analytic continuation and stability of operator semigroups

A N A L Y T I C C O N T I N U A T I O N A N D

S T A B I L I T Y O F O P E R A T O R S E M I G R O U P S

By

RALPH CHILL AND YURI TOMILOV*

Abstract. We prove a new criterion for the analytic continuation of functions across a linear boundary. As corollaries, we obtain new conditions for convergence of orbits of operator semigroups on Banach spaces with Fourier type.

1 Introduct ion

The study of analytic extendability of a function across a (in most cases linear)

boundary is a classical subject in analysis. Besides its natural value for function

theory [6, 11 ], it is of importance for the theory of partial differential equations

[24, 15, 16], harmonic analysis (study of ideals in Banach algebras via the Carleman

transforrn) [7, 23, 29], operator theory [35, 39], and mathematical physics [9]. The

first systematic treatment of analytic extension criteria seems to be [10].

One such criterion is provided by the well-known edge-of-the-wedge theorem;

see [38]. It says that two analytic functions f + and f - which are defined in wedges

(open regions) and which coincide in the distributional sense on their common edge

admit a common analytic extension to a neighbourhood of the edge. This principle

is convenient if some a priori information on the behaviour of the integral means

of the two "wedge" functions f • near the edge is available (e.g., local Hardy

space conditions, as for instance in [44, Theorem B] or in [10]). However, if both

functions f • coincide only in the sense of (nontangential) pointwise convergence,

the classical edge-of-the-wedge principle cannot be applied for several reasons.

Nevertheless, it is possible to find an appropriate edge-of-the-wedge theorem even

in this setting if the f • satisfy certain growth restrictions near the edge. See, for

example, [40, Theorem 4.4] where a corrected version of [44, Theorem E] was

proved (and also the comments in [5]).

*This work started during a visit of the first author at the University of Torun. The kind hospitality is gratefully acknowledged. The second author was partially supported by a KBN grant and by the NASA-NSF Twinning Program.

331 JOURNAL D'ANALYSE MATHE/vlATIQL1E, Vol. 93 (2004)

332 R. CHILL AND Y. TOMILOV

In this article, we present a criterion which implies that an analytic function f

defined on the region {z E C �9 -1 _< R e z , I m z <_ 1} \ (-1, 1) extends analytically

through the interval ( -1, 1). If F(z) := f ( z ) - f(2), then we assume that F can be

estimated in the form

(1.1) IF] < IGl. tHI,

where G satisfies an integral condition, and H gatisfies a pointwise convergence

condition. Very formally, this can be considered as a "product" of the two types

of conditions above. While one condition is on nontangential behaviour of H, i.e.,

behaviour in sectors perpendicular to the real axis, the other one is on behaviour of G along the parallels to the real axis. This "orthogonality" of the conditions leads

to difficulties in applying known arguments (a situation which is close in spirit can be found in [I, p. 158]).

Our interest in such problems stems from the study of the asymptotic behaviour

of operator semigroups. In this theory, functions with an estimate of the form (1.1)

appear very naturally when considering weak local resolvents and when using

the resolvent identity. We give several applications of our extension criterion to

stability of bounded operator semigroups on Banach spaces with Fourier type. In

this situation, the main task is to relate the growth of the resolvent of the semigroup

generator near the imaginary axis, the type of the underlying Banach space and the

asymptotic properties of semigroup orbits, thus leading to theorems of tauberian

character. Such a relation is expressed finally in terms of certain properties of

images of the semigroup generator. This part of the paper can be considered as a continuation of the research started in [12]. We discuss also similar results in the

context of discrete operator semigroups.

2 Pre l iminaries

We start by recalling some very basic facts from the theory of Hardy spaces of

analytic and harmonic functions. Throughout, we denote by I~ the open unit disc and by "IF the unit circle.

Def in i t ion 2.1. Forp > 0, HP(D) is the set of all analytic functions f on D for which

(2.1) (f02~ ) l /p

[IflIHp := sup If(reir de < oo. 0_<r<l

Defin i t ion 2.2. For p _> 1, hP(l~) is the set of all harmonic functions f on D

A N A L Y T I C C O N T I N U A T I O N 333

for which

( [2 , , )1/, (2.2) ]]ft]hp := sup ]f(rei•)] p de < oo.

O < : r < l \ J o

It is well-known that i fp > 1, HP(I~), resp. hP(D), are Banach spaces equipped

with the norms []. ]]Hp, resp. []. [Ihp. Moreover, the map

I : f ~ - ~ P f

is an isometric isomorphism from LP(T) onto hP(l~) for every 1 < p < c~, where

P f stands for the Poisson integral of f . For f E LP(~), 1 < p <_ cx~,

lim (Pf)(r~) = f(~), a.e. ~ E V, r---~ 1 -

SO that a function in LP(q~), 1 < p _< c~, can be identified with the boundary value

o f P f E hV(lD); see [3, pp. 103-105, 116].

For f belonging to either of the introduced spaces, we have the estimate

C C (2.3) [f(z)l <- (dist(z,OID))l/P - (1 - Iz l ) l /p ' z e 1~,

for some C > 0 (see, for example, [3, p. 121]). Clearly, similar definitions and

properties can be stated if ID is replaced by any other open disc in C.

Our further reasoning is based on two maximum principles. The first is the

so-called Gabriel inequality, which can be considered as an integral maximum principle for harmonic functions.

Theorem 2.3 (Gabriel's m a x i m u m principle). Let f~ C ~2 be a simply

connected domain. Suppose that F1, F2 C f~ are two closed convex curves such that ['2 is contained in the interior o f the domain f~l bounded by F1. Then for every p > 2, there is a constant Cp (not depending on FI, F2 and ~) such that for every

harmonic function u on ~,

(2.4) ~r2 [u(z)[P ,dz[ < Cp fr l [u(z)[" [dz[.

The inequality (2.4) was first proved by R, M. Gabriel in [19]. A modem

approach to Gabriel-like inequalities can be found in [22] (in particular, see [22, Theorem 1 ] and the comments following it).

