migroup Forum Vol. 54 (1997) 356-363 1997 Springer-Verlag New York Inc.

RESEARCH ARTICLE

Analytic Families of Semigroups Shmuel Kantorovitz Communicated by J. Goldstein

Abstract The Cauehy Problem for linear partial differential equations with coefficients depending analytically on a parameter, motivates the study of analytic families of infinitesimal generators of semigroups. The relations between the three natural ways to understand "analytieity" of the family are clarified. This is then applied to obtain some general results on the stability of the analyticity of the family under perturbations of the generators. Let ~ be a region in C (that is, an open, connected, n o n - e m p t y subset of the complex p l a n e C). For each z E f~, let A(z) be the generator of the C0semigroup {T(t; z); t ~ 0} in the Banach space X . In general, the constants of exponential growth of the semigroups (that is, the constants a, M such that ]IT(t;z)l[ ~ Me ~* for all t > 0 ) d e p e n d on z. The following standing uniformity hypothesis is made:

Uniformity hypothesis: For each compact subset K of Q, there exist cons t a n t s a = a ( g ) ~_ 0 a n d M = M ( K ) > 1 such that (1)

]IT(t; z)l I <_Me ~'

for all t > 0 a n d z E K . We consider the following possible "analyticity assumptions" on the family, with respect t o its p a r a m e t e r z:

1. Generator analyticity: for all x in a c o m m o n d o m a i n D for all A(z) ( z E ~), A(.)x is analytic in ft. 2. Resolvent analyticity: for each subregion n0 with compact closure K C ~/, t h e resolvent R(~; A(.))x is analytic in f~0 for all )~ > a = a(K) and x E X . 3. Semigroup analyticity: T(t; .)x is analytic in ~ for each t > 0 and x E X . We shall clarify t h e relations between these three concepts. We s t a r t with the following simple lemma, which is contained implicitly in the p r o o f of [4; T h e o r e m 1.54]. Lemma. Let a < b ~_ oc. Let {xs(.) I s E ( a , b ) ) be a family of X-valued functions, analytic and uniformly bounded on compact subsets in a region ~, such that xs(.) -+ x(.) weakly, pointwise in ~, as s -+ b. Then x(.) is analytic in ~ . Proof. For each x* E X * , the family of complex analytic functions {x*xs(.) I s E (a, b)) is uniformly b o u n d e d on c o m p a c t subsets in Q, hence is normal. T h e r e exists therefore a sequence x*xsk (.) converging uniformly on compact subsets of to a function f analytic in Q. Since x*xs(.) -+ x'x(.) pointwise in Q, we have x'x(.) = f , thus x'x(.) is analytic, for each x* E X * , which means t h a t x(.) is analytic in ~/ (cf [3; T h e o r e m 3.10.1]). •

KANTOROVITZ T h e o r e m 1. Semigroup analyticity is equivalent to resolvent analyticity. Proof. 1. Assume semigroup analyticity. Let ~0 be a subregion of ~ with compact closure K C ~ , and let a = a(K), M = M ( K ) as in (1). By (1), R(A; A(z)) exists for £ > a, and is given by the absolutely convergent Laplace transform (2)

R(£; A(z))x =

//

e-XtT(t; z)x dt,

(~ > a;z E ~0;x E X). For z, w E ~o, )~ > a, and x E X , (3)

IlR(£;A(z))x - R()~;A(w))xll <_

L

e-xtllT(t;z)x - T(t;w)xlldt.

