.17.1]). Ifhis holomorphic and bounded inBr(x), then
h(x+y)= ∞ n=0
1
n!Dˆnh(x)(y), (2.9)
for allyinBr(0).
Example 2.5. The holomorphic functional calculus associates classical holomor- phic functions with holomorphic functions on domains in spaces of operators.
Specifically, letWbe a complex Banach space and letX=ᏸ(W) be the Banach space of all bounded linear operators onW. Suppose f(λ) is a complex-valued function holomorphic in the disk|λ|< r. Then
f(λ)= ∞ n=0
anλn, for|λ|< r, (2.10) and hence
f(A)=∞
n=0
anAn (2.11)
is defined for|σ(A)|< r, where
σ(A)=lim
n→∞An1/n (2.12)
is the spectral radius. It is not difficult to show that the set Ᏸ=
A∈X:σ(A)< r (2.13) is open and it follows fromExample 2.3that f :Ᏸ→Xis holomorphic.
More generally, letΩbe an open subset of the complex plane and let Ᏸ=
A∈X:σ(A)⊆Ω. (2.14)
Define
f(A)= 1 2πi
Ꮿf(λ)(λI−A)−1dλ, (2.15) where Ꮿ consists of a finite number of positively oriented rectifiable Jordan curves inΩwhose interiors are a disjoint coverσ(A). It can be shown (see [14, Theorem 5.2.4] or [5, page 568]) that this defines f(A) independently of the curveᏯand the domainΩand that f :Ᏸ→Xis holomorphic.
Example 2.6. To justify hypotheses in later theorems, we exhibit a function that is holomorphic everywhere inX=c0, bounded onBr(0) for everyrwith 0< r <1 but unbounded onB1(0). Define
h(x)= ∞ k=0
xkk, forx= xk
∞
0. (2.16)
Then his defined and holomorphic everywhere inX by the weak definition.
Also, since|xkk| ≤ xkfor anyx∈X, it follows that h(x)≤ 1
1− x, x∈B1(0). (2.17)
Now for each positive integerndefine anx∈B1(0) by xk= 1
√n
2, 0≤k≤n, xk=0, k > n.
(2.18)
Then
h(x)≥ n k=0
xkn≥ n k=0
1
2. (2.19)
Hencehis unbounded inB1(0).
3. Holomorphic fixed-point theorems
A setSis said to liestrictly insidea subsetᏰof a Banach space if there is some >0 such thatB(x)⊆Ᏸwheneverx∈S. The following theorem may be viewed as a holomorphic version of the Banach’s contraction mapping theorem.
Theorem3.1 (Earle-Hamilton [6]). LetᏰbe a nonempty domain in a complex Banach spaceXand leth:Ᏸ→Ᏸbe a bounded holomorphic function. Ifh(Ᏸ)lies strictly insideᏰ, thenhhas a unique fixed point inᏰ.
Proof. We construct a metricρ, called the CRF-pseudometric, in whichhis a contraction. Define
α(x, v)=supDg(x)v:g:Ᏸ−→∆holomorphic, (3.1) forx∈Ᏸandv∈X, and set
L(γ)= 1
0αγ(t), γ(t)dt, (3.2) forγin the setΓof all curves inᏰwith piecewise continuous derivative. Clearly αspecifies a seminorm at each point of Ᏸ. We viewL(γ) as the length of the curveγmeasured with respect toα. Define
ρ(x, y)=infL(γ) :γ∈Γ, γ(0)=x, γ(1)=y, (3.3) forx, y∈Ᏸ. It is easy to verify thatρis a pseudometric onᏰ.
Letx∈Ᏸandv∈X. By the chain rule,
D(g◦h)(x)v=Dgh(x)Dh(x)v, (3.4) for any holomorphic functiong:Ᏸ→∆. Hence,
αh(x), Dh(x)v≤α(x, v). (3.5)
By integrating this and applying the chain rule, we obtainL(h◦γ)≤L(γ) for all γ∈Γand thus the Schwarz-Pick inequality
ρh(x), h(y)≤ρ(x, y) (3.6)
holds for allx, y∈Ᏸ.