Remark 2.4. Our main result (Theorem 3.1 below) can be improved if (2.4) also holds for p > 1. It is known that (2.4) holds for p > 1 if u is analytic on

f~ [19, 22]. Thus, by passing to conjugate functions we may reduce the problem


whether the estimate (2.4) holds forp > I to the problem of extending the M. Riesz

inequality (1 < p < 2) for Lp-norms of conjugate functions to general domains

with Lipschitz boundary. This latter problem requires the verification of the

Muckenhoupt condition on the modulus of a conformal mapping of the domain f21

to the unit disc. Unfortunately, we have difficulties with this approach even when

f~l is a convex polygon. In general, the question whether (2.4) holds for 1 < p < 2

seems to be open. Note that a sharper version of the inequality (2.4) appears to be

false even i fp = 2 [18].

R e m a r k 2.5. Recall that if the inside curve F2 is convex, then the measure #

defined by #(E) ".'= length (I'2 f3 E) (E C 11~ 2 Borel set) is a Carleson measure; see,

for example, the discussion in [2 1 ]. Hence, if the outside curve F 1 is Lipschitz, then

the inequality (2.4) for 2 < p < ~ can be derived from Dahlberg's characterization

of Carleson measures; see [14, Theorem 1] and the remark following it. If F 1 is

a C 1-curve, then it is possible to extend the range of allowed p in Theorem 2.3 to

p > 1. For C2-curves this result was obtained by HSrmander [26, Theorem 3.1].

However, the reasoning in the proof of our Theorem 3.1 requires the consideration

of a Lipschitz curve F1. Moreover, in order to include p = 2 in Theorem 3.1, we

need to know that the inequality (2.4) holds for some p < 2.

The second maximum principle which is crucial for us gives the estimate of

a subharmonic function on the unit disc in terms of its radial boundary value. In

the following general form, the result is due to Dahlberg [13, Theorem 1]. If u is

harmonic, a proof may be found in Wolf [43, pp. 95-97].

T h e o r e m 2.6 ( D a h l b e r g ' s m a x i m u m principle) . Let u be a nonnegative

subharmonic function on D. Suppose that

1. f o rever ) , ~ E 72 there exists l im~x_ u(r~) =: / (~) , and f E LI(T);

2. we have

sup u ( z ) = o ( ( i - r ) - 2 ) , r i - .

Izl=r

Then

(2.5) u < P f on D,

where, as before, P stands for the Poisson integral.

Observe that if u is a harmonic function on D, if ]u] satisfies the conditions of

Theorem 2.6, and if in addition f E LP(72), then u E hP(/l)). This fact will be used

later.

ANALYTIC CONTINUATION 335

3 Analytic continuation

This section is devoted to the proof of our main result, an analytic extension

criterion.

We define the rectangle

(3.t) R := {z E C : - 1 _< Rez < 1, - 1 < Imz _< 1};

and for 0 E (0, rr/2) we let

Z0 := {z E C : 0 < argz < 7r- 0}.

By C we denote constants which may vary from line to line.

T h e o r e m 3.1. Let f : R\I~ -+ C be analytic, and define F : R\IR -+ C by

F(z) = f ( z ) - f (2) (z E R\~) . Assume that

1. there exists a constant m E [0, 2) such that

sup I : (a +i/3)1 -- O(I/~l-'~), D ~ 0; aE(-1,1)

2. there exist a measurable function G : R \ R ~ C and a continuous function

n : R\Ii~ --+ C such that IFI <_ IGI. Inl and

(3.2) sup lie(" + i/~)ll~:, < ~ f o r s o m e 2 < p <_ oc, ~E(0,1)

and there exists 00 E (0, 7r/2) such that

(3.3) Jim l log(Imz)H(z) l = 0 forevery a E ( -1 ,1) . zEc~+~g O

Then the function f admits an analytic extension to 1L

P r o o f . In order to simplify the notation, we define the function

/ : ( 0 , 1 ) ~

/3 ~ l(f~):--Ilog(/3/2)l.

We define the set S C ( -1 , 1) of singular points by

S := {a E ( -1 , 1) : the function f does not have an analytic

extension near the point a}.


If the set S is empty, then the claim is proved. We assume therefore that the set

S is nonempty and we show that this leads to a contradiction.

S tep 1: Define, for every n E N,

S n : = { a E S : sup I I ( Imz)n(z ) l<n} zEa+EOo

= N { a E S : sup II(Imz)H(z)i <_ n}. t3>0 *E'~+~0~

I m z >/~l

By continuity of the functions l and H, the set S,~ is closed for every n E N. By

assumption (3.3),

S - - U s n . nEN

Thus, by Baire's category theorem, there exists no such that Sn0 has nonempty

interior in S, i.e., there exists an interval (a, b) C (-1, 1) such that

SA (a,b) = Sno M (a,b) r 0.

Since the set (a, b)\S is open, it is the countable union of disjoint intervals

(ak, bk), i.e.,

(a, b)\S = U (ak, bk). kEN

We define in the following for every k E N and every 0 E (0, ~r/2) the rectangle

(3.4)

the set

(3.5) T ~ := {z E Rk :

and the disc

Rk := {zE R : a k < R e z <bk},

]arg(z--ak)] <O and ]arg(z-bk) - r I <0},

{ z Ibk~ ak + bk < (3.5) Dk := z E Rk : 2 ~ "

Note that the set Tk ~ is a rhombus for all suffucuently large k.

For every k E N, the function f is analytic in the rectangle R~ (and, in particular,

also in the set Tk ~ and the disc Dk), and therefore the function F is harmonic in Rk. Moreover, there exists a constant M > no such that

(3.7) ]l(Imz)H(z)] <_ M forevery z e U Rk\:F~ ~ kEN

Note that the constant M may be strictly larger than no since the end points of the

interval (a, b) need not belong to Sn0.


x

a~,

' ' Rk t

S j /

r

j r

/

Figure 1. The rectangle Rk, the rhombus T ~ and the disc Dk.

Step 2: We define for every k �9 N and every 0 �9 (0, 7r/2) the sets

A ~ fqR and B ~ fqR;

these are, in general, triangles if 0 is close to 7r/2 and k is sufficiently large.

