THE SPECTRAL MAPPING THEOREM FOR EVOLUTION SEMIGROUPS ON L ASSOCIATED WITH STRONGLY CONTINUOUS COCYCLES p
Y. LATUSHKIN AND R. SCHNAUBELT Abstract. In this note we show the spectral mapping theorem for the evolution semigroup on Lp (; ; X ), 1 p < 1, associated with a strongly continuous
cocycle on a Banach space over a continuous ow on a locally compact metric space.
1. Introduction In this note we prove the spectral mapping theorem (1) (T (t)) n f0g = et(G) ; t 0; for the evolution semigroup T = (T (t))t0, T (t) = etG , de ned on the space E = Lp(; ; X ), 1 p < 1, by
1
(T (t)f ) () = d'?t () p ('?t; t)f ('?t); 2 ; f 2 E: (2) Here and throughout the paper we impose the following assumptions: is a locally compact metric space, is a - nite regular Borel measure on being positive on open sets, and X is a Banach space. Further, ' : R ! : (; t) 7! 't is assumed to be a continuous ow and : R + ! L(X ) is a strongly continuous, exponentially bounded (semi)cocycle over '. This means that (; t + s) = ('s; t)(; s) and (; 0) = I for 2 and t; s 0, the function (; t) 7! (; t) 2 L(X ) is strongly continuous, L(X ) is the set of bounded on X linear operators, and k(; t)k Mewt for t 0, 2 , and constants M 1 and w 2 R . We assume that the measure is quasi-invariant with respect to the ow 't, i.e., the Radon-Nikodim derivative d 't=d belongs to L1() and is uniformly bounded for t 2 R . In addition, we suppose that the set of aperiodic points of 't (i.e., 't 6= for all t 6= 0) has full measure in . This implies that the aperiodic points are dense in . In Lemma 1 we will see that T is a strongly continuous semigroup on E = Lp(; ; X ). We denote the generator of T by (G; D(G)) and set T = T (1). Let us recall that cocycles arise as solution operators for equations like (3) u0(t) = A('t )u(t); 2 ; t 0; where A() is a linear operator. Such equations occur, for instance, if one linearizes a non-linear equation over an invariant set or if one considers the \hull" of a given non-autonomous equation, see, e.g., [20] and the references therein. Moreover, in Y.L.: Supported by the National Science Foundation under the grant DMS-9622106 and by the Research Board of the University of Missouri. R.S.: Support by Deutsche Forschungsgemeinschaft is gratefully acknowledged. 1
the special case = R, 't = + t, and = 0 equation (3) yields the evolution equation u0(t) = A(t)u(t), t 0. Our result generalizes spectral mapping theorems for evolution semigroups obtained by a number of authors and is proved by a di erent method. Such theorems were rst shown in [5, 6, 9] on the spaces L2(; ; X ) and C (; X ) for a nite dimensional space X and a smooth compact manifold . The main tool was the so-called Ma~ne Lemma, see [6] and (4) below, which does not hold in in nite dimensions. Also, was assumed to be 't -invariant, which leads to additional restrictions on the dynamics of 't and the spectrum of T (t), cf. [5, Prop. 1.4]. In [14] the case of a uniformly continuous cocycle, p = 2, and a Hilbert space X was considered. The main instrument to prove (1) was the Gearhart-Pru spectral mapping theorem (see, e.g., [16, 22]) that does not hold for Banach spaces. For Banach spaces and strongly continuous cocycles the spectral mapping theorem for evolution semigroups was proved in [12, Thm. 3.3] (see also [19]) on the space C0(; X ) of continuous functions vanishing at in nity. One of the motivations for proving this result in [12] was to develop a technique to obtain the spectral mapping theorem for the group generated by the kinematic dynamo operator on the space of divergence-free vector elds, see [7]. Though this goal was achieved, the proofs in [7, 12] work only for the sup-norm. Lp-proofs for strongly continuous cocycles on Banach spaces were obtained in [11, 17, 18] (see also [22] and, for X = C n , [2, 3]) in the special case of the ow of translations 't = + t on = R and (; t) = U ( + t; ), where fU (t; s)gts is an evolution family (roughly speaking, the propagator for a nonautonomous abstract Cauchy problem). In the above mentioned papers the spectral mapping theorems were essential steps to characterize exponential