Now by hypothesis there exists an>0 such thatB(h(x))⊆Ᏸwhenever x∈Ᏸ. We may assume thatᏰis bounded by replacingᏰby the subset
∪
Bh(x):x∈Ᏸ. (3.7)
Fixtwith 0< t </δ, whereδdenotes the diameter ofh(Ᏸ). Givenx∈Ᏸ, define h(y)ˆ =h(y) +th(y)−h(x), (3.8) and note that ˆh:Ᏸ→Ᏸis holomorphic. Givenx∈Ᏸandv∈X, it follows from
Dh(x)vˆ =(1 +t)Dh(x)v (3.9)
and (3.5), withhreplaced by ˆh, that
αh(x), Dh(x)v≤ 1
1 +tα(x, v). (3.10)
Integrating this as before, we obtain
ρh(x), h(y)≤ 1
1 +tρ(x, y), (3.11)
for allx, y∈Ᏸ.
Now pick a pointx0∈Ᏸand let {xn}be the sequence of iterates given by xn=hn(x0). Then{xn}is aρ-Cauchy sequence by the proof of the contraction mapping theorem. SinceX is complete in the norm metric, it suffices to show that there exists a constantm >0 such that
ρ(x, y)≥mx−y, (3.12)
for allx, y∈Ᏸ. SinceᏰis bounded, we may takem=1/d, wheredis the diam- eter ofᏰ. Givenx∈Ᏸandv∈X, define
g(y)=m(y−x), (3.13)
where∈X∗with =1. Theng:Ᏸ→∆is holomorphic andDg(x)v=m(v).
Henceα(x, v)≥mvby the Hahn-Banach theorem. Integrating as before, we
obtain (3.12).
The Earle-Hamilton theorem still applies in cases where the holomorphic function does not necessarily map its domain strictly inside itself. In fact, the
following interesting fixed-point theorem is a consequence of two applications of the Earle-Hamilton theorem.
Theorem 3.2 (Khatskevich-Reich-Shoikhet [15, 19]). Let Ᏸ be a nonempty bounded convex domain in a Banach space and leth:Ᏸ→Ᏸbe a holomorphic function having a uniformly continuous extension toᏰ. If there exists an>0 such thath(x)−x ≥wheneverx∈∂Ᏸ, thenhhas a unique fixed point inᏰ.
For example, the hypothesis thath(x)−x ≥ for allx∈∂Ᏸis satisfied whenᏰcontains the origin and
sup
x∈∂D
h(x)
x <1. (3.14)
Proof. Given 0< t <1 andx∈Ᏸ, define
ft(y)=(1−t)x+th(y) (3.15) and letδ >0 be such thatBδ(x)⊆Ᏸ. SinceᏰis convex, ft:Ᏸ→Ᏸis holomor- phic. To show that ft(Ᏸ) lies strictly insideᏰ, take=(1−t)δ. Lety∈Ᏸand letw∈B(ft(y)). Then
z=w−th(y)
1−t (3.16)
is inᏰsincez∈Bδ(x), so
w=(1−t)z+th(y)∈Ᏸ. (3.17) HenceB(ft(y))⊆Ᏸfor ally∈Ᏸ.
By the Earle-Hamilton theorem, fthas a unique fixed pointgt(x) inᏰ. Since the CRF-metric is continuous, the proof of the contraction mapping theorem shows that the iterates of ftat a chosen pointy0∈Ᏸare holomorphic and locally uniformly Cauchy inx. Hence the limit functiongt:Ᏸ→Ᏸis holomorphic by [14, Theorem 3.18.1]. Now anx∈Ᏸis a fixed point forgtif and only ifxis a fixed point forh. Thus, by the Earle-Hamilton theorem, it suffices to show that gt(Ᏸ) lies strictly insideᏰfor somet >0.
Sincehhas a uniformly continuous extension toᏰ, there exist>0 andδ >0 such thath(x)−x ≥wheneverx∈Ᏸand
d(x, ∂Ᏸ)=infx−y:y∈∂Ᏸ< δ. (3.18) SinceᏰis bounded, there is anMwithx ≤Mfor allx∈Ᏸ. Ifx∈Ᏸ,
hgt(x)−gt(x)=(1−t)hgt(x)−x, (3.19) so
hgt(x)−gt(x)≤2(1−t)M. (3.20)
Choosetclose enough to 1 so that 2(1−t)M <. Ifd(gt(x), ∂Ᏸ)< δfor some x∈Ᏸ, then
≤hgt(x)−gt(x), (3.21) a contradiction. Thus,Bδ(gt(x))⊆Ᏸfor allx∈Ᏸ, as required.