Rk

" , ' - , ' Dk

a'k"

Figure 2. The sets As and Bk.

We show in this step that for every k E N, every s E (2, p) and every 0 �9 (00, 7r/2),

(3.8) F e LS(OAek)

and

(3.9) F e LS(OBg).


In fact, we prove only the first relation; the second one can be proved similarly.

Assume first that

A~ = {z E C : z - ak E Eo and lmz < 1},

i.e., the set A ~ is in fact a triangle. k In order to prove F E LS(OA~), we show first that the function F is s-integrable

along the two non-horizontal sides of the triangle OA ~ In fact, we show for every

fixed s E (2,p) and for almost every 0 E (00, 7r - 00) that

f l / s i n 0

(3.10) IF(ak + rei~ dr < oo. ,tO Indeed, by assumption (2), inequality (3.7) and H61der's inequality,

fa~-oo fl/sinO IF(ak + reiO)ls dr dO o Jo

< [ ' [,cotoo IF(ak + + i )1" d,a 30 d - 3 cot 0o

~ ~ folfl_lfl(p_s)/p ( f 'Br176 )s/P IH(a, + a + ifl) a(ak + a + i/~)1 p da d~ \ J - 3 cOt ao

(f?l )s/p < C sup IG(a +ifl)l p da , 3~(od)

where C is a constant depending only on 00, p, s and M.

Since the right-hand side of this inequality is finite by (3.2), the function under

the integral on the left-hand side must be finite almost everywhere. Thus we have

proved that (3.10) holds for almost every 0 E (00, ~r - 80).

Since the integral in (3.10) is the integral of IFI s along one non-horizontal side

of the triangle OAf, and since the integral of lFl s is bounded along the upper side of

that triangle, we deduce from (3.10) that for almost every 0 E (00, 7r/2) the relation

(3.8) holds.

Now fix 0 E (0o, ~r/2) such that the relation (3.8) holds. Note that for every

e E (0, �89 the function F is harmonic in a neighbourhood of the cut-offregion

A~ A {z e C: e < I m z < 1 - c} =: A~(e).

Hence, by Gabriel's inequality (2.4), there exists a constant C > 0 such that for

every e E (0, �89 and every 0' e (0, 7r/2),

fo tF(z)l ~ l d z t<C fo IF(z)t s [dzl. (3.11) A,~' (e) - - A~(e)


Observe that, by assumption (2), we have

fa~+e cot Oo lim ] IF(o~ + ic)l s dc~ = O.

~.-+0+ J ak-e cot 0o

From this equality, the inequality (3.11) and the continuity of F in R \ (a, b) we

obtain

foAg' IF(z)l" tdzl = limsup f IF(z)l" ldzl e~o+ JOA~'(~)

C limsup / IF(z)l s ldzl < e--~O+ JOAn(e)

~--- C j/0A ~ I F ( z ) l ~ ldzt.

This implies that (3.8) holds for every 0' E (0, 7r/2), if it holds for 0 E (00,7r/2).

Thus we finally obtain that the relation (3.8) holds for every 0 E (00, 7r/2).

If A ~ is not a triangle as we first assumed, then we need only some obvious

technical changes in the proof above in order to see that the relation (3.8) holds also

in this case. This follows from the fact that the function F is uniformly continuous

away from the real axis.

S tep 3: We show that for every k E N and every s E (2,p),

(3.12) F E hS(Dk).

For this, we remark first that for every s E (2,p) and every k E N,

(3.13) [ IF(z)l" Idzl < o0. Jo Dk

Indeed, for every 0 E (00, 7r/2) the integral of the function IFI" over the arcs

ODk M A ~ and ODk M B ~ is finite (A ~ and B ~ are defined as in Step 2). This follows

from (3.8), (3.9) and Gabriel's inequality if one applies again the reasoning with

appropriate cut-off regions as it was done in Step 2. On the remaining bit of OD k in

the upper half plane (if any), the function IFI" is bounded, and thus also integrable.

Since the function F is antisymmetric with respect to the real axis, the relation

(3.13) follows. Second, we observe that the assumption (1) implies that for every 7 E ( �89 1/m)

and every r E (lk/2, lk)

fo2~r [ ( + ) ['Y - -m f2~rSo f ak + re ir de < C I k "~ ] I sin r dr < c%


where, as before, lk = (bk - ak)/2 is the radius of the disc De. By the definition of

H "r, this implies that for every 3' E (�89 1/m),

f E H-r(Dk).

Hence, by the definition of F and (2.3), for every 7 E (�89 l /m), there exists a constant C > 0 such that

(3.14) IF(z)[ <~ C(Zk -lzl) -a/-r, z e Dk.

Third, we note that the radial limit

(3.15) lim F ( ak + bk r--+l- \ 2

+ rlke w ) =: F*(e i~ exists for every 0 E [0, 27r).

For 0 E [0,2~r)\{0,Tr}, this follows from the continuity of F in R\IL whereas for

0 = 0 or 0 = 7r, it follows from the fact that the function F is identically 0 on the

interval (ak, bk).

The claim (3.12) now follows from (3.13), (3.14), (3.15) and from Dahlberg's

maximum principle (Theorem 2.6).

S tep 4: Let 01 E (00,7r/2) be such that tan01 = 2tan0o. We prove that for

every s E (2,p) and almost every 0 E (0o,01),

(3.16) E IIFII~,(OTZ) < oo. kEN

Note that for every k large enough, the set T~ is a rhombus, so that its boundary

consists of four line segments, one of which is the curve

bk - ak } p 0 : = z E C : arg(z - ak) = O and ]z -- ak] <_ "2~os-O "

Let lk := (bk - ak)/2 be the radius of the disc Dk. Then we obtain from Fubini's theorem

fa81 ft~/cos 0 dO = E IF(ak + rei~ dr dO

k E N o dO

l t an 01

<- E tF(ak + reW)[Sr-1 r dr dO. k EN o JO

By changing from polar coordinates to cartesian coordinates and using the


assumption (2), we obtain

< E fl-x IF(ak + c~ + ifi)l s da dfl kEN J0 J/3 cot 01

f0 /~ tan01 1 [Zc~176176 +a+i f l )H(ak +a+ifl) l(~)l s dad~ -< fi J cot0, kEN

k~EN/lktanO, f f l cot 00 < MS 1 IG(ak + a + ifl)l" da dfl -- . f l l ( f l ) s JecotO,

< MS 1 ]C(a + ifl)l ~ da dfl - f l t ( f l ) s

s _<C" sup IG(a + ifl)[ s da < oo. ~e(O,1)

Since the right-hand side of this inequality is finite, the left-hand side is finite,

too; and therefore the function under the integral must be finite almost everywhere.

