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RENDICONTI DEL ClltCOLO MATEMATICO DI PALEmMO Serie II, Tomo XXXV (1986), pp. 58-81
ON T H E F U N C T I O N A L . C A L C U L U S OF N O N - Q U A S I A N A L Y T I C
G R O U P S O F O P E R A T O R S AND COSINE F U N C T I O N S
ERICH MARSCHALL
Dedicated to Heinz-G/inter Tillmann on the occasion of his 60th birthday
Introduction.
The theory of genera;ized scalar operators of Colojoara and Foias, was
partly generalized to unbounded closed operators in [21]. The generator of a
temperate group of operators {U(t):tE IR} is such an operator ([21], 4.5).
But, if it is the generator of a group of operators and M ~ is the algebra of measu- Y
res ~ with f~(t)dlv.l(t)<~,, ~(t): =max(1, llu(t)ll), t h e n ' t x ( T ) x : =
=fU(t) d~t(t) is a functional calculus for T. This calculus was examined and
used by a lot of authors (e,g. [2], [3], [7], [8], [11], [15], [16], [19], [25], and others). Now M ~ is normal which means it contains decompositions of the unity, if U (.) is non-quasianalytic, and in this case M ~ is admissible in the sence
of [6], resp. [21]. Even in [19] this was used to construct spectral subspaces
and a decomposition of the generator. This decomposition can now be sharpened.
If U (.) is unformly continuous, hence the generator iT is bounded, then
Te[C**(IR)], fff IIu(t)ll=o(lt[ ), ~>_0 ([6], 5.4.5.), and most results of [3] and [16], e.g. thv theorem of support and the spectral mapping theorem, follow
from [6], in particular o '(U(t))=exp(it~r(T)) can be proved with the help
of the calculus for analytic functions. If U (.) is only strongly continuous, hence
ON THE FUNCTIONAL-CALCULUS OF NON-OUASIANALYTICj ETC. 5 9
T is unbounded, the situation is much more complicated, because the algebra M ~
is not inverse closed. But M ~' consists of local multipliers of the inverse dosed
subalgebra F", and [22], 1.3. yields the spectral mapping theorem ~r(U (t))=
=exp (it (o" (T) N C), VtE IR, because U (t)E[W"], hence U (t) is decomposable
VtE IR. Our results hold for cosine operator functions which are treated in
the third part, too, and with the help of the simple translation group on
LP(0, 1), l < p _ ~ , we obtain some important classical theorems on the con-
vergence of Fourier series (e.g. the theorem of Bernstein 2.5).
1. Notations are used as in [21] and [22]. Let {O}~X be a complex
Banach space and �9 (X) the Banach algebra of bounded linear operators on X.
61(X) be the set of closed linear operators in X and for F'E61(X) let
�9 = ~ U { ~ } be the extended spectrum of T, p (T): = ~ \ o" (T) the
resolvent set, and R (T, ~): = (Ix-- T)-~, t~ E p (T) N IE. Let ~ (X): = { T E 61 (X):
p (T) ~ ~ } and $ (X): = { T E 61 (X): T has the single valued extension property
(s.v.e.p.) }. If TE $ (X), then O'r (x) resp. pr (x): = <E \ o'r (x) are the local
spectrum resp. local resolvent set of x'EX. T~,TzE ~g (X) commute, if R (TI, IJ,0
commutes with R(T2,1x2) for some, hence all Ixi'Ep(Ti) N ~ ([21], Def. 1.13.
An algebra A of complex valued functions on 12c C is called admissible
([21], Def. 2.2.), if it (1) contains the constants, (2) A is normal, i.e. for every
open covering ~ c ~ Gic ~ of ~ there are (piE A, 1 <_]<n, with
O~<pi~ 1, ~ <pim 1, i=1
and ~ . c G i . Where _] :={zEl2: / (z)~O} is the support of tEA, and the
closure is always taken in the topology of ~ . (3) if l E A and lxef, then
f~'EA where f~(z):=zf(z) if Ix=o o, and f,(z):=f(z)/(lx--z) if t x ~ .
TE 61 (X) is of class A (TE [A]), if there is a homomorphism e r : A ~ ~ (30
with (4) e r ( 1 ) = l and (5) Tr162 for each lEA with z/~A.
60 ERICH MARSCHALL
An admssible a',gebra is called inverse clo~ed( if it consists of continuous
functions ort S2=~, and 1 / tEA if f (z)~OVzEfL In [21] the following admissible algebras were treated:
1.) Bm(IR), 0__<m<~, the algebra of complex valued functions on IR
with continuous derivations up to order m and with the uniform convergence of
these derivations, hence with the norm:
m 1 II/11 : . ,,sup, R IJ"(s)l" for m<oo .
A
2.) Cm(K), the C'-functions on the C~ K c C, with K fq C
circle or a straight line. In particular:
3.) Cm(S1): ={/:S1---, C:g( t ) : =I(e ' )EB'~(IR)},
with the norm II/ll.=llgll~, and
4.) c . , (J~) : ={k rR: =JR u{ oo }~ C:/IIR,~ JR ,~ m' OR);?(s): =f (l/s)}=
s,: ={z~ C: Iz[= 1 },
,. 1-be it ={/ : IR - - C :g (t): = / t= - - / -27) ~ S '~ (IR)}
with the norm P-(h:---IltlIRIl-§ which is equivalent to II111-:--Ilgll-.
BIn(C) and Cm(C) are similarly defined ([21], Def. 3.1.), and let
5.) H (K) be the algebra of locally holomorphic functions on K = 7 ( c C with
the usual inductive topology.
1. The calculus for non-quasianalytir groups.
For the following we refer for instance to [24], App. E and 1.3. Let M
be the Banach algebra of regu'~ar Borel-measures on IR with total variation as
norm and convolution as product. Let G be the set of real valued measurable
functions to on IR which are bounded on every compact interval and:
(G.1): l=to(O)<_~o(t+s)<_m(t)to(s) Vt, sE IR
ON THE FUNCTIONAL.CALCULUS OF NON-OUASIANALYTIC, I~TC. 61
and
(G.2): J" logto(t) ( l+ ta ) -~d t<oo . IR
If toEG, let M~: ={IxEM: j" to(t)dlIxl(t)<.o} IR
L 1" = { Ix E Mto: ~ is absolutely continuous }. to"
If IxEL 1" there exists an unique measurable function g which will be to ~
identified with Ix, such that f Ig(t)Joa(t)dt< oo and Ix (E)=fg( t )d t for every IR
Borel set E c I R . Obvious.~y b:M~'~M; blx(E): = f to( t )d~( t ) is bijeetive, and E
because of (G.1) M ~' is again a Banach algebra with convolution and norm
IlIxll.=fto (t) d lixl (t). Moreover, L~ is a closed ideal in M~':
v
6.) M~: ={J:IR -o C; /(s): =Ix(s): = f dt'dIx(t):IxEM ~} with the IR
norm Ilfllto'=llIxll~ M ~ 'consists of uniformly continuous functions and is a
Banach algebra under pointwise multiplication.