Considerably stronger results have been obtained for the case whereᏰis the open unit ball of a Hilbert space.
Theorem3.3 (Goebel-Sekowski-Stachura [9, Theorem 11]). LetBbe the open unit ball of a Hilbert space and leth:B→Bbe holomorphic. If there is a pointx0
inBsuch that the sequence{hn(x0)}of iterates lies strictly insideB, thenhhas a fixed point inB.
Obtaining fixed points for mappings that are only nonexpansive rather than contractive is more difficult. A fundamental result due, independently, to Brow- der, G¨ohde, and Kirk in 1965 is that a nonexpansive self-mapping of a closed bounded convex set in a uniformly convex Banach space has a fixed point. In analogy, Goebel, Sekowski, and Stachura [9] showed that the CRF-metricρ in the open unit ballBof a Hilbert space is uniformly convex and obtained a fixed- point theorem for holomorphic self-mappings ofB. All such mappings areρ- nonexpansive by the Schwarz-Pick inequality (3.6).
Theorem3.4 (Goebel-Reich [8, Theorem 25.4]). LetBbe the open unit ball of a Hilbert space and leth:B→Bbe an arbitrary function satisfying
ρh(x), h(y)≤ρ(x, y), (3.22) for allx, y∈B. Ifhhas a continuous extension toB, thenhhas a fixed point inB.
Corollary3.5 (Goebel-Sekowski-Stachura [9]). Ifh:B→Bis a holomorphic function that has a continuous extension toB, thenhhas a fixed point inB.
Extensions of Theorems3.3and3.4and ofCorollary 3.5to Cartesian prod- ucts of the Hilbert ballBcan be found in papers of Kuczumow and others. It would be desirable to obtain extensions to more general spaces. See [4,7,12] for basic material about function-theoretic metrics on domains in Banach spaces.
See [16] for an extensive survey of fixed-point theorems for holomorphic map- pings.
4. The linear numerical range
LetA∈ᏸ(H), whereHis a Hilbert space. Thenumerical rangeofAis defined by
W(A)=
(Ax, x) :x =1, x∈H. (4.1)
Like the spectrum, the numerical range associates a set of complex numbers withAthat reflects the properties ofA. However, unlike the spectral radius, the numerical radius is an equivalent norm onᏸ(H). The following is a list of some basic properties. (See [10].)
Theorem4.1. LetA∈ᏸ(H). Then (1)W(A)is convex,
(2)Ais hermitian (i.e.,A∗=A) if and only ifW(A)is real, (3)σ(A)⊆W(A),
(4)|W(A)| ≤ A ≤2|W(A)|.
The notion of the numerical range was successfully extended to operators on an arbitrary Banach spaceXby Lumer [17]. To give an equivalent form of his definitions, define
J(x)=
∈X∗: =(x)=1 (4.2)
forx∈X withx =1 and note thatJ(x) is nonempty by the Hahn-Banach theorem. LetQ(x) be a nonempty subset ofJ(x) for eachx∈Xwithx =1.
Now letA∈ᏸ(X). Define numerical ranges ofAby V(A)=
(Ax) :∈J(x),x =1, W(A)=
(Ax) :∈Q(x),x =1. (4.3) Clearly,W(A)⊆V(A). IfQ(x) is taken to beJ(x) for allx∈X withx =1, thenW(A)=V(A).
In the case whereX is a Hilbert space, the setsQ(x) andJ(x) coincide and consist of the single functional(y)=(y, x) by the Riesz representation theorem.
ThusW(A) andV(A) reduce to the previous definition of the numerical range.
Since there is in general no adjoint operation onᏸ(X), an operatorA∈ᏸ(X) is defined to behermitianifV(A) is real. The following theorem is a well-known extension ofTheorem 4.1. (See Bonsall and Duncan [1,2] for this and many other properties of the numerical range.)
Theorem4.2. LetA∈ᏸ(X). Then (1)V(A)is connected,
(2)Ais hermitian if and only ifeitA =1for all realt, (3) coσ(A)⊆V(A),
(4)|V(A)| ≤ A ≤e|V(A)|,
(5) sup ReW(A)=limt→0+(I+tA −1)/t.