This means that for almost every/9 E (/90,/gx),

~ / ~ IF(z)l~ 'dz' < ~176

Proceeding in a similar way for the other three sides of the sets T ~ we obtain

for almost every/9 E (/90,01),

S t e p 5: We show that

L If(z)l s Idzl < kEN T~

(3.17) lira F(. + ifl) = 0 weakly in L s(a, b). ~--+0+

Define 01 as in Step 4, and choose 0 E (0o,01) such that the relation (3.16)

holds.

Fix k E N. For every e E (0,/k/2), we define the cut-off region

Ck~ := {z E TO: Rez E (ak + r -- e)}.

Note that

lim e--+O+ s IF(z)l~ [dzl = ~T: IF(z)l~ Idzl"

342 R. CHILL AND u TOMILOV

In order to see this, it suffices to show that the integral of IF[ ~ along the vertical

lines in the boundary of C o (e) tends to 0 as e -+ 0+. This follows from the fact

that F C hS(Dk) for every s E (2,p) (Step 3) and, for example, from Gabriel's

inequality. Alternatively, one may also use the Poisson integral representation of

F. Note that the length of the vertical lines tends to 0 as e tends to 0.

In the argument using Gabriel's inequality, we consider, for example, the in-

tegral along the vertical line on the left and let 2 < s < s' < p. From HSlder's

inequality and Gabriel's inequality, we obtain for every e > 0 small enough

f etanO I F ( a , + e +i/3)1 ~ d/3 - e tan 0

(f.t.nO ~'1" <- (2etanO)(S'-sl/s' \d-etan0 IF(ak + e + i/3) s d/3)

< ce ("'-~)/"' IIFII~.,(D~),

with a constant C which depends only on O, s and s'. Letting e tend to 0 implies

the above claim.

Hence if/3 E (0, 1), then again by Gabriel's inequality,

IF(a + i/3)1" da < C L IF(z)] ~ Idzl ~<~ - C ~ ( s ) a+i~ET~

for all sufficiently small e. The constant C depends only on s. Letting e tend to O,

we obtain

J'/-<o<~, IF(a + i/3)1" da ___ 6' Jo f IF(z)l' Idzl

for every/3 E (0, 1), where the constant C again depends only on s.

The assumption (2), the construction of the set Sn0, the above inequality,

inequality (3.7), and inequality (3.16) from Step 4 imply that for every/3 E (0, 1),

L b ]F(~ + i/3)[ s da

= / ]F(a + i/3)]s da + / IF(a + ifl)]~ da J(a ,b)ns J(a,b)\S

fo rL:' < IG(a + ifl)I s da + IF(a + i/3)1 s da ,b)NS kEN


c/. < IG(~ + i~)l ~ d~ + IF((~ + i~)l ' dc~ -- 1 ~_<(,<_b~ kEN ~+i~r~

f , + iZ)I' IF(~ da k e n a + i ~ E T 0

--- 2 \ l o g 2 / j_lm(~+iB)l sac+m IF(z)l~ Idzl - r2

< C,

where C _> 0 is some constant independent of /7. Consequently, the net

(F(. + i~));~e(0,1) is bounded in L~(a, b).

Let now ~ E 79((a, b)\S). It follows from the definition of S and from the fact

that ~ has compact support in (a, b)\S that F(. + i~)lsupp~ converges uniformly to

0 as fl --+ 0+. In particular,

lira F (a + i~)qo(c~) dc~ = O. ~---}0+

Since the net (F(. + i~))ne(o,1) is bounded in LS(a, b) and 7)((a, b)\S) is dense in L s" ((a, b)\S) (s* being the conjugate exponent of s), this implies

lim f F(a + i~)g(a) da = 0 ~-+o+ J(~,b)\s

for every g E L ~" (a, b). On the other hand,

lira [ F(a + iB)g(a) da < lim f IG(a + i$)H(c~ + i~)l Ig(~)l da ~-+0+ J(a,b)nS -- ~-+0+ J(a,b)nS

=0

for every g E L s* (a, b). This follows from the assumption (3.2) and the fact that

lim Iig(a + i ~ ) g ( t ~ ) i l L . * ( ( a , b ) n S ) = 0 f~-~0+

by the dominated convergence theorem. Thus we have proved (3.17).

S t e p 6 : We prove that the function f extends analytically through the interval

(a,b). This follows, in fact, from the classical edge-of-the-wedge theorem. It suffices

to show that for every test function qo E 79(a, b), one has

lim f b L b ~ o + Ja ( f (a + ifl) - f ( a - ifl)) ~(fl) dfl = ~-~o+lim F (a + ifl)qa(fl) d~ -- O.


This follows directly from (3.17) in Step 5.

We have finally obtained a contradiction to the assumption that S fq (a, b) is

nonempty. Therefore, the assumption that S is nonempty was not true. The claim

is proved. []

R e m a r k 3.2. Theorem 3.1 would be improved if the assumption (3.3) on the

function H could be replaced by the weaker assumption

(3.18) }im H(z) = O. ZE~o0 +C~

Indeed, the stronger assumption (3.3) on the function H is needed only in Step 4.

It is possible to replace it by (3.18) in other places of the proof by applying the

theory of the Dirichlet problem on Lipschitz domains (and its consequences for

Hardy type spaces on such domains) developed in [14] (see also [31]). However,

we have not been able to weaken it just to (3.18).