The integrand converges pointwise to 0 when z --+ w, since T(t; .)x is analytic, hence continuous, in ~. It is majorized by 2M]lxile -(~-a)t E Ll(0, c~) (for > a). Dominated convergence implies therefore that R(~; A~.))x is continuous in ~0 for each A > a. If F is a triangular path lying together with its interior in ~0, then Fubini's theorem shows that R(A; A(z))x dz =

e -xt

T(t; z)x dz dt = 0,

by Cauchy's theorem, since T(t; .)x is analytic in ~. It follows that R()~; A(.))x is analytic in ~0, by Morera's theorem (for each x E X and A > a). In particular, we have "resolvent analyticity". 2. Conversely, assume resolvent analyticity. For any subregion ~0 as before, and its corresponding parameters a, M as in (1), consider B(.) := A(.) - aI. For each z E ~0, B(z) generates the Co-semigroup S(t; z) := e-atT(t;z), which satisfies (4)

IIS(t; z)l I < M

for all t >_ 0 and z E fl0. Hence (5)

II~n(~;B(z))ll < M

for all A > 0 and z E ~t0. Fix t > 0 and x E X . The family of X-valued functions xn(.) := ( n l t ) R ( n l t ; B(.))x (n = 1, 2, ...) are uniformly bounded in ~0 (by M llxll, according to (5)). Since R(A;B(.)) = R(~ + a;A(.)), the functions x,(.) are analytic in ~0 for all n > no (for some no E N), and converge pointwise to S(t; .)x (strongly in X ), by the well-known exponential formula for semigroups. By the Lemma, S(t; .)x is analytic in ~0, and therefore T(t; .)x = eatS(t; .)x is analytic in ~'/0, hence in ~. • The next theorem relates "generator analyticity" to the other two (equivalent) concepts. In order to deduce semigroup analyticity from generator analyticity, the common domain D needs to be sufficiently rich. On the other hand, an obvious necessary condition for generator analyticity is the uniform boundedness of ]IA(.)xH on compact subsets of ~/, for each x E D. This leads us to the following 357

KANTOROVITZ

T h e o r e m 2. "Semigroup analyticity" together with uniform boundedness of A(.)x on compact subsets of ~ (/or each x in a common domain D ) implies "generator analyticity" on D . Conversely, "Generator analyticity" on a dense R(.; A(.))-invariant common domain D implies "semigroup analyticity".

Proof. 1. Assume "semigroup analyticity" and uniform boundedness of A(.)x on compact subsets of f~ (for each x E D). Fix x E D and 3 > 0. Let (6)

xt(z) := t - ' [T(t; z)x - x I

(0 < t < Q.

Then each xt(.) is analytic in 12, by the semigroup analyticity hypothesis. If K C f~ is compact, and a, M are the corresponding parameters as in (1), then for all z E K ,

/o'

Ilxt(z)ll = IIt-1

T(s; z)A(z)z dsll _

t -1

/o'

Mea'ds IIA(z)xll

<_Me~IIA(z)~II, so that {xt(.); 0 < t < 3} is uniformly bounded on compact subsets of ft. Also xt(.) --4 A(.)x strongly, pointwise in f~, as t --+ 0 + . By the Lemma, it follows that A ( . ) z is analytic in ~ . 2. Assume generator analyticity on the dense R(.; A(.))-invariant common domain D. Specifically, this means that for each compact K C ft, there exists a = a ( K ) > 0 such that R(,k; A ( z ) ) D C D for all .k > a and z E K . Let ~0 be a subregion of ft with compact closure K C ft, and let a, M be the p a r a m e t e r s associated with K as in (1). Fix x E D and ,k > a. For z, w E f/0, z fixed, and y := R(A; A ( z ) ) x , we have y E D , and therefore

(7)

R(,k; A(z) )x - R(),; A(w) )x = R(~; A(w) )[A(z) - A(w)]y.

Hence

IIR(~X;A(z))x

-

R()q A(w))xll < A-~allA(z)y - A(w)Yll ~ o

as w ~ z, by continuity of the analytic function A(.)y in ft. Thus R(~; A(.))x is continuous in ft0 for each x in the dense subset D C X . Since IIR(~; A(w))t] < M for all W E f t 0 , it follows that R(A; A(.))x is continuous in ft0 for all ,k-a xEX. Next, denote v = [A(.)y]'(z), for z E i20 f i x e d and y as before. We have for all w E f~0 l[ R(,k; A ( z ) ) x - R(,k; A ( w ) ) x _ R(,k; A(z))v[[ Z--W