7.) Fto: ={ / : I[~---- C : / I IR=w+Ix ; w E C , IJ, E Lifo}= C<~ 1.1~. f Ei ~ can uniquely be decomposed into /~0+/0 with f,.(s)-~f(**) and /oEL~.
In fact, the theorem of Riemann-Lebesque shows that /o(o*)=O. Moreover,
II111. = I1(/(~)~ +~) " II o= If (~)1 + I llollto.
8.) W ~ : = { f : S l ~ : / ( e / ' ) = ~ anem', I1/llto:= ~ la.lto(n)<oo} nE 7/ nE 7/
is called Beurling algebra and is a Banach algebra, too. If ct___O, then w ~ ( t ) : - - ( l + [ t l ) ~ G , and we write F ~ resp. tlfll~ etc. instead
of F% resp. l ltlloo etc. If h, L > 0 and to E G, then toh,L (t): =to (L t) k E G. Moreover,
to2 E G whenever to1, to2 E G, and if to1___ c to,, then Mtoz is continuously and,
since the measures with compact support are dense, also densely embedded in
M% If there are C,c>O with ctol<to2<__Cto~,to~ and to2 are called equivalent
and Mto.=M% For to~( t )=( l+l t l ) ~ we have:
LEMMA 1.1. Let ~t>_O and n, mE [No with m > < x + l / 2 and g>_n, then:
a) H (11~) ~ C" CIF~) ~-> F'c_M~cB" (IR)
62 ERICH MARSCHALL
and
b) H (S0 ~-~ C" (St) ~-~ W ~ ~ C" (St).
,'3_ Proo/: Let /ECm(IR) with / (** )=0 and let ~ : = m - ~ > l / 2 . Now:
m m ^
tl/11"-- f (1 -I-ItJ) m l/(t)[ d t__< 11(1 "4-Itl)-'[Io j~_-0 ( j )l[(i t)i / (t)l[o~< (l+lt])~ m ~cZj=o(~.) II/~ co, i f IC~EL2(IR)VO<j~m.
But,
(1 Is) d id s t [.l/s) lim s / ( s )= lira s- l = lira _s_2
lsl--~* lsl--~* Is1-*** Is)--,** = l im~ (I/s)=~" (0)
and/ 'E L 2 (IR), because / is bounded. Moreover, by induction (see also [21]) we N i
obtain /~ with rn>_j+l, hence /t,~, l<_]<m, is in n = t
L 2 (IR), too. C m (SOcW " follows similarly (el. the proof of [21], 4.3.e). The other inclusions are obvious. The density claims are proved in [21] and 1.3.
To obtain similar results for general toe G, we need the following technical lemma:
LEMMA 1.2. a) Let toe G. There are constants CI, C2 >_ 1 and a monotone
insreasing sequence o] positive numbers {sk}/~No with so= 1,k=t2 (ksO- l< ~ , o
and sk<H k /or some H > I, such that with Mp: = p ! II Sk we have: k = t
~ [tl'= i) to( t )<Ct~o Mp :Cttoo(t) and to, EG
ii) to(t)<C2 k=1 ~ ( l + ~ ) = : C 2 o ~ ( t )
b) Let tEC**(IR) and t~>~EE(IR)Vpa, INo, then tCL~ and
ll/II,o~<Cs~, (2H)~ II/o,)llL2 o=o Mr,
i/ the sum converges.
ON THE FUNCTIONAL-CALCULUS OF NON-QUASIANALYTIC, ETC. 63
- ltl" c) There is a sequence {Np}pE Nowith ff~ (t): = ~ r - E G and o~<--C4ff~
p=O z "l p
such that if ff&6'~(IR) with II~,ll,.~C,N~ and f is locally hotomorphic on
{z~ I I~ : l z -~ t<~} /o r some ~>0, then /d/E L~ and
i] the supremum is finite/or some [3> 1/2.
Proo]: a) The existence of such a sequence is proved in [5], 1.8. p. 13. Moreover, i) and ii) obviously follow from [5], 1.8. iii)= >ii) resp. iii)= >iv).
It is easy to compute that to~ fulfils (G.1)(of. [5], p. 11). Since to,(t/2)_<~ 1 k=0
ltl" s u p s , [5], 1.8. i i )= >iii) yields ~ logt~(t /2) d t < hence (G.2). j, ,,-p 1 l + t 2 ~"
g ^ , l , + l , l , + , , ^ b) 11111% =,,= ~, I Itl" If (t)] d t=,=0 ~ ~Me IR f 1 + Itl If (t)l d t <
1 ^ ^ ~2 ~ I1(1 + ]tl)-' IlL= <llltl" t (t)l l~+ IIItl" / Ct)l I~) p=O grip
_<c( liP"IlL'+ sup o pEIN0 Mp(2H) p = Mp+l
--<C sup (p-t-1)s,+l ~ (2H)" llfO,~ll~ ~ pEIN0 (2H) p+I -- p=o Mp
_<C sup p ~, (2H) p
e) By [5], 1.5. iii) there exists a monotone increasing sequence {rk}k~lN ~
>rk > with to=l, ~ , (k r , ) - l< oo, and 1_~_ . . O.
Ddine Np: = p ! ~I rk and ~ ( t ) : - - ~ , . . . . . then t~___~. k=l p=o Np
64 ~ c a M~SCraLL
Let / be holomorphic on an e-neighbourhood of IR N ff, then b. implies:
11~/ll-,---c ~, ~ I l f" l l , ,- < F=O "'Jp
<C :~ (2H)" (ll(l+l=l)-'fll,-, sup (1+1~1) ~ I(ff/)"' (= ) I 'P - < v=o Mv sEIR
<--C~ ~ (2Hy v=o Mp sEIR rl~, j=o
<Cii ~ (2H)" .~up (1+1~1)"1~: (':)N,c~(p-i):(i),-," sup I/(,)l---
v=o Mv s61R n_~ i=o / ~ I,-,I~
_<c, <~ +,), c~ ,up (~ +l=lV I/(=)1 ~: , -~-<2 z-z), (1 +1 ) ,_< lz-sl=, p=o lv~v
~lR nff
s,,p + r l)' It ___ r c , I ~ - I R n ~ J = , =
P r k < sup (4H)" (1 + I ) , 11 - - _
pEINQ ~ k=O sk
<_c~,,c, sup (1 + Izl) ~ It (z)l. I z - IR N d~/=~
V
If f ie M E, then ~ fulfils 1.2. c. In fact, let ~=lx, then
" Itl" . Itl' db l (t)<__g, f ~ 0 - - ~ - a bl (t)=NJll~ll = tW"IIL'-< IR f aR
For to=o~, we can define 6 : = ( 1 +lt[) m. But, even for to(t)mI~ we cannot
allow [3< 1/2 as the remarks below 2.4. will show.