In the above, the symbol co denotes the closed convex hull. Following Lumer, we can deduce from part (5) of the above theorem that the closed convex hulls ofW(A) andV(A) are equal no matter what choice ofQis taken. Thus, in par- ticular,|W(A)| = |V(A)|.
The theorem we extend is a characterization of dissipative operators. This can be viewed as a one-sided formulation of part (2) of the previous theorem.
Theorem4.3 (Lumer-Phillips [18]). LetA∈ᏸ(X). The following are equiva- lent:
(a) sup ReW(A)≤0, (b)etA ≤1for allt≥0,
(c)(I−tA)−1 ≤1for allt≥0.
5. The holomorphic numerical range
We first consider the case where the domain is the open unit ballB ofX and h:B→Xis a holomorphic function that is uniformly continuous inB. Thenh has a uniformly continuous extension toB. Moreover,his bounded inBso
h =suph(x):x∈B (5.1)
is finite. In analogy with (4.3), define numerical ranges ofhby V(h)=
h(x):∈J(x),x =1, W(h)=
h(x):∈Q(x),x =1. (5.2) Theorem5.1 [11]. (a) sup ReW(h)=limt→0+(I+th −1)/t.
(b)IfPnis a homogeneous polynomial of degreen >1, then
Pn≤nn/(n−1)WPn. (5.3) As was shown in [11], a consequence of part (a) ofTheorem 5.1is that the closed convex hulls ofW(h) andV(h) are the same no matter what choice ofQ is taken. It was also shown in [11] that the constant in part (b) of the theorem is best possible.
Following [13], we now extend the numerical range to functions defined on more general domains. LetᏰbe a convex domain inXand supposeᏰcontains the origin. For eachx∈∂Ᏸ, let
J(x)=
∈X∗:(x)=1,Re(y)≤1∀y∈Ᏸ. (5.4) It follows from [5, Corollary 6, page 449] thatJ(x) is nonempty. LetQ(x) be a nonempty subset ofJ(x) for eachx∈Xwithx∈∂Ᏸ. Ifh:Ᏸ→Xhas a contin- uous extension toᏰ, then we define
V(h)=
h(x):∈J(x), x∈∂Ᏸ, W(h)=
h(x):∈Q(x), x∈∂Ᏸ. (5.5) Otherwise, consider
hs(x)=h(sx), 0< s <1. (5.6)
The functionhsalways has a continuous extension toᏰand hence we may define L(h)=lim
r→1−supRehs(x):∈Q(x), x∈∂Ᏸ, r≤s <1 (5.7) since the indicated supremum is decreasing inr. Ifhis uniformly continuous on Ᏸ, thenhhas a uniformly continuous extension toᏰ. Hence,W(h) is defined andL(h)=sup ReW(h).
Lemma 5.2 (cf. [11, Lemma 2]). If h:Ᏸ→X is holomorphic and bounded on each domain strictly insideᏰ, thenhsis uniformly continuous onᏰfor eachswith 0< s <1.
The following lemma is the key to our extension of the Lumer-Phillips the- orem to holomorphic functions. Throughout the remainder of this section we assume that the domainᏰis bounded as well as convex.
Lemma 5.3 [13]. Let g:Ᏸ→X be holomorphic and bounded on each domain strictly inside Ᏸ. If L(g)<0, then the equationg(x)=0 has a unique solution x∈Ᏸ.
Proof. ByLemma 5.2, it suffices to prove the lemma for the case wheregis uni- formly continuous. Let
p(x)=inf{r >0 :x∈rᏰ} (5.8) be the Minkowski functional forᏰ. It is not difficult to extend the proof of part (a) ofTheorem 5.1(given in [11]) to the case whereBis replaced byᏰand p replaces the norm, that is,
I+tg =suppx+tg(x):x∈Ᏸ. (5.9) (In fact, we can avoid inverses in that proof by arguing as in [1, page 83].) By hypothesis there is an>0 with sup ReW(g)<−. Hence there is at >0 with
px+tg(x)<1−t, (5.10) for allx∈Ᏸ. Thus (I+tg)(Ᏸ) lies inside (1−t)Ᏸand so strictly insideᏰ. By the Earle-Hamilton theorem,I+tghas a unique fixed point inᏰsog(x)=0 has
a unique solution inᏰ.
Theorem5.4 [13]. Leth:Ᏸ→Xbe holomorphic and bounded on each domain strictly insideᏰ. IfL(h)<1, thenhhas a unique fixed point inᏰ.