< IIR(A; A(w){ A ( z ) y - A ( w ) y _ v}ll + ii{R(.k; A(w)) - R(),; A(z))}vll Z--W

M

A ( w ) y _ [A(.)yl,(z)l I + iiR(,k; A(w) )v - R(,k; A(z)vll. < y-:-~_all A ( z ) y- - -~ When w --+ z, the first term above tends to 0 by analyticity of A(.)y at z (since y E D ) ; the second t e r m tends to 0 by the continuity of R(,k; A(.))v at z, which we verified earlier. 358

KANTOROVITZ Thus R()~; A(.))x is analytic in ft0, for each x e D and ~ > a = a(a0). For x C X arbitrary, we use the density of D to obtain a sequence xn in D such that xn --~ x. Let r = sup,, I[x,lI. Then

Mr

IIR(A; A(.))x.II < ,~--S--d_ a on Q0 (with the corresponding parameters a,M), and, for each fixed ~ > a R(X; A(.))xn --~ R($; A(.))x strongly, pointwise in f/0- Since each vector function R(,k; A(.))xn is analytic in Q0 (as we proved above), it follows from the Lemma that R(~; A(.))x is analytic in ~20, for each ~ > a = a(fto). By Theorem 1, this is equivalent to semigroup analyticity. • We consider now a special case, with obvious applicability to the Cauchy problem for linear partial differential operators, whose highest total order mixed derivatives have coefficients independent of the parameter z, while all other coefficients are analytic in z in some region. Let A be the generator of a Co-semigroup T(.) on the Banach space X , let Ft C C be a region, mad let {B(z); z E f t ) be a family of closed operators satisfying the hypothesis

T ( t ) X C D(B(z))

(H1)

for all z C ~ 2 and t > 0 . By the Closed Graph Theorem, B(z)T(t) C B ( X ) for all z e gt and t > 0. Ift,t+h>0,

B(z)T(t + h)x - B(z)T(t)x = [B(z)T(t)][T(h)x - x] -~ 0 when h --+ 0, showing that B(z)T(.) is strongly right continuous on (0, oc). It follows that IIB(z)T(.)tt is bounded on compact intervals, for each fixed z e f~. A routine argument implies then that B(z)T(.) is strongly continuous and IIB(z)T(.)II is measurable on (0, c¢) (cf. [2, Chapter VIII]). If to denotes the type of T(.) to := lim log llT(t)ll t ~

t

'

then, for any t > e > 0, e)log IIT( t - e)l[ log IIB(z)T(t)ll < logB(z)T(e)ll + ( 1 - ~t-e t so that lim sup log liB(z)T(t)ll < ~o. t.--~c~

t

Together with the boundedness of IIB(z)T(.)ll on compact subintervals of (0, c¢) (observed above), this implies that, for each e > 0, a > w and z E ft, there exists a constant M = M(e,a,z) > 1 such that IIB(z)T(t)ll < M(e,a,z)e a~ for all t > e. In order to control the growth of [IB(z)T(.)II on the whole ray [0, c¢), uniformly with respect to z in compact subsets of ft, we make the following hypothesis: 359

KANTOROVITZ (H2) For each compact K C ~2, there exist constants a = a(K) > 0 and M = M ( K ) > 0, and a positive function h = hK E L 1(0, 1), such that

IIB(z)T(t)ll < Me at

(z E K ; t > 1)

and tlB(z)T(t)ll < h(t)

(z E K ; 0 < t < 1).