COROLLARY 1.3. Let oJEG, then H(IR)~-->P and H(SI)c--~W".
Proof: H ( I ~ ) c P follows with 1.2. e. and d~----1 and 13=1.
We have to prove that H:={to / : fEH(IR)} is dense in L~(IR).
Let 9EL'*(IR) with fto/~"-=0 VfEti(ll~). Now, ~'e-"dUdt= 0
ON THE FUNCTIONAL-CALCULUS OF NON-OUASIANALYTIC, ETC. 65
= ( z - i s ) - I E H ( I R ) VRez>O, implies f t o ( t ) e -Z tg ( t )d t=O VRez>O, and 0
the uniqueness of the Laplace-tranfsorm yields to (t) (p (t) = O, hence (p(O=O,
for almost all t>O. Similarly we obtain (p(t)=O for almost all t<O and H
is dense in L ~ (JR) by the Hahn-Banach-theorem.
Let / (s) = ~ an s" E W ~. The partial sums are in H (SO and converge nE~7
to t in W ~'. Let / ( s ) = ~ a,s~EH(S~), then ]a~l<Crl~l for every nE 7/ and nET/
some 0 < r < l . Now:
(*) 1 < l i m ~/to (n) _<exp (lira 1--- logto~ (n)) =
. ~ (n) ~_ 1 = exp ( hm ------7-~, ) = exp ( lim
to~nj ,~.~=~ ks~+n )=1.
Hence,
[an[ to (n) < oo and f E W ~'. nEZ
W ~ is just the Belling algebra treated in [6], 5.2. p. 139, because (1) of
[6], p. 140 equals (G. 1) and (2) follows with (*) from (G.2) Therefore, W ~'
is topologically admissible ([6], Def. 3.5.1. p. 90) by [6], 5.2.7. and 5.2.12,
hence, W ~ is admissible in the sence of [21], Def. 2.2. Moreover, W ~ is inverse
dosed, because, by a theorem of Banach algebras for fEW ~' with f ( s ) ~ 0
V s E $1, 1/rE W ~', if there are no multiplicative linear forms besides 8s (/'): = f (s),
sES:. But this is just [6], 5.2.2. p. 141. Now we prove analogous results for i ~
THEOREM 1.4. Let to'E G. M ~ and F ~ are admissible and F ~ is inversed
closed ( see also [19], 1.1.4. and 1.1.5.).
Proo]: This follows from 1.1. if to=tin. Using [5], 2.5. or [19], 1.1.1.
it is easy to construct decompositions of unity in F ~' for every open covering of II~.
Let / E F ~ and lxE ~ \ [ and t5 as in 1.2. c. Let ~ E ~ with tp--1 near / and
l x ~ . Now 1.2. c yields ravEL t ' hence, f ~ = f ~ E I ~ and F ~' is admi~ible.
Therefore, the larger algebra M ~ is admissible, too. Again we have to show that
6 6 ERICH MARSOHALL
^
8, (]): = ] (s), s'E IR, are the only complex homomorphisms of F% Now [9], 1.41.
yields that h0: = h lLt~ vanishes or ho= 8, for some s'E I R, because [9] , 2.11. proves
that F ~ is an algebra of type F ([9], Def. p. 5). In fact, using (G. 1) and the
local boundedncss of to it is easy to show that (G.2) is equivalent to
~n-21ogto(n t )<oo Vt'EIR which is assumed in [9], 2.11. (cf. also [4], n=l
p. 823.).
An A-calculus * r for T is called regular, if SE ~ (X) commutes with
e r ( t ) k / tEA, whenever S commutes with T ([6], Def. 4.17.).
COROLLARY 1.5. The continuous F ~ - , M ~ - , and W~-calculi are unique and
regular.
Proo]: Since the H ( I R ) - resp. H (Sl)-calculus is unique and regular, this
is also true for the F ~ - resp. W~-calculus by 1.3. If ]EM ~, then [ i ( s ) -
= (i--s) -1 ] (s) E F ~ and ~)r (f) = (i-- T) a)0 qi) with r the unique and regular
F~-calculus for T. Hence ~ r is unique and regular, too.
Therefore, we write J (T) instead of * r (3 for the continuous P ' - - , M ~ ' - ,
and W~ The following lemma is essential for these calculi:
LEMMA 1.6. Let TE [F ~]
a) 3{kb,,}nElNcF~ with k b ~ c [ - n , n ] and l i m ~ ( T ) x = x V x E ~ ( T )
b) x~ (C) =x;iiR-~=~-~75 c) o~ is not an isolated point of o'r (x) for any x'E X.
Proo]: a) Let t5 be as in 1.2. and ff E F~'with d?-- 1 near zero and ~bc [ - 1, 1].
Let t~,, (s): =~b (s/n). If x = R (T, i) yE ~ (T), then
~, (T) x = ((i-- .)-~ ~,~) (T) y ~-:-> R (7, i) y=x ,
because 1.2. c with (3=3/4 implies:
II(i-.)-* ~ - ( i - . ) - , II = ~ c I1~- ~11: sup { (1 + [zl) 3/'
li-~l : I~I>'T -~}~--~~
ON THE FUNCTIONAL-CALCULUS OF NON-QUASIANALYTIC, ETC. 67
N o w ,
A ^ A
tl,l,.lt z =lt~'4'-II"=lln<~'l'cn')tl~'=ll<~c" In)4'11~'< ll~>4,ll,,=li4,lt ~,
since ~(t)-- t~(l t l) is increasing. Therefore, {dh,(T):nE,lN}c-I (X) is bounded
and {d/,,(T)x} converges VxE~)(T)" #%
b) X r ( C ) = X r ( I R ) , since o ' (T)cIR. r by [21],
2.6. b and the assertion follows from a).
c) If o ' r (x)=KU{<~} with K c c C , let ~'EI :~ with r near K and
q)c C. Now x,: = x - - ~ ( T ) x ~ O , x, EXT ({ ~ })=:Xo, and To: =TIXoE [P ] with
#(To)=J(T)IXo V f E F ~'. Moreover, o'(To)={<~}, hence dg,(To)x=0 VxEX0, #%
arid a. implies ~) (To)={0} that means o'(To)= C which is impossible.