This led to the additional term log a in the Theorems 4.2 and 4.9 from the next

section about the stability of operator semigroups. The question whether in general

it is possible to replace (3.3) by (3.18) remains open.

In our next result, we show that if the set of singular points of f is sufficiently

thin, then it is possible to replace the condition (3.3) by the weaker condition

(3.18). For this, we give the definition of a set withfinite 7-entropy.

Defin i t ion 3.3. A closed set E C [a, b] is called a set with finite 7-entropy

('7 > 0) if [a, b]\E = Un%l In for a sequence (In)n>_1 of open intervals such that

oo

(3.19) Z II,~l I l o g l / n l r < oo. r~-~l

R e m a r k 3.4. Note that every closed set E C [a, b] has finite 0-entropy. Sets

with finite 1-entropy are simply called sets with finite entropy in the literature (see

[25] and also [28], where such sets appeared under a different name). Note that

if E is a set with finite entropy, and if E has Lebesgue measure zero, then E is a Carleson set.

Note that if E is such that ~n~__l II, d ~ < oo for some a E (0, 1), then E is a set

with finite "r-entropy for every ~, >_ 0. If, in addition, E has Lebesgue measure 0,

then it is also called a C1-~ set [27]. Such sets have also been studied in [36].

Let R be defined as in (3.1).

ANALYTIC C ONTINUATION 345

T h e o r e m 3.5. Let f : R\I~ ~ C be analytic, and define F : R\I~ --~ C by

F(z) = f ( z ) - f (2) (z �9 R \~ ) . Assume that

1. there exists a constant m �9 [0, 2) such that

sup If(c~ +ifl) I = O(lAI-m), /3 ~ 0; ~e(--1,1)

2. there exist a measurable function G : R \ ~ -~ C and a continuous function

H : R \ R --+ C, such that IFI _ IG[ IHI and

sup Iia(" + i3)llL, < o0 for some 2 < p < oo, /AE(0,1)

(3.20)

and

(3.21) !irn IH(z)l = 0 for every a E (-1, 1) and f ixed 00 �9 (0, 7r/2); zE~+~O 0

3. the set S o f singular points o f f on (-1, 1) is a set with finite 7-entropy for

some 7 > P/(P - 2).

Then the function f admits an analytic extension to R.

Proof . The proof is similar to the proof of Theorem 3.1. Let the set S of

singular points be defined as in the proof of Theorem 3.1, and assume that S is

nonempty. We show that this leads to a contradiction.

S tep 1: In the first step, we let for every n E N,

Sn := {c~ E S : sup In(z)l <_ n}. zE a+I ]%

The sets Sn are closed by the continuity of H, and by assumption (3.21),

S = t J S,~. nEN

Hence, by Baire's category theorem, there exists no E N and (a, b) C (-1, 1) such

that

S M (a,b) = Sno M (a,b) # O.

Let the rectangle Rk, the set Tk ~ and the disc Dk be defined as in (3.4)--(3.6).

Then, for every k �9 N, the function f is analytic in the rectangle Rk. Moreover,

there exists a constant M >_ no such that

(3.22) Ig(z)l < M for every z �9 U Rk\T~ ~ kEN

346 R. CHILL AND Y. TOMtLOV

Step 2 and S tep 3 are identical as in the proof of Theorem 3.1. They prove

that for every k E N and every s E (2, p)

F ~ hS(Dk). Step 4: Let 81 E (0o,7r /2) be such that tan01 = 2tan00. We show that for

every s E (2,p) small enough and almost every 0 E (0o, 01),

F 8 (3.23) ~ 1 1 NL,(OT D < 00. kEN

As in Step 4 of Theorem 3.1, we obtain

J;;i _;!o,.,=o, r oo 0, IF(z)l" Idzl dO < 8 -1 IF(ak + a + i~)I: da dE, o ~ J~ cot 01

where F0k and Ik are defined as in the Step 4 of Theorem 3.1. Hence, by HSlder's inequality and the inequality (3.22),

io' (z i ) <: 8 -1 l(o,/~ tan 01) (fl) __ jc~176 IF(ak + ~ + i f l ) l s da 48 \kEN dJ cot 01

~ MsLI~-I( E t~OI)(P-s)ip (Jfa b ) sip IG(a + ifl)l p da dr le Ei~l

fl<l k tan 01

By assumption (3), there exists 7 > P/(/P - 2) such that for every/~ E (0,1),

Ilogni < c IkllogZ i < C < 0 o . tan 01 -

~E~ kEN fl<lle t a n 01

Note that the constant on the right-hand side of this inequality is independent of

B e (0,1). The preceding two inequalities and assumption (3.20) imply that for s e (2,p)

small enough (so that 7 > P/(P - s)),

/o~ fr~ IF(z)l 8 ldzl d O < ex).

Hence, the function under the integral is finite for almost every 0 E (0o,0~).

Proceeding in a similar way for the other four sides of T ~ we have thus proved that inequality (3.23) is true for almost every 0 E (8o, 01).


One may now proceed as in S t e p s 5 and 6 of the proof of Theorem 3.1 in

order to obtain that the function f extends analytically through (a, b). This is a

contradiction to the construction of Sn0 and (a, b). Hence, the assumption that S is

nonempty is false, and the claim is proved. [3

Theorem 3.1 has the following analogue for the case of unit disc.

Define R := {z E C : 1/2 < tZl < 2, e iargz E I} ,

where I E T is an open interval. For 0 E (0, lr/2), let

E o : - { z E C : - O < a r g z < O } .

T h e o r e m 3.6. Let f : R \ T --+ C be analytic, and define F : R \ T -+ C by

F(z ) = f ( z ) - f (1 /5) , z E R \ T . Assume that

1. there exists a constant m E [0, 2) such that

sup [f(rei~ = 0(11 - [ r l l - rn ) , r --+ 1; 0e[o,2,r]

2. there exist a measurable function G : R \ T --4 C and a continuous function

H : R \ T ~ C, such that IFI <_ IGI �9 IHI and

sup IlG(rei')llL~(t) < oo for some 2 < p <_ c~, re(l,2)

and f o r somef ixed 0o e (0, lr/2)

lim l l o g [ 1 - 1 z [ t g ( z ) [ = O for every ~ = e i'~ E I. z E ~ O 0

Then the function f admits an analytic extension to R.