This hypothesis is stronger than the hypothesis/-/2 in [4;Part I, Section F]. It follows in particular (cf. L e m m a 1 there) that D(A) C D(B(z)) for each z E ~2, and by the Hille-Phillips perturbation theorem (cf. [2; Chapter VIII] or [4; T h e o r e m 1.38, p. 34]), the operator

A(z) := A + B(z) with domain D = D(A) generates a C0-semigroup T(.; z). Our final hypothesis on the family {B(z); z E f~} is (H3) B(.)x is analytic in a , for each x E D. We can state now 3. Let A(.) = A + B(.) with A and B(.) as above. Assume the hypothesis (111-113) are satisfied. Then semigroup analyticity holds for the family of semigroups iT(.; z); z E f~} generated by the operators A(z), z E ~. Proof. Let K C f~ be compact, and let a, M, h be the parameters associated with K as in (H2). For r > a, set

Theorem

q(r; z) :=

f0 c~ e-~t llB(z)T(t)li dr.

It was observed above that the integrand is a measurable function of t on (0, cx~). By breaking the integral into two integrals over (0, 1) and [1, oc) respectively and applying the hypothesis (H2), we obtain the estimate q(r; z) < Q(r), for all z E K and r > a , where Q(~) :=

fo I e - " h ( ~ ) d~ +

M.... e - ( r - a ) r--a

is independent of z. By Dominated Convergence, Q(r) ~ 0 when r -+ oc. We can then fix r = r ( g ) > a such that Q(r) < 1. For this r, q(r;z) < 1 for all z E K , and it follows from the proof of Theorem 1.38 in [4; Part I, Section F] (cf. the estimate on p. 39 for IIS(.)II, which is precisely liT(.; z)ll, in our notation) that for all

zEK,

M ert < Mt ert~ IIT(t;z)ll < 1 - q ( r ; z ) -

where M

! .~

M _ _ 1 - Q(r)

depends only on the compact subset K of f~ (and is necessarily > 1 ). Thus the uniformity hypothesis (1) is satisfied by the family iT(.; z); z E f~}. 360

KANTOROVITZ The common domain D := D(A) of the operators A(z), z E D is dense, as the domain of a C0-semigroup generator. For each compact K C f~, if r and q(.) are as before, it follows from Lemma 1 , p. 34, in [4; Part I, Section F] that for z E K, A > r and x E X ,

IIB(z)R(A;A)xll

= II

fO~ e-XtB(z)T(t)xdtll

<_ q(r;z)llx}l <_ Q(r)llxll.

Since Q(r) < 1, this implies that the series R(A; A)~=o[B(z)R(A; A)]" converges in B ( X ) when A > r, and a routine calculation shows that its sum is precisely R(A; A(z)) (cf. proof of Lemma 2, pp. 35-36, in [4; Part I, Section F] for the details). In particular, R(A; A ( z ) ) X C R(A; A ) X = D, hence D is certainly R(A; A(z))-invariant for all A > r and z E K . Finally, for each z E D, A(.)x = Ax + B(.)x is analytic in 12 by (H3). The conclusion of Theorem 3 follows then from Theorem 2. •

Corollary.

Let A be the generator of a Co-semigroup region in C, and let B(.) : ~ -+ B ( X ) be such that B(.)x is each x E X . Then semigroup analyticity holds for the family iT(.; z); z E f/} generated by the operators a(z) := A + B(z),

T(.), let f~ be a analytic in ~ for of Co-semigroups z E fl.

Proof. Fix a > w, where w denotes the type of A. There exists then a constant M1 (depending only on a) such that llT(t)ll <_ Mle at for all t :> 0. For each compact K C f/, sup IIB(z)xll <

zEK

for all x E X , by continuity of the analytic function B(.)x in 12. By the Uniform Boundedness Theorem,

sup lIB(z)[[ :-- M2(K) < oo. zEK Then for all t > 0,

[[B(z)T(t)l I <_M1 M2(g)e at = M ( K ) e at. Thus the hypothesis (H1-H3) of Theorem 3 are satisfied, and the Corollary follows. • We consider now the special case of contractions Co-semigroups. The proof below relies on two weU-known results, a perturbation theorem of Kato and the Trotter P r o d u c t Formula. These can be found in any of the usual books on semigroups, such as [1] oi" [5], but a specific page reference is given to [4] for the author's convenience.