THEOREM 1.7. Let to'EG, a) For T E 61 (X) the following statements are
equivalent:
i) T IF ~'] and D ( T ) = X
if) TE [M ~ and (D (T)=X
iii) T is the infinitesimal generator o/ a strongly continuous group of
operators { U (t): t E IR } with I lu (t) ll < c ~, (t) v t~ JR.
b) SE[I/I/~], ill S E ~ ( X ) , 0r and Ils:ll~,cto(n) Vn,~Zz.
Proo]: ,,(i) ~ (if)" Let f E M <~ and { 3, } c F ~ as in 1.6. ~,~ f'E F ~' and
{ (d / , , f ) (T ) :nEIN})c~ (X) is bounded. Since (~b,,f)(T~x=(dh, oJ)(T)x
VxEXr(IR) and n>_no(x), and XT(IR) is dense, we have:
Dr(f): = f (T) : =s--lim(~b./) (T)E ~ (X) and llf(T)ll~ sup fl(~b./)(T)II~cll/II~ nEIN
Therefore, ~ r is the continuous M~-calculus for T.
,,(if) ~ (iii)" is proved similarly to [21], 4.13. In fact, U(t): =exp( i tT ) ,
tEIR, is a group of operators with Itu(t)ll_<ll~{I Ilgll~_<co, ct). Since l,,,h (z): = h - ' c,,+h~, "~ d,~ (e ~ --e ~ )h~--~-~iz V w ~ C uniformly for z E K c c I E . Therefore, g/~f~,~h~--~--~i~di, eiW'iEM ~ by 1.2. I~ xEXr( IR) , then x ( w ) : =
68 E R I C H M A R S C H A L L
=(~,,e i~') (T)x with ~,---1 near O'r(X) is entire and an ext~ntion of U(.)x. Hence x( . ) is continuous on IR and x'(t)=(izdb, e itz) (T)x=iTU( t )x . Since Xr(IR) is dense and U( . ) is locally bounded, U( . ) is strongly continuous, iT commutes with U (.), hence with its generator A. Moreover, we have i T x=A x on Xr (IR) which is dense, and [21], 1.3. yields i T=A.
v
,,(iii) ~ (ii)" If f= IxEM ~ and xEX, let ~r(DX: = f U(t)xdtx(t). ~r IR
is a continuous homomorphism with I[~rl[<__C and ~ r (1)=I .
q~r( ( i z - - . ) - l )x=i fe~ 'U( t )xd t=R(T, iz) VRe z<O by [14], Th. 11.5.2, 0
and ~ r is the M~-caleulus for T (cf. [21], 2.4.).
,,(ii) ~ (i)" is trivial and b) follows with [6], 5.2.-5.3. because condition
(2) of [6], p. 140 can be obtained from the Beurling's condition (B) ([6], p. 1495
as equation (*) in the proof of 1.3. shows.
Remarks 1.8. a) Let { U (t): t E IR } be a strongly continuous group of opera-
tors with (*) f (1 + tz) -1 log llU (t)ll d t < oo, and let iT its infinitesimal generator, IR
then to(t):=max(1,llU(tSll)EG and TE[M~*]. If t0EIR, then Ilu(t05nll - -
[[U(nto)[[<<-to(nto)=to,.,o(n). Hence U(to)E[W~'t0] and t(U(to))=q*d'o.)(13
V/EW'~ Groups of operators which satisfy (*) are called non-quasianalytic
([4], [19]).
b) U( . ) is called temperate, if IIU(t)l l=O (It[ ~) for some ~>__0.
In this case to and to~,t 0 are equivalent and TE [C" (IR)], m = [ ~ + 3 / 2 ] , A
by 1.1. and 1.7. Moreover, f o ( T ) x - - f t o ( t ) U ( t ) x d t VtoEC"(IR) with IR
,'3
t0(oo)=0 and to the L:-Fourier-transform of t0. This strengthens [21], 4.5.
and proves the formula used in [12], 2.4.
c) 1.1. implies: TE[B"(IR)]=>TE[F"]'->TE[G"+:(IR)]. If T is
bounded, hence U( . ) is uniformly continuous, then the M~-calculus is not
larger than the F~-calculus (see [3] and [16]) and [21], 3.6. yields A
TE [C "+1 (IR)] = > T E [8 "+2 (IR)], hence T is generalized scalar, if and only ff
IIu(tSll--o(Itl ~) for some r as in the unitary case 1.1. b (of. [6], 5.15. If T
is unbounded, this is not true as the simple translation-group shows:
ON THE FUNCTIONAL-CALCULUS OF NON"QUASIANALYTIC, ETC. 6 ~
^
Let X: =LP (IR), l < p < oo, X: =C~ (IR), let ~)(T): ={xEX:x is absolutely
continuous and x 'EX} and Tx: =--ix/, x E ~ ( T ) , then iT is the infinitesimal
generator of the isometric translation group U (t) x (s): = x (t+s), tE IR, xEX, A
hence TE [M~ Now Tr [B~*(IR)] if p # 2 by [12], 2.4. Moreover, T3E [CI(IR)]
(ef. [21], 3.8., 4.2.), but iT 3 does not generate any semi-group if X=C~
(el. [10], Ex. 21 p. 656). Therefore, TE [C 1 (IR)] # > T E IF ~] for any t~EG.
d) Let TiE[M~I], ]E{1,2}, be dense',y defined and commuting, then
i T~+T2 generates the strongly continuous group H(t): =U~(t)U2(t) (see e.g.
[12], 1.1.), hence Ta+Tz'E [M~'~,]. But the above example shows that even T 2
is not of class F '~ for any ~ E G. Moreover, in general, sums and products of
densely defined operators of class C 1 (IR) are not even decomposable. In fact,
let X: =Lt( IR) , T, and U( . ) as above. There is a measure i.tEM ~ such that v
the convolution operator F ~ = I x ( T ) is not decomposable (el. [11]). We
can assume that Ix has compact support, hence ~'EB** (IR) and v
( i - . ) - t i xECl ( IR) . Therefore, TI:=i--T and T2:=R(T, OF~ are of class
C a (1~), densely defined, and commuting. But T~ T2 = T1 T2 = F~ is not decompo-
sable, and since 2 T a T 2 = ( ~ ~ - - ~ - - - - - - ~ 2 , sums are not decomposable, too.