P r o o f . Let

/~ := {z E C : - l o g 2 <_ Imz < log2, a < Re z < b},

where a, b E I~ are such that I = {e iz : a < z < b}. Define the function g : /~ ~ C

by

g(z) = f(Hz), z k .

Then g satisfies the assumptions of Theorem 3.1. Hence, g admits an analytic

extension to the interval (a, b). This, however, is equivalent to the fact that f admits

an analytic extension to I. []

348 R. C H I L L A N D Y. T O M I L O V

4 Appl icat ion to stability of operator semigroups on Banach spaces with Fourier type

Let X be a complex Banach space, and (T(t))t>_o be a C0-semigroup on X. We

say that (T(t))t>o is stable if for every x E X,

lira I IT ( t )x l l = 0. t--4oo

Because of many interesting connections with various areas of analysis, the

problem of characterizing the stability of C0-semigroups has attracted considerable

attention in recent years (see the recent accounts [2], [34]). However, the stability

of semigroups is still far from being understood. Necessary and sufficient stability

conditions are available only in exceptional cases.

In this section, we present some new sufficient algebraic stability conditions

which are close to being optimal in Banach spaces with Fourier type; cf. [12].

We start by introducing some notation. For every closed linear operator A on

X with dense domain D(A), we denote by R(A, A) its resolvent defined on the

resolvent set p(A) and by a(A) the spectrum of A. By s we denote the Banach

space of all bounded linear operators on X.

Recall that a Banach space X has Fourier type p E [1, 2] if the Fourier transform

on the vector-valued Schwartz space S(R; X) extends to a bounded linear operator

from LP(II~; X) into Lq(~; X) (q := _x_l~, i.e., if the Hausdorff-Young inequality p - 1 /

holds.

By the Lemma of Riemann-Lebesgue, every Banach space X has the trivial

Fourier type p = 1. Thus, by interpolation, if X has Fourier type p E [1, 2], then

it has Fourier type p' for every p' E [1,p]. If a Banach space X has Fourier type

p E [1, 2], then the dual X* has the same Fourier type p. A Banach space has

Fourier type p = 2, i.e., Plancherel's Theorem holds, if and only if it is isomorphic

to a Hilbert space. The space Lp(f~,d#) has the Fourier type m i n { p , p / ( p - 1)}.

Most of the results relevant to the Fourier type can be found in the survey [20].

Our approach to proving stability of semigroups is based on the following

characterization of stability in terms of complete trajectories. Recall that a function

F : ]I~ --+ X is called a complete trajectory for a semigroup (T(t))t>o if

F(t + s) = T(t)F(s), s E IE, t E ~+.

L e m m a 4.1. Let (T(t) )t>o be a bounded Co-semigroup in a Banach space X.

Then the semigroup (T(t))t>o is stable i f and only i f (T*(t))t>o does not admit a nonzero bounded complete trajectory.

1With the usual convent ion for p = 1 o r p = oo. Note that l i p + 1/q = 1.


This statement was proved in [4, Theorem 3.1] (see also [41]).

The following result shows the interplay between the stability of a semigroup,

the growth of the resolvent of the generator and the geometry of the underlying

Banach space. Compare this theorem with [8, Theorem 5] and [40, Theorem 3,4].

T h e o r e m 4.2. Let X be a Banach space having Fourier type p E [1, 2], and

let q = p/(p - 1). Let (T(t))t>o be a bounded Co-semigroup on X with generator

A. Assume that for every fl E 11~ there exists an open interval I 9 /3 and a dense set

M C X such that for every 13' E I and every x E M,

(4.1) lim I logal Ilal/qR(a + il3',A)xll = O. c~---~0+

Then the semigroup (T(t) )t>_o is stable.

P r o o f . For p = 1 and p = 2, the claim actually follows from [5, Theorem 2.4]

and [40, Theorem 3.4], respectively. Hence we can assume that p E (1,2).

Let F : ~ ~ X* be a bounded complete trajectory for the adjoint semigroup

(T*(t))t>o. Replacing F by the weak* convolution F �9 qo for some ~ E LI(I~), we

can without loss of generality assume that F is uniformly continuous. Note that

F �9 ~ is nonzero if F is nonzero and ~ is chosen appropriately.

The Carleman transform fi" of F, defined by

{foe -ZtF(t) dt, Rez > 0,

F ( z ) : = - f ~ e-ZtF(t) dt, R e z < 0 ,

is analytic in C\ilIL We show that it is in fact entire.

Indeed, choose/3 E IL and let I 9 /3 and M C X be as in the theorem. Let

x E M, and put f ( t ) := (F(t), x) (t E I~). Then the Carleman transform ] = (F', x)

is analytic in C.\i~, and it satisfies the estimate

(4.2) sup I](a + i/3')1 = O(1/a), a --+ 0 + . /3'ER

We show that ] extends analytically through iI. By [5, Lemma 6.1], for every

a > 0 and every/3' E IL

(4.3) P ( - a + i/3') = R(a + i/3', A*)F(O) + 2 a R ( a + i/3', A * ) F ( - a + i/3').

Moreover, the fact that F is a complete trajectory implies that

R(z,A*)F(O) = -f'(z) w h e n R e z > 0.

350 R. C H I L L A N D Y. T O M I L O V

Hence, for a > 0 and/3' E IL we have

I](a + ii3') - ] ( - a + il3')1 -- 12<al/P/~(-a + i13'), a l / qR (a + i13', A)x) I

_< Ia(a + i~')11g(a + i~3')1,

where

G(a + ir := 112al/PP(-a + il3')11 and H(a + i/3') := Ilal/qR(a + il3',A)xll.