dichotomy of cocycles (or evolution families) in terms of spectral properties of T or G. These results can be used, e.g., to show robustness of dichotomy, see [12, 13, 21]. The main tool in our approach in the present paper is to \localize" approximate eigenfunctions of T in order to construct approximate eigenfunctions of G. This gives a replacement for the Ma~ne Lemma in in nite dimensions. In a sense, this idea goes back as far as to [15]. To achieve this localization we use weighted shift operators (T ) acting on the sequence space `p = `p (Z; X ) by the rule (T )v = (('k?1; 1)xk?1)k2Z; v = (xk )k2Z 2 `p: The appearence of the operators (T ) is related to a deep algebraic structure in the background; in fact, de ne a representation of a certain weighted translation algebra of operators of the type (2) in `p, see [1], [8], [11], [12], [13], [14], and the references therein. 2. Proofs Descriptively, our strategy of the proof of (1) is as follows. By standard arguments (see, e.g., [12, Thm. 3.3]), involving rescaling, the spectral inclusion theorem, and the spectral mapping theorem for the residual spectra, see [16, A-III], it suces to show that 1 2 A(T ) implies iR A(G) for the approximate point spectrum de ned by A(B ) = f 2 C : ( ? B )fn ! 0 for kfnk = 1g for a closed operator B . So we assume 1 2 A(T ) and give an explicit construction for an \almost-eigenfunction" 2
g of the generator G. This function is supported in a long and thin tube, i.e., a \ ow-box" formed by trajectories of 't. As a preparation we prove in Proposition 4 that uniform injectivity of I ? T on p L (; ; X ) is equivalent to that of I ? (T ) on `p uniformly for all 2 . This result is of interest by itself, but we need it as a replacement of the Ma~ne Lemma, mentioned in the Introduction. The Ma~ne Lemma holds for dim X < 1 and says that 1 2 A(T ) on the space of continuous functions if and only if there exists a point 0 2 and a vector x0 2 X , kx0 k = 1, such that (4) supfk(0 ; t)x0 k : t 2 R g < 1; see, e.g., [6, p.158]. In nite dimensions, see [6, 7], the \almost-eigenfunction" g is constructed by propagating x0 along the orbit through 0 and \spreading" it in a tube around the orbit. Instead, in in nite dimensions and Lp, we construct g below by choosing a point 0 where I ? 0 (T ) \almost" looses its uniform injectivity, and spreading the corresponding almost-eigenvector v = (xk )k2Z 2 `p for 0 (T ) in a tube around the orbit through 0. To start, we remark that for a given ow 't a quasi-invariant regular Borel measure with supp = always exists, see e.g. [1, x9]. For brevity we denote the RadonNikodim derivative by J t () = dd' (). Notice that ? (5) J t+s () = J t ('s)J s() and J t () ?1 = J ?t('t ) > 0 for t; s 2 R and a.e. 2 . We rst show that T is in fact a C0 {semigroup, cf. [14]. t
Lemma 1. Under the assumptions listed in the Introduction, (2) de nes a strongly continuous semigroup T = (T (t)) 0 on E . Proof. Clearly, T1 is a semigroup in L(E ). De ne isomteries T0(t) on E by t
T0 (t)f () = J ?t() f ('?t ). Since kT (t)f ? f k Mewt kT0(t)f ? f k + k(('?t ; t) ? I )f k; f 2 E , it suces to show strong continuity of T0(t). We claim that (6) for tn ! 0 9 a subsequence tk so that J t () ! 1 for a.e. : Otherwise, there is tk ! 0 and a compact set W of positive measure so that jJ t () ? 1j > 0 for 2 W . We may assume that J t () 1. Hence, ('t W ) ? (W ) (W ). However, ('t W ) ! (W ) by the dominated convergence theorem. Therefore (6) must hold. Now, if T0 (t) were not strongly continuous then we would nd a continuous function f with compact support and a sequence tn ! 0 so that kT0 (tn)f ? f k > 0. Using (6) we easily get a contradiction. 2 Next, we give two technical lemmas concerning ows. Lemma 2. Let 0 2 be aperiodic and n 2 N . There exists an open set U and a set such that 0 2 U , U is compact, and for all 2 U there is a unique number t() 2 (?n; n) with := 't() 2 . Moreover, the map 7! t() is continuous. p
k
k
k
k
k
3
The proof of this lemma can be found in [4, Thm. IV.2.11]. The set U is called tube of length n with section at 0 . The next result is a simple consequence of the Halmos-Rokhlin lemma.
