RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie H, Tomo XXXV (1986), pp. 317-329

R E M A R K S ON N O R M A L OPERATORS ON B A N A C H SPACES

ERICH MARSCHALL

A closed densely defined operator T on a Banach space X is called normal. iff T r [C~ i.e. there is a homomorphism (functional calculus) f - - ~ f ( T ) of the continuous functions on (~ :=~U {oo} into the bounded linear operators

with IIf(T)II <_ sup{lf(z)l:~ ~dr} and which extends the analytic calculus. It is shown that this definition agrees with that of T.W. PALMER (Trans. Amer. Math. Soc. 133 (1968) 385-414). The functional calculus yields simple proofs of properties which are well-known for normal operators on Hilbert spaces. In the second part the methods are applied on semi-groups of normal operators resp. cosine operator functions.

Let H be a Hilbert-space and N be a bounded linear operator on H, then it is well known that the following are equivalent:

(i) N N " = N ' N

(ii) IlNzl[ = [[N'zl[ for every z E H

(iii) N = R + i J with R J = J R and R = R*, J = J"

(iv) There is an isometric *-isomorphism

O r : C~ ..-* B( I t ) with ON(l) = I and ON(z) = N.

(v) There is a strongly a-additive and self-adjoint spectral measure E(.) on the Borel-sets of IF with

N = f XdE(X) * g

o(/v)

ON in (iv) is uniquely determined as the inverse of the Gelfaad homomorphism of the U'-algebra gener?ted by N, N ' , and I.

318 m,acn ~RscrtA~

(v) can be obtained by extending cI,~v to the algebra of all bounded measurable functions (cf. [5], 6.10.).

Moreover, by the theorem of Stone a Closed Operator in / / is self-adjoint, iff it is the infinitesimal generator of a strongly-continuous group of unitary operators.

(i) or (ii) can be used to define unbounded normal operators on a Hilbert-space (see [14], p. 121). To define bounded or undounded normal operators on Banach-spaces, (i), (ii), and (iii) are meaningless. Since the star in (iv) is superfluous (cf. 1.2. a below), one considers (iv) and (v). A generalization of (v) are the scalar type specral operators of Nelson Dunford, respectively prespectral operators (cf. [5], [12]). But in this case the Banach space should be weakly sequentially complete (w.s.c.), respectively a dual space. So (iv) seems to be the best way to generalize normal operators to general Banach spaces. Unfommately in the literature a closed operator is usually called hermitian or self-conjugate ([4], [5], [12], [13], and others), if it generates a strongly-continuous group of isometric operators. These operators should preferably, and in this note will be, called infra-hermitian (i.h.), because except for Stone's theorem they have only few properties in common with self-adjoint operators. For instance even T 2 is not always i.h. if T is ([11], 1.8.d., see also [12], remarks p. 406). Moreover, the functional calculus and some spectral properties show, that i.h. operators are far removed from self-adjoint operators (cf. [11], 1.8. and 2.4.).

By the definition of T.W. Palmer [12] a closed densely defined operator N on an arbitrary Banach space is normal, iff N = .R+id, R " J " are i.h. for every n ,m E IVo, and all these operators commute, that means the generated groups commute.

We will show (1.11.), that this is equivalent to (iv).

We use notations as in [9], [10], and [11]. Especially let dr :=artg{oo} and u(T) be the extended spectrum of T, that means oo E u(T), iff T ~ ~(X)\/3(X). ~I(X) is the set of closed operators and B(X) the Banach-algebra of bounded operators on X.~(X) := {T E ~I(X) : p(T) : = ~ t r ( T ) ~'~}. TI,T2 E s commute iff some resolvents R(Tj, t~j) := ~ j - T j ) -I , /~. E p(Tj), j E {1,2}, commute ([9], 1.2. and [12]). Let $(X) := {T E/I (X) : T has the single valued extension property}. For T ~ $(X) and z E X ~,r(z) is the local spectrum and p r ( x ) : = t ~ r ( z ) the local resolvent set. If F c dr let Xr (F) := {z ~ X : or(x) C F}. %(T), ~,,(T), and oc(T), are the point spectrum, residual spectrum, and continuous

REMARKS ON NORMAL OPERATORS ON BANACIt SPACES 319

spectrum, respectively.

An algebra A of complex valued functions on f l c d : is called admissible in the sense of Colojoara and Foias (see [9], def. 2.2.), if A contains decompositions of the unit and the rational functions locally belong to A. T ~ J(X) is of class A (T ~ [A]), iff there is an homomorphism OT:A---,/3(X) which are locally holomorphic on r It is clear, that for

= ~ (:dr C ~ C(fl) is an admissible algebra which even is inverse

closed ([9], Def. 2.7.).