Together with the above remark it is therefore very useful that p (T)E [C" (C)]
if TE [C" (IR)] for every polynomial p (cf. [21], 3.8.).
e) 1.6. yields condition ~-~ of [22] for V = ~ and the F~-calculus can be
extended to the algebra of C-local multipliers which contains M ~. If U (-) is
temperate, then C~(IR)cI:~(C), rn=[~z+3/2] (this sharpens [25] where
~z=O is treated). Moreover, in this case [22], 2.4. is applicable and the calculus
can be extended again. If U (.) is temperate, we can prove some spectral properties:
THEOREM 1.9. Let TE [F'], ~z_>O, then
Xr ({ix}) =Ker (I.L-- T)E~+~ = Ker (I~-- T) t~+i V~E C,/EIN.
Proot: It is well-known that Ker(~,--T)kcXr({~}) VkE[N (of. [21],
2.1. f. and b). Lot xEXT({t~})C~) (T~). We have to show that y: = ~ - - T ) ' x
7 0 ERICH MARSCHALL
vanishes, r: = [~] + 1. Since t r (T )c IR and the translation with t E IR is iso-
metric in i =~, we may assume that Ix= 0.
Let ~,EC~(IR) with ca--1 near zero and <pcIR. Let q~,,(s): = ~ (ns), nEIN.
9~(T)y=y , because o ' r (y)co ' r (x)c{0}. But q ~ ( T ) y = ( ( - - s ) ' t p , , ) ( T ) x ~ O , because
l[ s~ q~[[== II (1 q-ltt)~(@(n .) (i.)~) ̂ (t)llz~= 1 II(1-~-Itl)~-~-~-~(t/n)tlz,=
A A
THEOREM 1.10. Let TE [F ~ or T'E [l/g ~ then Ker (IX--T) N ~g (Ix--T)={0}.
In particular, the range of tx--T is not dense, if Ix is an eigenvalue of T.
Proof: Let TE [F~ We can assume that i~=0. Let {q~"}nEIN cC**(IR) as
in 1.9., then {~p~ (T): n E IN } c ~B (X) is bounded because
A A
I1 :110= II IIL, = II 1 I n , ( . /n) l l , , = I1 110.
Moreover, s g,,---0 in F ~ by 1.2. or the proof of 1.9. If X = T y E ~g (T),
then ~p,(T)~c=(sq~n)(T)y~0. Hence, l img , , (T )x=0 V~'E~R(T). But if n.-~oo
x E Ker (T) we have r (T) x = x for every n E IN and the assertion follows. 1 ~t--I
Let TE[W~ We can assume that IX=I. Let tp~(z) :=--Y~z k, then n k=o
{tt~ (T): n E IN }c ,~ (X) is bounded because II , li0= 1. Moreover,
(1 - - z )~ ,~ ( z )= l (1--z n) --, 0 in W ~ n
If x=( I - -T )yE~R( I - -T ) , then c p , ( T ) x = ( ( 1 - z ) c p , ) ( T ) y ~ O . Hence,
lim cp,~ (T) x = 0 V x E �9 (I-- T).
1 n--1 But if T x = x , we have ~ ( T ) x = - - ~ T k x = x and the assertion follows.
n k=o
Remarks 1.11. a) The proof of 1.9, is similar to the prooof of [22], 2.2.,
and if ~ = 0 , then 1.9. was proved in [8] (el. [8], 8.16. and p. 219). 1.9. holds
ON THE FUNCTIONAL-CALCULUS OF NON -QUASIANALYTIC, ETC. 71
for TE [W~], too, but we omit the proof which is more complicated. (Note that
the stronger result, asserted in [17], Th. 2 and its corollary is false if q=~z_>l.
In fact, ~ ( T ) : { 1 } and 117 11=o (Inl) donot imply T=I.) Moreover, 1.9. $
can not be generalized to every to E G as the Voltera operator Vx (s): = f x (t)dr, 0
x E L p (0, 1), shows. V is injective, but Xv ({ 0 }) = L p (0, 1) because tr (V) = { 0 }.
Since ][ei'~l]=~2 Itl~lJV~[l __<~2 Itl~=:to(t)~n, we have VE [F~]. e=o P f p=o ( p f)2
A b) 1.10. holds for TE [C~ , too, but it does not hold for T;E [ P ] , if
g > 0 . In fact, let l > g > O and p),g-1. Let X: ={f : Ilfll: =[Ifl+ltl)- fll < oo },
u ( . ) the translation group, and iT its generator, then [tu(t)[l<_(l+Itlr, the
constant functions are in Ker T, and the range of T is dense, because the adjoint
operator T" is injective.
c) Usin~ the notation of the << state >> of an operator (of. [13] and [22]),
1.10. means that T~ is not in the state 113. Hence, o'p (T)={IxE C:T~EIII3} and
the following theorem 2.1. is even a mapping theorem for the states, if tz=0.
Moreover, the adjoint operator T',E IF~ resp. T'E [W~ too, and the state
diagram [13], p. 66 yields that T~ is not in the state II12, hence o ' r (T)=O, if
~ = 0 and X is reflexive. For TE [VIP] this was similarly proved by E. Lorch
(ef. [18], Th. 4).
2. Spectral mapping theorems and decompositions.
Since F ~ is inverse closed we have tx ( [ (T) )=f (o ' (T) )=f (o ' (T)NC) for
every t ~ F~. The measure in 1.8. d shows that this equation does not hold for v
every IxEM ~ (el. also [11] and [24]). But in [22] we proved that
f(o'(T)N C ) c t r ( f { T } ) for every local multiplier and that equality holds, ff
f{T} is decomposable. Now we are espeeiaUy interested in U(t)=expit{T},
t E IR. Since there is a strongly continuous group of operators such that the
spectrum of its generator consists only of { oo } (of. [14], p. 655), a spectral
72 ERICH MARSCHALL
mapping theorem does not hold for general groups. But if U (.) is non-quasia-
nalytic, then U(t) is decomposable VtE IR by 1.8. a and [21], 2.6., and [22],
1.3, together with [14], 16.7, yields the following complete fine structure
spectral mapping theorem which generalizes the theorem for the translation groups
on LP(IR) or on U'(0,1), l_.<p<oo, for which it can be computed directly.