Lemma 3. Consider 0 g 2 L1 (; ; R), j = 1; ; m. Then for every > 0 and n 2 N there is a measurable set F such that ' F \ ' F = ; for k 6= l, k; l 2 f?n; ; ng, and j
Z
nU
k
gj () d() ;
l
j = 1; ; m;
S
where U = jkjn 'k F . Proof. Fix > 0 and n 2 N . There is a set 0 of nite measure so that R 1 1 n0 gj () d() =2 for j = 1; ; m. Choose 0 < 2 L (; ; R ) \ L (; ; R) with = 1 on 0. Set := . Notice that gj 2 L1 (; ; R). We now apply the Halmos-Rokhlin lemma, see e.g. [10, Thm. 1.11], to the aperiodic map '1 on the nite measure space (; ). So, for each > 0, we derive the existence of a measurable subset F of S satisfying 'k F \ 'l F = ; for k 6= l 2 f?n; ; ng and ( n U ) , where U = jkjn 'k F . Hence, Z
nU
Z
gj () d()
()gj () d() + gj () d () + 2
\(nU ) Z 0
nU
Z
n0
gj () d()
for j = 1; ; m and > 0 small enough. Setting F := F yields the assertion. 2
Proposition 4. Under the above assumptions the following assertions are equivalent. (a) k(I ? T )f kE c kf kE for all f 2 E and a constant c > 0. (b) k(I ? (T ))vk`p c kvk`p for all 2 , v 2 `p, and a constant c > 0.
Proof. (a))(b). Consider rst an aperiodic point 0 2 and a sequence v = (xk ) 2 `p with xk = 0 for jkj > n. Fix > 0 and choose an open set U 3 0 of nite measure such that 'k U \ 'l U = ; for k 6= l 2 f?n; : : : ; n + 1g and
(7)
k('k ; 1)xk ? ('k 0 ; 1)xk k for jkj n; 2 U: 1
?
We de ne f () := J ?k () p xk if 2 'k U , jkj n, and f () := 0 otherwise. Clearly, f 2 E and (8)
kf k = p E
n X k
=?n
kxk k
Z p 'k U
J ?k () d() = (U )kvkp` : p
4
On the other hand, using (5) and (7) we compute
k(I ? T )f k = p E
= =
Z
f ( )
Sn+1 Z
k
U k
Z Z
=?n
n+1 X U k
?
xk ? J 1('k?1) J k?1()
? p1
kxk ? ('k?1; 1)xk?1kp d()
=?n +1 ?
kxk ? ('k?10 ; 1)xk?1k
=?n
+ k('k?10 ; 1)xk?1 ? ('k?1; 1)xk?1k p d()
n+1 X ? U k
p ('?1 ; 1)f ('?1)
d()
('k?1; 1)xk?1 p J k () d()
n X U k
Z
?
=?n 'k U n+1 X
? k ? p1
J ( )
? ?1 1 J () p
kxk ? ('k?10 ; 1)xk?1k + p d()
=?n n+1 X ?