Note that the boundedness of the function F and the Hausdorff-Young

inequality imply

(4.4) sup IIG(a + i')IILq(R) < oo. a > 0

Moreover, from the assumption (4.1), the resolvent identity and the bounded-

ness of the semigroup (T(t))t>o, we obtain for every 190 E (0, 7r/2) and every ~' E I,

lim sup flog a I H(a + i~") a-+O+

l~It'-Btl_<a tan 0 0

< lim sup I log al Ila'/q(R(a + i f l" ,A) - R (a + i~',A))xll ~.--~0+

I ~ " - - B ' l < a tan e o

_< lira sup Ilog al Ila t a n O o R ( a + i l 3 " , A ) a l /qR(a+i l3 ' ,A )x l l c,--.* 0+

l~"-B*l_<~ tan 0 0

< tan00 sup IlZ(t)ll limsup I log al Ilal/qR(a + il3', A)x[I t > 0 a - - ~ 0 +

=0 .

It follows from this inequality and the estimates (4.2) and (4.4) that we can apply

Theorem 3.1 in order to see that the Carleman transform ] extends analytically

through the interval iI.

Hence, by [37, Proposition 0.5 (i)], the support of the distributional Fourier

transform of f = (F, x) does not intersect the interval I for any x E M. Since the

function F is bounded and M is dense in X, this implies that the support of the

distributional Fourier transform of F does not intersect the interval I.

Since/3 E ~ was chosen arbitrarily, the support of the distributional Fourier

transform of F is empty, i.e., F = 0.

Hence there exists no nonzero bounded complete trajectory for (T * (t))t>o. By

Lemma 4.1, the semigroup (T(t))t>_o is stable. []


let q = p / (p - 1). Let (T(t))t>o be a bounded Co-semigroup on X with generator


A. Assume that ia(A) M 1~ has Lebesgue measure 0 and that ia(A) fl [a, b] is a set

with finite 7-entropy for some 7 > q/(q - 2) and every a < b. Assume in addition

that for every 13 E ~, there exists an open interval I 9 13 and a dense set M C X

such that for every 131 C I and every x E M,

(4.5) lim a l/q R(ce + i13', A )x = O. or--+0

Then the semigroup (T(t) )t>_o is stable.

P roo f . The claim for p = 1 and p = 2 was proved without any restriction on

a(A) in [5, Theorem 2.4] and [40, Theorem 3.4], respectively. We may therefore

assume thatp E (1, 2).

Observe first that the nonnegativity and concavity of the function f : (0, +co) -~

N, x ~ x(log x)-r on the interval (0, e 1-'~) imply that f is semiadditive on the same

interval, i.e.,

ak) _< Z S ak) k~N kEN

for every summable sequence (ak) C (0, oo) such that ~k~r~ ak < e 1-'r. Hence, a

closed subset of a set with Lebesgue measure 0 and finite 7-entropy is again a set

with finite 7-entropy. Indeed, let S C S' c I~ be two compact sets, and assume that

S I has Lebesgue measure 0 and that it is a set with finite 7-entropy. Let a < b, and

let [a, b] \ S I = Un~N I~ and [a, b] \ S = U,~N 1, for two sequences (I~) and (In)

of mutually disjoint open intervals. Since S' has Lebesgue measure 0, we find for

every interval I,~ a family of intervals ( I~)m~m. such that In \ S I = Umbra, I~ and

II.I = ErnEM. I t ' l . Since S C S', the index sets M , are mutually disjoint. The claim follows from semiadditivity by noting that only finitely many of the I,~ can

have measure greater than e 1--r.

Let F : II~ --+ X* be a bounded complete trajectory of (T*(t))t>o. As in the

proof of Theorem 4.2, we may assume that F is uniformly continuous. It follows

from [41, Proposition 3.7] that the Carleman spectrum of F,

sp(F) := {13 E IR : i13 is singular point for #}

(~" being the Carleman transform of F), satisfies the relation

(4.6) isp(F) C or(A) M ilk

Hence, by assumption, sp(F) ~ [a, b] is a set with finite 7-entropy for every a < b.

As in the proof of Theorem 4.2, one can show that sp(F) is empty. For this, we

use the assumption (4.5) and the analytic extension criterion from Theorem 3.5.

Since sp(F) is empty, the function F must be 0.


Hence there exists no nonzero bounded complete trajectory for (T * (t))t>o. By

Lemma 4.1, the semigroup (T(t))t>_o is stable. E]

From Theorems 4.2 and 4.3, we obtain as corollaries two range conditions

implying stability. In order to state these corollaries, we introduce some more

notation.

Def in i t ion 4.4 ([331, Definhqon 3.1.1, Corollary 5.1.12). Let A be a closed

linear operator with dense domain D(A) such that

(4.7) (0, oo) C p(A) and sup AIIR(A, A)[ [ < co. A>0

For each 7 E (0, 1), define the 7-thfractional power of - A to be the closure of

the operator

sin77rTr fo ~176 dA, x E D(A). (4.8) (-A)Tx . -

For arbitrary 7 > 0, set

( -A) ~ := (-A)'r-['~](-A)[X].

Note that if A generates a bounded C0-semigroup, then for every/3 E ~ the operator A - i t satisfies (4.7). Thus, the fractional powers of i/3- A are well-defined

in this case. Note that for every fl E IR and every v > 'y ___ 0,

(4.9) Irn(i/3 - A) v C Im(ifl - A) ~.

Observe that if A satisfies (4.7), then for every A > 0, the operator R(A, A)

satisfies (4.7), so we may define the fractional powers of R(A, A) by Definition 4.4.

Moreover, by [32],

(4.10) supA':IIR(A,A)'ql < co, 7 E (0,1). A>0

The following variant of the classical Mean Ergodic Theorem has been proved

in [32] (see also [42] where bounded semigroups were considered).

T h e o r e m 4.5. Let A be a closed linear operator with dense domain such that

(4.7) holds. F/x 7 E (0, 1], x E X. Then x E Im A if and only if

lim a'rR(a, A)'rx = O. a--tO+

I f Im A is dense in X, then for everyfixed 7 E (0, 1],

Im(-A) "r = {x E X : lim R(a, A)'~x exists in X} . a-+O+


From this theorem and Theorem 4.2, we obtain the following stability criterion.

C o r o l l a r y 4.6. Let X be a Banach space having Fourier type p E (1, 2].