THEOREM 2.1. Let i T be the infinitesimal generator of a strongly continuous group of operators { U (t): t E IR } with f (1 +F) -a log [[U (t)ll ct t < oo, then:
a) r ( t))=exp(it(~(T)N ~ ) ~ S l VtE IR
b) crv.)(x)=exp(it(~rr(x)N C)) VtEIR, x E X
c) ~p(U(t))=exp(it~p(T)) V t ~ O
d) Ker(k- -U( t ) )=LH{Kerf~--T):exp( i t~)=) ,} V k E o'p (U (t))
e) ~,(U(t))cexp(itcr,(T))c~,(U(t))Utrv(U(t)) VIEIR
f) tr (T) =erp (T) U { oo }___ > 0" (V (t)) = 0-p (U (t)) = 0"~ (U (t)) U o'c (U (t)) k/t E IR
and o 'p(T)=O r for one hence all t ~ 0
g) tT(T)-~.O'r(T)U{ oo } :::~ r VtEIR
and t r , (T)= ~3 ~ o',(U(t))=O VtE IR
o'(T)=o'c(T) r , ( U (t))=crc(U (t)) for one hence all t#O. h)
Proof: a) and b) follow from [22], 1.3., since U (t) is decomposable and
V A = A ~ C U(t) IXr (A)=exp ( i tT I Xr (A)) is decomposable, too. e), d), and
e) are well-known (of. [14], 16.7.2.- 16.7.5.) and f), g), and h) are consequences
of a), c, and e). But there is also a simple proof of c), d), and e) using the
M~'-calculus. In fact, let 0 ~ t E I R , XESI, and e%=)~,j'E~_, then hi(s):= X - d "
�9 -- EM ~' by 1.2. and )~--U(t)=(lx/--T)hl(T)Dhj(T)(I~--T) Hence, l~j-- s
Kcr (t~i--T)cKerO~--U(t)), exp( i t~(T))c~p(U(t ) ) , and exp ( i t o ' , (T ) ) c
Co'r (U (t)) U o" v (U (t)). Let 9J E M ~ with 9J---- I near [p . / -= /2 t, t~-1-=/2 t] and
ON T H E F U N C T I O N A L - C A L C U L U S O F N O N - Q U A S I A N A L Y T I C p E T C .
~iC(ixi--'l~/t, Ixi"]-~/t), then gi (s): =~1 (s) (lxj--s) (X--ei") -1E M ~ and
(ixi--T) qJi (T)=gt (T) (~,--U (t)).
73
Hence, if y ~E Ker (~,-- U (t)), then ~i (T) y E Ker (Ix/- T). Let { ~, } c M ~ as in
1.6. Now b) implies oT(y)c{IXt:/E 7]}U{o*}. Hence
n 0
~ ( T ) y = ~ ~, (T) ~pi (T) yE L H ( Ker (IXi-- T): j E 7/} / = - n o
and 1.6. yields d). Moreover, o ' j ,(U(t))cexp (i top(T)), because not all ~pi(T)y
vanish, if y~O. If ~,Eor(U(t)), then there is O~x'EX" with U(t)'x'=~,x'.
9j (T') x ' = 0 V i E 27 yields Or, (x') = { oo } which is impossible by 1.6. (note
that T 'E[M~], too). Hence x 0 : = g i 0 ( T ' ) x ' ~ 0 for some i0E7] and
g, j0Eop(T')co 'p(T)Uo,(T) . But lxi0r op(T), because of c).
Remarks 2.2. a) If the group is bounded, then 2.1. a. is contained in [7].
Moreover, [21], 2.6. yields o ( T ) = s u p p ~ r (.) which is often called the Spectral
Theorem (el. [8], 8.19).
b) By definition in [22] **Eoc(T), since ~ ( T ) = X , and 2.1. shows that
this definition is useful.
c) The dual group U(-) ' is not strongly continuous in general, but 2.1.
applies to the group U'(t): = U ( t ) ' l X ~ with X| = ~ (T')cX" and generator
iT '=iT'IX" (el. [14], 14.4. and [23]). Moreover, 2.1. a., b., c. hold for U (.) ' , too.
In view of [22], 1.3. it is usefu~ to know that f{T} is decomposable. For
instance, the trivial spectral mapping theorem for the local spectrum shows that a closed operator F is decomposable, iff F -1 is decomposable, 0r
Hence /{T} is decomposable, if 1 / (1x-f )EF ~' for some IXE IE. The following theorem, which generalizes [11], Cor. 10 and [7] in the
case of groups of operators, is only proved for temperate groups, but it holds
for any to E G, too, if we use a suitable nuclear algebra of ultra-differentiable
functions insteact of C*~(~) in [6], 4.3.3. Let I~EM ~, then tx decomposes into
the continuous and into the discret part of IX. The continuous part decomposes
74 E R I C H M A R S C H A L L
into the singular part and into a part absolutely continuous with respect to the
Lebesque measure.
THEOREM 2.3. Let T'E [Ms], ~z_>_0, and ['EM ~. if the singular part of A
tx=f vanishes, hence lx=Ixa+Ixd with lxaELZ and txa discrete, then f(T) is strongly-decomposable.
Proo/: Since C" (IR) is dense in F ~ by 1.1., there is a sequence { f , } c C m (11~)
and a sequence of measures {/.t~} with finite support IJ~=~akSt~ such that k = l
M" v
fn+l-~-- ' f in M s. Now 8tk(T)=U(tk)(~[~l~], hence br(tk)E[em(Sl)] and therefore generalized scalar. Moreover, f~ (T) is generalized scalar by [21], and
Xe
[6], 4.3.3. yields that ff~+V-~)(T) is generalized scalar, hence decomposable. ' v
Now [1], 2.8. implies that f (T)= l im (f~+v~) (T) is decomposable, too. If Y c X
is spectral maximal for [(T), it is invariant for g(T) VgEM ~, and O0(g): =
=g(T)IY is the unique MS-calculus for To: =a~0{s}. The first part of the
proof shows that f(T)lY=f(T0) is decomposable, too.
Now T,E [ P ] is decomposabe itself, but if T is unbounded, then oo Eo'(T)
and one of the covering open sets must contain oo, hence the complement of
a compact interval. In [21] we proved that T is <<o'-decomposable at oo>), if
T'E [B ~ (IR)]. If T'E [F ~'] this is not true in general, but a slightly weaker form holds, which is nevertheless stronger than the decomposition property used in
[19] wich corresponds to (i) and (ii) of our theorem.
THEOREM 2.4. Let TE [F ~] and densely defined. Let 8 > 0 and {U~.: =
=(ai, bi):i'E 7] } be an open covering of IR with ai+2>__bi>__ai+t+28 Vi'E 27, then there are spectral maximal subspaces Yi for T such that:
(i) Yic | and o'(TIYi)cUiNo'(T) VIE27
(ii) X = ~ Yi iET/
(iii) If xE ~ (T), then there are yj~. Yi, j,E7/, such that ~, Yi converges
unconditionally to x.
Proof: Let t5 as in 1.2. and 9'EF ~" with q~ml near zero and ~cU~(O). m
Now llq) )ll =_<N,, fl, IT: the remarks below 1.2.).