= (U )
k
kxk ? ('k?10 ; 1)xk?1k + p :
=?n
Then (8) and (a) imply
cp kvkp` = (cU ) kf kpE (1U ) k(I ? T )f kpE p
p
n+1 X ?
k
=?n
kxk ? ('k?10 ; 1)xk?1k + p :
Letting ! 0, we derive k(1 ? )vk`p ckvk`p . Thus we have proved (b) for nitely supported v 2 `p and aperiodic , which are dense in by assumption. The assertion now follows by straightforward approximation arguments. (b))(a). Fix f 2 E and > 0. By Lemma 3 there are n 2 N and a measurable set F such that the sets 'k F , jkj n, are disjoint and (9) (10)
Z
kf k + p E
S
jkjn 'k F
Z
+1 F
'n
kf ()kp d();
kTf ()kp d() :
For almost every 2 F we de ne v() = (xk ()) 2 `p by (
xk () :=
?
1
J k () f ('k ); if jkj n; 0; otherwise: p
5
Then (5), (b), and estimates (9){(10) yield kf ? Tf kp Z E
p ? 1
S
f ( ) ? J ?1 ( ) p ('?1 ; 1)f ('?1 ) d( ) = = =
jkjn 'k F Z X n
f ('k )
Z
F k
=?n
n X F k
Z
=?n
?
? ?1 k p1 J (' )
kxk () ? ('
k
?1 ; 1)x
? + c
Z
p
p ?1 ; 1)f ('k?1)
J k ( ) d( )
?1 ()k
k(I ? (T ))v()kp` d() ? p
F
k
('
k
p
Z
+1 F
'n
d()
Z F
k('n; 1)xn()kp d()
kTf ()kp d()
kv()kp` d() p
ZF
kf ()kpd() = ? + cp S k jkjn ' F p ?(c + 1) + cpkf kpE : Since > 0 is arbitrary, assertion (b) is proved. 2 We can now show our main result. Let T = f 2 C : jj = 1g. Theorem 5. Suppose the assumptions listed in the Introduction. Then the spectral mapping theorem et(G) = (T (t)) n f0g holds. Also, (T (t)) = T (T (t)) for t > 0 and (G) = (G) + iR . Proof. Step 1. Assume 1 2 A(T ). Proposition 4 implies that for all n 2 N there are vectors vn 2 `p and points n 2 such that k(I ? n (T ))vnk`p < n1 kvnk`p : (11) Since the aperiodic points are dense in we can assume that n are aperiodic. Fix n 2, v = (xk ) := vn and set k := 'k n, k 2 Z. Because of 0 (T )v 2 `p there is N 2 N such that N n and N ?1 1 X X 1 p (12) k(k ; 1)xk kp : k(k ; 1)xk k 2 k=?1 k =?N Due to Lemma 2 we can choose a tube U of length 2N with a section at 0 such that (U ) < 1, (13) k('k?1; 1)xk?1 ? xk k 2 k(k?1; 1)xk?1 ? xk k and 1 k( ; 1)x k k('k?1; 1)x k 2 k( ; 1)x k (14) k ?1 k ?1 k ?1 k ?1 k ?1 2 for all 2 and jkj 2N . Finally, (11) yields 1 1 p X X 2 p k(k?1; 1)xk?1 ? xk k < np (15) k(k?1 ; 1)xk?1kp: k =?1 k =?1 6
We now construct an approximate eigenfunction g of G. Set U0 = f = 't : 2 ; 0 < t 1g and U1 = f't : 2 ; 43 < t 1g. Choose a smooth function : [0; 1] ! [0; 1] such that = 0 on [0; 41 ], = 1 on [ 34 ; 1], and k 0k1 3. Choose a smooth function : R ! [0; 1] such that = 1 on [?N; N ], supp (?2N; 2N ), and k 0k1 N2 . For 2 U we set = 'k+ , where 2 and t = k + 2 (?2N; 2N ) with k = ?2N; : : : ; 2N ? 1 and 0 < 1. For each 2 R we de ne a function g 2 E by ?