Proof. This theorem follows from the previous lemma withg=h−IsinceL(g)=
L(h)−1<0.
The above theorem is an extension of the Earle-Hamilton fixed-point theo- rem since it is not difficult to show thatL(h)<1 whenhmapsᏰstrictly inside Ᏸ.
Our main result is the following extension of the Lumer-Phillips theorem.
Theorem5.5 [13]. Leth:Ᏸ→Xbe holomorphic and uniformly continuous. Then sup ReW(h)≤0if and only if(I−th)−1 is defined on Ᏸand is a holomorphic mapping of Ᏸinto itself for eacht≥0.
Proof. Suppose that sup ReW(h)≤0. Since sup ReW(h) is positive homoge- neous, it suffices to consider the case wheret=1. Lety∈Ᏸand takeg=y+h.
Then
sup ReW(g)≤p(y) + sup ReW(h)<1 (5.11) by hypothesis. Hence by the previous theorem, the functionghas a unique fixed point f(y) inᏰ. It can be shown as in the proof ofTheorem 3.2that f is holo- morphic inᏰ. Moreover, for each y∈Ᏸ, the only solution of (I−h)(x)=y withx∈Ᏸisx= f(y). Hence (I−h)−1= f exists onᏰand is a holomorphic mapping ofᏰinto itself.
Conversely, suppose that (I−th)−1is defined onᏰand mapsᏰinto itself for allt >0. Sincehis uniformly continuous onᏰandᏰis bounded, there is an M >0 withh(x) ≤Mfor allx∈Ᏸ. Also, given>0 there exists aδ >0 such thath(x)−h(y)<wheneverx, y∈Ᏸandx−y< δ. Clearly,
(I−th)−1(x)−x=th(I−th)−1(x) (5.12) for anyx∈Ᏸandt >0. Let 0< t < δ/M. Ifx∈Ᏸ, then
(I−th)−1(x)−x< δ, (5.13) so
h(I−th)−1(x)−h(x)<. (5.14) Now the Minkowski functional pforᏰsatisfiesp(x)≤ x/r for allx∈X, wherer >0 is chosen so thatBr(0)⊆Ᏸ. Sincepis subadditive, it follows from (5.14) and the identity
(I+th)(x)=(I−th)−1(x) +th(x)−h(I−th)−1(x) (5.15) that
p(I+th)(x)≤p(I−th)−1(x)+t
r, (5.16)
for allx∈Ᏸ. Hence
I+th ≤1 +t
r, (5.17)
for 0< t < δ/Mand thus sup ReV(h)≤/r.
The above arguments obtain the forward implication ofTheorem 5.5as an easy consequence ofLemma 5.3. Conversely,Lemma 5.3is an easy consequence of the forward implication ofTheorem 5.5. To see this, by hypothesis andLemma 5.2we may suppose thatg:Ᏸ→Xhas a uniformly continuous extension toᏰ and that
δ≡ −sup ReW(g)>0. (5.18)
Puth=g+δIand note that sup ReW(h)≤0. Hence byTheorem 5.5, the equa- tion (I−th)(x)=0 has a unique solution x in Ᏸ for each t≥0. Since g=
−δ(I−th) whent=1/δ, it follows thatg(x)=0 has a unique solutionxinᏰ.
It is shown in [13, Corollary 9] that ifᏰis the open unit ball ofXand ifL(h) is finite for the choiceQ=J, thenhis bounded on each domain strictly insideᏰ.
Thus, in the case mentioned, this hypothesis may be omitted fromLemma 5.3 andTheorem 5.4.
See [13] for further details and applications to Bloch’s theorem and Cartan’s uniqueness theorem.
In the case where the underlying space is finite dimensional, there is a version ofLemma 5.3where the domainᏰdoes not need to be convex and the numer- ical range is computed as in a Hilbert space. (For purposes of comparison, we reformulate the theorem.)
Theorem5.6 (Shih [20]). SupposeXis finite dimensional and let(·,·)be an in- ner product on X. LetᏰbe a bounded domain in X containing the origin and let g :Ᏸ→X be a holomorphic function with a continuous extension toᏰ. If Re(g(x), x)<0for allx∈∂Ᏸ, theng(x)=0has a unique solutionxinᏰ.
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Lawrence A. Harris: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
E-mail address:[email protected]