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KANTOROVITZ T h e o r e m 4. Let f~ C C be a region. For each z E ~, let S(.;z) and T(.;z) be Co- semigroups of contractions, with generators A(z) and B(z) respectively. Suppose that for each z E Q, (1)D(A(z)) C D(B(z)) , and (2) there exist a(z) E [0, 1) and b(z) > 0 such that

IIB(z)xll < a(z)IIA(z)xll + b(z)Ilxll for all x E D(A(z)). Then C(z) := A(z)+ B(z) generates a Co-semigroup of contractions U(.; z) for each z E ~, and semigroup analyticity for S(.;.) and T(.;.) implies semigroup analyticity for U(.; .). Proof. For each z E ~ , B(z) is dissipative (as the generator o f a C0-semigroup of contractions) and A(z)-bounded with A(z)-bound < 1 (cf. [4; Definition 1.28, p. 24]). By Theorem 1.30, p. 25, in [4], C(z) := A(z) + B(z) generates a C0-semigroup of contractions U(.; z), which is explicitly given by the Trotter Product Formula (cf. [4; Theorem 1.37, p. 33]): (8)

V(t;z)x = ,_. lim [S(t/n; z)T(t/n; z)lnx

(x E X),

strongly in X . Observe that if F,G are B(X)-valued such that F(.)x and G(.)x are analytic in 12 for each x E X , then F(.)G(.)x is analytic in 12 for each x E X . Indeed, fix z E f~, and let 6 > 0 be such that z + h E ~ for all complex h with ]h I < 6. By continuity of the analytic functions F(.)x in f~,

sup IIf(z + h)xll < oo Ihl<_~ for all x E X , and therefore, by the Uniform Boundedness Principle, M := sup IIF(z + h)l I < ~ . Ihl_<~ For 0 < Ihl < 5 and all x E X ,

Hh-~{F(z + h)G(z + h)x - f ( z ) G ( z ) x } - F(z)[G(.)x]'(z) - {F(.)[G(z)x]}'(z)I I

< IIF(z 4- h ) { h - l [ G ( z 4- h)x - G(z)x] - [G(.)xl'(z)}ll +[[h-~[F(z + h) - f(z)][G(z)x] - {f(.)[G(z)x]}'(z)[[

+ll[F(z + h ) - F(z)l[G(.)x]'(z)ll. The first summand on the right of the inequality is < M IIh-l[G(z + h)x G(z)x] - [G(.)x]'(z)][ -4 0 as h -4 O, by analyticity of G(.)x at z. The second summand -4 0 by analyticity of F(.)[G(z)x] at z. The third summand -4 0 by continuity of F(.)y at z, for the fixed vector y := [G(.)x]'(z). Thus F(.)G(.)x is analytic in 12 for each x E X . Suppose now that semigroup analyticity holds for S(.; .) and T(.; .). By induction on k, it follows from the above observation that [S(t/n; .)T(t/n; .)]kx axe analytic in Q for a l l t > 0 , x E X , n E N and k E N . For t > 0 and x E X fixed, let x,(.) = [S(t/n; .)T(t/n; .)]~x. Then x,,(.) axe analytic and [[xn(.)I[ < ]]xil in ~ , for all n E N, and by (8), x,(.) -+ U(t; .)x strongly, as n -4 co. By the Lemma, it follows that U(t; .)x is analytic in 12, for each x E X . • 362

KANTOROVITZ References

[1] Davies, E.B., "One-Parameter Semigroups," Academic Press, London, 1980. [2] Dunford, N., and Schwartz, J.T., "Linear Operators," Volume I, Interscience Publishers, New York, 1958. [3] Hille, E., and Phillips, R.S., "Functional Analysis and Semigroups," Amer. Math. Soc. Coll. Publ. 31, Providence, R.I., 1957. [4] Kantorovitz, S., "Semigroups of Operators and Spectral Theory," Pitman Research Notes in Mathematics Series, Longman, Harlow, UK, 1995. [5] Pazy, A., "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983. Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel Received June 11, 1995 and in final form August 30, 1995

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