1. Normal operators on Banach spaces.

Definition.l.1. Let T be a closed densely defined operator on the

Banach space X, then

(i) T is normal, iff T 6 [C~ and [[~'TI[ = 1

(ii) T is n-hermitian, iff T is normal and r ~ :=/Rt./{co}

(iii) T is n-unitary, iff T is normal and o(T)c S~ := {z ~ a : : Izl = 1}

(iv) 2" is normal-equivalent, iff T ~ [G~ but I1~11 = M > 1.

Remark 1.2. a) The following proposition shows that the definition is correct because Or is unique. Moreover, if X is a Hilbert space, then [2], 3.2.6. yields that Or is positive and, therefore, a *-homomorphism. Hence T is normal in the usual sense.

b) If T is only normal-equivalent, then IIIxlll := sup{ll ' t 'r(f)xll : f ~ C~ Ilfll <-1} is a n o r m on X which is equivalent to the original norm I1" II, and on (X, lll" lid T is a normal opera tor (cf. [5], [12]).

c) If T is normal and X is a dual space that means X = Y' for some Banach space Y, then the proofs of [5] in the bounded case and [12] in the unbounded case show that T is even prespectral of scalar type and class F = Y. If X is w.s.c, a normal operator is spectral of scalar type

([5], 6.10.).

We shall, therefore, treat only normal operators and the obvious modifications for normal-equivalent, prespectral, or spectral operators are

left to the reader.

d) For some parts we even would not need that T is densely defined,

and some proofs become trivial, if T is bounded.

320 ERICH ~SCHAtL

PROPOSITION 1.3. Let T �9 A(X) be a normal operator.

a) The C~ Or is unique and regular, that means i f A �9 B(X) commutes with T, then A commutes with O r ( f ) f o r every ! �9 6"~

b) T is even o f class C~ and the C~ is isometric.

C) O'(OT(f)) = f(o'(T)) for every f �9 C~

Proof." a) It T is bounded this is proved but not remarked in [3], 4.1.5. It follows also by duality and [5], 5.9., 5.13. If T �9 A(X)\8(X) let Kct::r then T[Xr(K) is normal and by the first part of the proof and [9], 2.5.a f--* Or(f)lXr(K) is the unique and regular C~ Hence, AOr(f)x = Or(f)Az for every z �9 Xr( r [.J Xr(K) and every bounded

operator A which commutes with T. Since Xr(~/7) is dense ([9], 3.12. and 3.17.), Or is unique and A commutes with Or(f).

b) Let f e 6'~ and f :d ' - -*~ ' an extension of f with

II/ l l , := s u p { l / ( z ) l : z e u ( y ) ) = 11.711 = s up{I f ( z ) l : z � 9

Let O,(f):=~r(-7). We only have to show that O, is well-defined. For instance by [9], 2.9. we have ~(OT(-7))= -7(~(T)) for every -7 �9 G'~(d'). Hence, if 4 is another extension o f / ' , then ~,(Or(-7- 4)) = (-7- 4)0r(T)) = {0}. Now Or ( , f -4 ) is quasi nilpotent and by [9], 3.8. generalized scalar, even normal, and [3], 4.3.5. shows that ~r(-7) = Or(~). c follows again with [9], 2.9. But now:

IlOo(f)ll = I lOr(i) l l -< Ilfll = IlIII, = sup{lfG~)[ : t~ �9 a(Y)} =

= sup{l~l : t~ �9 f ( a (T ) ) = = ( o , ( f ) ) ) =: r ( o , ( f ) ) < IlO,(f)l l

and I l f l l , = I I o , ( f ) l l �9

COROLLARY 1.4. If T E A(X) is normal, then IlTzll _< Ilzll r r (x ) =: Ilxll

sup{l~l : , �9 ~r(x)} for every x �9 D(T) with r r (x) < oo. Moreover. IITII = r ( r ) i f T is bounded.

Proof." If T �9 B ( X ) , m e n IIT#I = IIO,(z)l l = Ilzll , = r ( ~ . closed, x �9 D(~, and ~r(z) = Kcr:r then To := T[Xr(K)

liT011 = r(To) = rz0:). Therefore .

If T is only is normal and

l[Tx[l = IITo:rll _< llTo[l llzll = rT(X ) I[:ZlI-

REMARKS ON NORMAL OPERATORS ON BANACH SPACES 321

Remark 1.5. In the following we always use the unique and larger Co(v(T))-calculus and write f(T) instead of ~ , ( f ) . It T is a normal operator on a Hilbert space or if T is spectral of scalar type, then the calculus ~ , can be extended to every measurable function, bounded or not (see [14], 7.14. and [6], Th. 11, p. 2238). This can not be expected in the general case, since one can not associate an operator to every characteristic function. But in [10] an A-calculus was extended to a large algebra of local multipliers. If T is normal, this can be sharpened, and we even

obtain the spectral-mapping-theorem which seems to be very useful, for instance in the second part of this note.