ON THE FUNCTIONAL-CALCULUS OF NON-QUASIANALYTIC, ETC. 7 5
Let ci: =(ai+bi+~)/2 and q~i: IR --- IR; q~i(s): = ~ ( s - c j ) on [ci-~,ci];
q)/~l on [ci, c i+l-~ ], 9 i ( s ) : = l - g ~ ( s - c i + l ) on [ei+l-~,C)+l]; and zero else-
where, q)i'E F ~ by 1.2. b. and ~] q)i(s)--I on IR. Let Yi: = X r (q)i). Yi is spectral tET/
maximal for T and we have (i) (cf. [21], 2.6. and 2.1.). (ii) follows from (iii).
If x 'E~ (T), then x = R (T, i)z for some z E X . Therefore, we have to show
that for any z E X ~ ( .q~i ) (T) z ---, ~ q)i (T) R (T, i) z = : ~ Yi converges for io, l - . ?E ?7 iE ?7
the net of finite subsets ]0 of 7/. This is true for z 'EXr (IR), because 0"r ( z )c c IR
is covered by Uit ..... Ui,. Since Xr(IR) is dense, it remains to show that
,p lL_<c for all finite subsets ]0c77. But this follows from 1.2. io0
c, because
i%
Remark 2.5. Let us suppose that U (2 ~) = I , hence U (.) is periodic.
Therefore, U (.) is bounded and T'E [M~ Moreover, 2.1. a yields ~ (T)c 7/U { ~, }
and [22], 2.2. yields ~r (T) N 77 =~rp (T). Moreover, Xr ({j})=Kern ( j - - T ) V j ~ 7/
by 1.10. Let Pi be the spectral projection onto the spectral set {j}, then
Pi (X) = Xr ({ j }) = Yi, if { Uj: j ~ ?7 } is suitable. Now 2.4. proves that ]~ Pi x---, x fEZ
unconditionally on @ (T), even on ~) ( ( - i - . ) ~ {T}), ~ > 1/2, as the proof
of 2.4. shows. If X: = ~ ( I R ) , the periodic continuous functions on IR, and
U ( t ) x ( s ) : = x ( t + s ) , t, sEIR, then ]~ Pi x ( s )= ~ yi (s )= ]~ aie iis is just the iE 7/ iE ?7 ~E ?7
Fourier series of x and converges unconditionally and uniformly, hence abso-
lutely (s=0), to x, if x E ( - i - . ) - a ( T ) ( X ) , ~3> 1/2. This is just the well known
theorem of Bernstein (of. [26], 3.1. p. 240). In fact, if x'EAr: ={x(~C~ (IR):
i-~ ? t -~-1 (e-t x (t + [x(s)-x(t)l<__C Is - t ] '} and O < [ 3 < y < l , then y(s): =1"(-[~)
+ s ) - x ( s ) ) d t E ~ ( I R ) and ( - i - - T ) - ~ y = x E f ~ ( ( - - i - - T ) ~ ) , became y ( s ) =
= ( - - i - - T ) ( - - i - - T ) ~ - l x ( s ) = - - i ( 1 - cl ) i1-~ F t ~ - Z e - t x ( t + s ) d t ) ds (" r(1-B) 0
which can easily be computed.
76 ERICH MARSCHALL
Moreover, the example [26], 3.1. shows that we can not allow 13<1/2
in 1.2. and that in 2.4. (iii) the sum dosn't converge for all x E X.
3. Cosine operator functions.
Cosine functions were treated in the last years by a lot of authors (ef. [20],
[23], and the papers cited there). A cosine operator function {C(t):tE'IR} is
a strongly continuous operator valued solution of the functional equation:
C ( t + s ) + C ( t - - s ) = 2 C ( t ) C ( s ) ; C (0)=I , t, s E IR.
T: =C"(O) is called the infinitesimal generator and is a closed and
densely defined operator. We can define admissible algebras and functional
calculi, similarly as in the first part and obtain together with [22] a spectral-
mapping theorem for non-quasianalytie cosine functions.
DEFINITION 3.1. Let t0•G be even and let
9.) N~: = { l : ( - oo,O] ~ C:g(s):=/(-sZ)~EM~},
10.) G ~ : - - { J : [ - o o , 0 ] --" G :g ( s ) := t ( - - sZ) 'EF ~} with the norms
Ilil[~ = ]lgll~, and
l l . ) C ~ ' : = { ] : [ - - 1 , 1 ] - - - G : g ( s ) : = f ( c o s s ) E M ~'}
={f : [ - 1 , 1] ---, C:h(z) : = f ( ( z + z - l ) / 2 ) ' E W ~} with the norm
II/11 = Ilgtl.= Ilhll .
THEOREM 3.2. N%G ~, and C • are admissible Banach algebras and
Ca w and C ~' are inverse dosed. The continuous calculi are unique and regular.
Proof: K: N ~ --- { g 'E M •: g even }, K J (s): = J ( - s z) is a homomorphism which
is bijective, isometric, and has dosed range. Consequently N ~' and G ~' are Banaeh-
algebras which contain the constants. Let {UI ..... U,~} be an open covering of
[--~o,0] and let {q~fil<_]_<n} c F ~ be a docomposition of the unity for
ON THE FUNCTIONAL-CALCULUS OF NON-OUASIANALYTIC, ETC. 77
Vi:={-+-vr'Zv:O>__vEUi}, l__<]_<n, which cover I[~. Let X i ( v ) : = ( t p i ( ~ +
+~Pi (-- Xl-S-v))/2, 1 < ] < n, and v E [ - ~ , 0], then Xi E G~',Xi c Ui, and ~ Xi (v) - 1 i=l
on [--~0,0] . Hence G ~' is normal. Let lEG% If ~ #lx~f , then ___V--------~r (])
and g (s): = K f (s) (--~/--tx--s) -1 (~/--ix--s)-l= K t (s) E I:% But g=K (f~), hence V,+s ~
f~EG ~'. If 0or then --s2Kf(s)=K(f~)EF% Hence f**EG ~ and N ~ are admis-
A
sible. Let l E G ~ with ] (v )#O V v ~ [ - ,o ,0] , then Kf(s)#O' VsE IR, and 1.3.
4 yields ~ , K ~ ( S ) = K ( ) (s)EI ~. Hence I - E G ~ and G ~' is inverse closed.
K(l t ( [ -o , ,O]) )={hEl t ( IR) :h is even} is dense in K(G~). Therefore, the
continuos G~ N~-calculi are unique and regular. The proof for C ~ is
similar.
LEMMA 3.3. Let TE [{3 ~]
a) 3{X,,}rtEi N c G ~' with X__~c[--n2,0] and lim X~(T)x=x
b) Xr ( ~ ) = X r ((-- ~ , 0 ] ) =@ (T)
c) oo is not an isolated point of ~r (x) /or any x E X.