Let (T(t))t>o be a bounded Co-semigroup on X with generator A. If, for some

E (I/p, 1],

(4.1 1) N Ira(i/3 - A) "r is dense in X, BcR

then the semigroup (T(t) )t>_o is stable.

Proof . The casep = 2 was proved in [12] with 7 = �89 so we may suppose that

p E (1, 2). By Theorem 4.5, for every/3 E II~ and every z E N~ea Ira(i/3 - A) "r,

lira R(a + i/3, A)~z exists in X. a-+0+

Moreover, the assumption (4.1 I) and the equality

Im (i/3 - A) = Im (i/3 - A)~

(see [33, p. 142]) imply that Im (i/3 - A) is dense in X for every/3 E I~. Hence, by

(4.10) and Theorem 4.5,

lim a l - ' rR(a + i~, A)z = 0 a---+O+

Since 7 > 1/p, this implies

lim I logat Ilal/q/~(a + i/3, A)zll -- 0 a-+O+

for every z E n Im(i/3 - A) "r. /~ER

for every z E n Im(i/3 - A) "~, /3ER

where q = p/(p - 1). Since the set Im(i/3 - A) "~ is dense in X by assumption,

the claim follows from Theorem 4.2. []

R e m a r k 4.7. It was proved in [12] that the condition (4.11) with 7 < 1/p

does not, in general, imply the stability of a bounded semigroup (T(t))t>o on a

Banach space X having Fourier type p E [1,2]. The question whether Corollary

4.6 remains true for 7 = 1/p (/9 ~ 1, 2) is open (compare with Remark 3.2).

Under an additional restriction on the spectrum a(A), the condition (4.11) with

"7 = 1/p does imply stability.

Corollary 4.8. Let X be a Banach space having Fourier type p E [1, 2].

Let (T(t))t>_o be a bounded Co-semigroup on X with generator A. Assume that


ia ( A ) M ~ has Lebesgue measure 0 and that ia (A) N [a, b] is a set with finite v-entropy

f o r some V > p/ (2 - p) and every a < b. I f

N Im(ifl - A) 1/p is dense in X , tiER

then the semigroup (T(t))t>o is stable.

Proof . The claim follows from Theorem 4.3, using arguments similar to those

in the proof of Corollary 4.6. Note that V > P~ (2 - p ) is equivalent to V > q~ (q - 2)

if q = p/ (p - 1) is the conjugate exponent. []

Statements similar to Theorems 4.2, 4.3 and their Corollaries 4.6, 4.8 are true

for bounded discrete semigroups (Tn)n>0 C s

For example, the following theorem is a discrete analogue of Theorem 4.2.


let q = p / (p - 1). Let T be a power bounded linear operator on X . Assume that

f o r every ~ E ~, there exists an open interval I C ~, ~ E 1, and a dense set M C X

such that f o r every ~' E I and every x E M,

(4.12) lim I log(r - 1)l II(r - 1 ) l / q R ( r ~ ', T)xll = O. r---r 1 +

Then, f o r every x E X ,

( 4 . 1 3 ) l im IIT"xll = O. n--4oo

Proof . We only sketch the proof, which is similar to the proof of Theorem

4.2, and leave the details to the reader; see also [5].

Recall that (4.13) holds if and only if (T*n)n>0 does not admit a nontrivial

complete bounded trajectory which is now a bounded sequence g : Z ~ X* such

that gn+m = T*ngm for every m E Z and every n E N (see [41]).

So let (gn)nez be any bounded complete trajectory for (T*n),~>0. The Carleman

transform of (gn)nez is defined by

( E~=I zl, g(n), Izl > 1 0(z) := -1 1 n

- ~ n = - ~ z - + ' f f ( ), Izl < 1

and is analytic in C\qr.

In a way similar to that in the proof of [5, Lemma 6.1 ], one obtains the identity

~(z) = R(z*,T*)g(O) + (z - z*)R(z* ,T*)~(z) , z E ~), z* := 1/2.


Using this identity, the Hausdorf f -Young inequality and, finally, Theorem 3.6,

we obtain that ~ extends to an entire function.

From limlzl~oo tlO(z)ll = 0 and Liouvil le 's Theorem, we obtain that t~ and thus g

are trivial. Therefore, there does not exist a bounded nonzero complete trajectory

for (T*n)n>O. []

I f T E Z:(X) is power bounded, then for every ~ E "I~ we can define the fractional

powers of the operators ~ - T by using Definition 4.4. Moreover, a discrete analogue

of Theorem 4.5 is true in this context as well as the inequality (4.10).

The proof of the following corollary is similar to the proof of Corollary 4.6 up

to an appropriate change of notation. We leave the details to the reader.

C o r o l l a r y 4 .10 , Let X be a Banach space having Four ier type p E (1, 2]. Let

T E E,(X) be a p o w e r bounde d operator. If, f o r some 7 E (1/p, 1],

(4.14) N Im(~ - T ) "r is dense in X , ~ET

then f o r every x E X

lira I IT~xl l = 0. n--+oo

Remark 4.11. One can also formulate a discrete counterpart of Theorem 4.3

and of Corollary 4.8 if a(T) M qF has Lebesgue measure 0 and is a set with finite

7-entropy. We leave the details to the reader. Note that asymptotic properties of

orbits of such T were studied in [30] and [17].

Acknowledgements. The authors thank A. Aleman, A. Granados and

N. Nikolski for useful discussions, and the referee for a careful reading of the

manuscript and for useful remarks.

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Ralph Chill ABTEILUNG ANGEWANDTE ANALYSIS

UNIVERS1T.~T ULM 89069 ULM, GERMANY

email: chillOmathematik, uni-ulm.de

Yuri Tomilov FACULTY OF MATHEMATICS AND COMPUTER SCIENCE

NICOLAS COPERNICUS UNIVERSITY UL. CHOPINA 12/18

87-100 TORUN, POLAND email: tomilov~mat.unl,torun,pl

(Received March 23, 2003 and in revised form August 13, 2003)

analytic continuation and stability of operator semigroups

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