1
g() = (t)e?it J ?t () ('k ; )[(1 ? ( ))('k?1; 1)xk?1 + ( )xk ] p
for 2 U and g() = 0 for 2= U . Step 2. We show g 2 D(G) and compute Gg. Consider 2 and h > 0 with '?h = 'k+ ?h 2 U . First, assume > 0. For h < and t = k + follows ?
1
(T (h)g)() = J ?h() p ('?h; h)g('?h) ? 1 = (t ? h)ei(h?t) J ?t+h('?h)J ?h() p ('k+ ?h; h)('k ; ? h) [(1 ? ( ? h))('k?1; 1)xk?1 + ( ? h)xk ] ? 1 = (t ? h)eihe?it J ?t () p ('k ; ) [(1 ? ( ? h))('k?1 ; 1)xk?1 + ( ? h)xk ]: If = 0, we obtain for small h > 0 1
?
T (h)g() = (t ? h)ei(h?t) J ?t () ('k?h; h)('k?1 ; 1 ? h)xk?1 ? 1 = (t ? h)eihe?it J ?t () ('k?1; 1)xk?1: p
p
This implies T (h)g() = g() = 0 for h small enough and 2= U , lim h?1 (T (h)g() ? g()) &0
h
?1 ( (t ? h)eih ? (t))e?it ?J ?t () p1 ('k?1 ; 1)x = hlim h k ?1 &0
? 1 = ig() ? 0(t)e?it J ?t () p ('k?1 ; 1)xk?1
for 2 U with = 0, and lim h?1 (T (h)g() ? g()) &0
h
? ?1 ih ? 1 ( t ? h ) h (e ? 1) ? h?1(eih = e?it J ?t() p ('k ; ) hlim &0
( ? h) ? ( ))) ('k?1; 1)xk?1 + h?1 (eih ( ? h) ? ( ))xk +h?1( (t ? h) ? (t)) (1 ? ( ))('k?1; 1)xk?1 + ( )xk ) ? 1 = ig() + e?it J ?t() ('k ; ) (t) 0( )(('k?1 ; 1)xk?1 ? xk ) ? 0(t)((1 ? ( ))('k?1; 1)xk?1 + ( )xk ) p
7
for 2 U with 6= 0. Since the above limits do not depend on we derive g 2 D(G) and 1
?
Gg() ? ig() = e?it J ?t() ('k ; ) [( (t) 0( ) + 0(t) ( )) (('k?1; 1)xk?1 ? xk ) ? 0(t)('k?1; 1)xk?1 ) p
for 2 U and Gg() ? ig() = 0 for 2= U . Step 3. It remains to show kGg ? igkE < Cn~ kgkE for a constant C~ independent of n and g. Let M := supfk(; )k : 0 1; 2 g and C := sup fJ t () : t 2 R ; 2 g. By our assumptions M; C < 1. Moreover, (U1) 41C (U0). Then (13) and (14) yield p
kGg() ? ig()kp CM p 8 k(k?1; 1)xk?1 ? xk k + N4 k(k?1; 1)xk?1k
p 1 C1 k(k?1; 1)xk?1 ? xk k + n k(k?1 )xk?1k for = 'k+ 2 U , where C1 is a constant not depending on k; n, and g. Consequently,
kGg ? igkE =
Z
kGg() ? ig()k d()
!1
p
p
S2N ?1
=?2N 'k U0 " 1 1 X k
C1
p
k
=?1
(' U0) k(k?1 ; 1)xk?1 ? xk k k
1
p p + n1 k(k?1; 1)xk?1k !1 1 p X 1 3 k(k ; 1)xk kp : < (CC1(U0 )) p n k =?1
(16)
In the last step we have used (15). On the other hand we have 1
?
g() = e?it J ?t () ('k ; )xk p
for 2 'k U1, k = ?N; : : : ; N ? 1. Moreover,
k(; 1)xk k(' ; 1 ? )k k(; )xk M k(; )xk holds for all 2 , 0 1, and x 2 X . Thus (5) and (14) imply kg()k = p
?