VxE@ (73
Proof. Let { ~ } c I :~ as in 1.6. and ~(v):=(hbn(q ' -~+~, , ( -V"Z~)) /2 , then x,,EG~,X,.c[--n2,0] and { ~ ( T ) : n E I N } c ~ 8 ( X ) is bounded.
II R (T, 1) (1--X~ (73)11-<c [[(1-v)-, (1-x~ (v))ll._<ctl(l+~ ( 1 - ~ n ) t l . ~ 0 and the assertions follows a~ in 1.6.
THEOREM 3.4. Let t~E G be even. a) For TE6[ (X) the following statements are equivalent:
(i) TE[G"] and @(T)=X
(ii) T,E [N ~] and @ (T)=X
(iii) T is the infinitesimal generator of a strongly continuous cosine fun-
ction {C(t):t~.lR} with IIc(Oll---c~(t) VtEIR.
b) S<[e'], ill SE~B(X)and IIs.ll<C~(n) V n ~ o with So:=l 81: =S, and S~: =2S8~_1-$._2 Vn>2.
78 ERICH MARSCHALL
v
Proof: a) ,,(iiii) ~ (ii)" Let I'EN ~, hence K(I)=IXE M ~'. Let ~ r ( / ) x : =
= f C (t) x d Ix (t) = 2 J'~c (t) x d Ix (t), x E X. Or is well-defined, linear, and conti- IR 0
V V
nuons with Iler[I---C. Let l,gEN ~ and K (/')=IX, K(g)='~. Since v is even, too,
we have:
Or (1") #Or (g) x - - .f C,(t) f C ('~) x d v ('~) d Ix (t) = f f C (t) C ('c) x d v ('r d Ix (t) = IR IR IR IR
= } ( J" .f (C(t+.~)xd,v(-c)dix(t)+ f [ C(t--r)xd vC--Od~(t))= IR IR IR IR
= f C ( t ) x d i x * ' ~ ( t ) = O r ( l g ) x . IR
Therefore, Or is a homomorprism, t (v): =(1 - v ) - l E G ~', K (t) ̂ (t)=e-tt!,
and t ( T ) x = ~ e - ' C ( t ) x d t = R ( T , 1)x by [20], equation, (0.8) and we con- 0
elude that Or is the N"-calculus for T. The other implications are proved as
in 1.7. and will not be used below.
b) ,,=:>" Let ~ ( v ) : = l , p l ( v ) : = v , and p , , ( v ) :=2vp , , -1 (v ) - -p , - z ( v )
V n > 2 . p,~.C, ~, p , ( S ) = S , , g , ( s )=p , ( cos s ) = eosns, and Ils.ll<_Ctlp.ll. =
VnE INo.
, , ~ " If f e e ~, then g ( s ) : = f ( c o s s ) E M ~ and g=~,a,(~",+6"_,)/2. r im0
Let Os (t): = ~ a ,S , . Os is well-defined and continuous, because n = 0
co I:Q
The definition of {S, :nE IN } yields that Os is homomorphic on the algebra
generated by {p,,:nEiN}. Since this algebra is dense, Os is a homomorphism
on C" with O s ( 1 ) = l and Os(S)=S.
Remarks 3.5. a) Let { C(t):t EIR } be a strongly continuous cosine function with
+t-')-'log lie (t)ll a t < o. and T its then (t): (1, l lc (t)ll)'Ec generator, OJ ~ m a x
and TE[N~]; in particular o ' ( T ) c [ - o o , O ] . For t0EIR let S:=C(to) , Hence
ON THE FUNCTIONAL-CALCULUS OF NON-OUASIANALYTIC, ETC. 79
S,=C (n to) and lls.ii ~<to in to)=~,,0(n) Vn En~ and 3.4. b. yields C (to) E [C'"to], in particular o ' ( C ( t ) ) c [ - 1 , 1 ] V tE IR.
b) If C( . ) is temperate, then 13"([-~, ,O])~-->G" and 1 3 " ( [ - 1 , 1 ] )
r Hence T and C(t),tE IR, are generalized scalar in this
case. This can also be proved directly. In fact, let IlC(t)ll=O((l+ltl)') and
i x = k z E C \ l R , Re k>O, then using R(T,~)x=~e-X'C(t)xdt w e obtain
(1 + [~12), ^ ^ IIR(T,~tI=o ( t t rn~ t=+ I .), r ~ l R . Hence TE [Ct=a+2(IR)] by [21], 4.3. (note
that IIR (T,~)II=o (IZm~l-"),~--" IR, is not true in general). I1R <c(t),~)II earl be estimated similarly.
c) One can show that 1.9., 1.19., 2 .3 , and 2.4. hold for cosine functions,
too. Moreover, we have Xr ({O})=KerT t*aj+I for Ix=0 and the sum in 2.4.
converges for xEf9 ((1 --T)~),I3> 1/4.
d) Similar to the p ro~ of 1.7. (ii) ~ (iii) we obtain that if TE [M ~']
and ~ (T) =X, then -- T z generates the cosine function C (t): = (U (t) + U (-- 0) /2.
Hence, --T2E IN ~-] with ~ (t): =max (to (t),to (--t)). But not every non-quasianalytic cosine function is the real part of a
group of operators.
The spectral-mapping theorems [14], 16.7. for semi-groups are proved
for cosine functions by B. Nagy in [23]. Similar to 2.1. we now obtain the
following complete result:
THEOREM 3.6. Let T be the infinitesimal generator of a strongly continuous cosine operator function { C (t): tE:IR } with
f (1 +t~) -' log IIc (t)]l dt< oo, IR
then:
a) o'(C(O)=cos( t~/-r C) VtE IR
b) r (x)=cos (t x/-crr (x) f3 ~)
0 ,p(C(t))=cos(t~--------*p(T))
VtE IR, xEX
Vt#O
8 0 ERICH MARSCI-IALL
d) Ker (k - -C ( t ) )=L H {Ker (Ix--T): cos (t xT-~)=~,} V~,~o'p (C (t))
e) o', (C ( t ) )ccos ( t ~ ) c C r r (C (t)) U o'p (C (t)) V tE IR
f) o '(T)=~rp(T)U{oo} ~ r162162 VrEIR
and o'~, (T) = O r162 tr,, (C (t)) = O for one hence all t ~ 0
g) ~r(T)=o'r(T)U{oo}~o'(C(t))=r162 VrEIR
and o', (T) = O =~ r (C (t)) = O V t E IR
h) o ' (T)=r r tr(C (t))=~,(C (t)) ]or one hence all t~O.
Prool: Let ct(v): =cos( tx / - -v)=~ v"t~ N ~' ,=o (2 n) l ; then c, ~ and c, (T) = C (t).
Now the theorem can be proved as 2.1.
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