?t p1
J ( )
('
k
p ; )xk
C (21M )p k(k ; 1)xk kp 8
1 k('k ; 1)x kp CM k p
for 2 'k U1, k = ?N; : : : ; N ? 1. Using (12) we conclude
kgk p E
Z
SN ?1 k
=?N
'k U
1
kg()kp d()
?1
C (21M )p ('k U1 )k(k ; 1)xk kp k =?N N X
(17)
1 X
0) p 8C3((2UM )p k=?1 k(k ; 1)xk k :
Combining (16) and (17) establishes the theorem.
2
References
[1] A. Antonevich, \Linear Functional Equations. Operator Approach", Oper. Theory: Adv. Appl. 83, Birkhauser, Boston, Basel, Berlin, 1996. [2] B. Aulbach, Nguyen Van Minh, P.P. Zabreiko, Groups of weighted translation operators in Lp and linear nonautonomous di erential equations, J. Math. Anal. Appl. 193 (1995), 622{631. [3] A. Ben-Artzi, I. Gohberg, Dichotomy of systems and invertibility of linear ordinary di erential operators, Oper. Theory, Adv. Appl. 56 (1992), 90{119. [4] N.P. Bhatia and G.P. Szego, \Stability Theory of Dynamical Systems", Springer-Verlag, Berlin, Heidelberg, New York, 1970. [5] C. Chicone and R. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Di . Eqns. 36 (1980), 28{39. [6] C. Chicone and R. Swanson, Spectral theory for linearization of dynamical systems, J. Di . Eqns. 40 (1981), 155{167. [7] C. Chicone, Y. Latushkin and S. Montgomery-Smith, The spectrum of the kinematic dynamo operator for an ideally conducting uid, Commun. Math. Phys. 173 (1995), 379{400. [8] D. Hadwin and T. Hoover, Representations of weighted translation algebras, Houston J. Math. 18 (1992), 295{318. [9] R. Johnson, Analyticity of spectral subbundles, J. Di . Eqns. 35 (1980), 366{387. [10] L.K. Jones and U. Krengel, On transformations without nite invariant measure, Adv. Math. 12 (1974), 275{295. [11] Y. Latushkin and S. Montgomery-Smith, Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. 127 (1995), 173{197. [12] Y. Latushkin and S. Montgomery-Smith, T. Randolph, Evolutionary semigroups and dichotomy of linear skew-product ows on locally compact spaces with Banach bers, J. Di . Eqns. 125 (1996), 73{116. [13] Y. Latushkin and T. Randolph, Dichotomy of di erential equations on Banach spaces and an algebra of weighted composition operators, Integr. Eqns. Oper. Th. 23 (1995), 472{500. [14] Y. Latushkin and A. M. Stepin, Weighted composition operators and linear extensions of dynamical systems, Russian Math. Surveys 46 no. 2 (1992), 95{165. [15] J. Mather, Characterization of Anosov di eomorphisms, Indag. Math. 30 (1968), 479{483. [16] R. Nagel (ed.), \One Parameter Semigroups of Positive Operators", Lecture Notes in Math. 1184, Springer-Verlag, Berlin, 1984. [17] F. Rabiger and R. Schnaubelt, A spectral characterization of exponentially dichotomic and hyperbolic evolution families, Tubinger Berichte zur Funktionalanalysis 3 (1994), 204{221. [18] F. Rabiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum 52 (1996), 225{239. [19] R. Rau, Hyperbolic linear skew-product semi ows, Z. Anal. Anwend. 15 (1996), 865{880. [20] R. Sacker and G. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Di . Eqns. 113 (1994), 17{67. 9
[21] R. Schnaubelt, \Exponential Bounds and Hyperbolicity of Evolution Families", Dissertation, Tubingen, 1996. [22] J. M. A. M. van Neerven, \The Asymptotic Behavior of Semigroups of Linear Operators," Operator Theory Adv. Appl. 88, Birkhauser, 1996. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
E-mail address :
[email protected]
Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, 72076 Tubingen, Germany
E-mail address :
[email protected]
10