Let T �9 ~I(X) be generalized scalar, i.e. T �9 [C**(~, let V c dr be open with ~,~(T)c: V and A := ~(T)\V be separated that means A contains no dense-in-itself subset 1), then Xr (A)= {0}([10],2.3.).

If, moreover, a , ( T ) c V , then X r ( V ) = X and we have 1 and 2 of [10]. If T is normal and a~,(T)tAa,(T)C V, then we even have an approximate unity, that is, the stronger condition 3: Them is a sequence {,b,,},,eav c CO(v(T)) with support ~b, =: r c V and lim ~b,(T)x = x for every x � 9

For, let ~b,, �9 G~ with ,b,, = 1 on ~U21,(A) and ~_.~nf'~[fl/n(A)= := � 9 1 6 2 : > -} u (oo}), then {~b.(T) : n �9 ,,W'} c 8(X) is bounded

and if = �9 Xr(V), hence or(=)crV, we havc ~b,~(Ir')= = z if n >__ n= by [9],

2.6. e. Since Xr(V) is dense in X, ~b.(T) convcrges strongly to the idcntity.

If f is continuous on a(T)nV, then following [I0], I.I.

D ( f i T } ) := i x �9 X : lim (f~bn)(T)x exists}

f i T } x "= lim ( f ~ n ) ( T ) x fo r x E D(ftT })

is a wall and densely defined closed operator. The map f �9

Si T) �9 A(X) is homomorphic, that means:

( f + g)iT} = f{ T) + gi T} and (fg)[ T) = f iT}giT) ([10], 1.1.g).

If A �9 B(X) commutes with T, then A commutes with (DA,)(T) by

1.3.a., hence A commutes with f{T}. Moreover, we obtain:

THEOREM 1.6. Let T �9 JI(X) be normal and V c d as above. If I �9 C~ then ${T} is normal, too, and ~y{r}(g) := (g~

g �9 C~ is the Co(dO-calculus for f iT} .

1) I f A is finite or i r a has only a finite set of accumulation points, then A surely is separated.

322 ERICH MARSCHALL

Proof" (9 o J){T} is closed and densely defined. If x �9 Xr(V), we have

11(9 o f ) { T } x I I = I1r o f ~ . . ) x l l

<__ Ilxll sup{l(9 o l ) ( z ) l : z �9 r <__ Ilxll II011.

Hence (9 o f){T} is cotinuous on Xr(V) and (9 o J'){T} �9 B(X), because Xr(V) = X. g---, (9of){T} is homomorphic by the above remarks. Moreover, r = I and II,~i(r)ll = 1. Let g �9 6'~ such that z g ( z ) � 9 C~ then:

r = f {T) (9 o f){T} = f{T}(9 o f){T} D (9 o f ) ( T } f { T }

= r )

and cI, l(r} is the C~ for f{T} ([9], Def. 2.2.).

COROLLARY 1.7. Let T �9 A(X) be normal and

f �9 C~ as in 1.6., then we have:

(i) a(f{T}) = I(a(T) n V)

(ii) at{r)(x ) = f(ar(x) N V) for every z �9 X

Moreover, f{T} �9 B(X) /ff f is bounded.

Vc~ a n d

Proof" By 1.6. f{T} is normal, hence decomposable, and the corollary follows by [10], 1.3. c. and g.

COROLLARY 1.8. (Palmer, see also [51, 5. 13.) Let T E ~I(X) be

normal, then there are unique, commuting, n-hermitian operators R and J

such that T = R + iJ.

Proof" Following [9], 3.18., let R := Re{T} and J := Im{T}. By 1.6. R and J are normal and 1.7. shows that they have real spectrum. The remarks above show that R and Y commute and T = R+ iJ. But the proof of [9], 3.18. yields O ( T ) c l)(R)n/9(3"), that is T = R+ iY. It remains to show uniqueness. Let T = R'+iY' with the same properties. Since R' and J ' commute, they commute with T, hence with R and J. But now R - R' and J - J ' are i-hermitian by [12], 3.9. If z E D(T), then R - R ' z = i Y - J ' z and

[9], 1.3. b. yields R - R' = i J - J'. Hence, a(R - R') c ~ N i ~ = {0, oo}.

But R - R' is generalized scalar ([9], 4.5.) and oo is not an isolated point of a ( R - R ' ) by [9],3.12.a. Hence a ( R - R ' ) = {0}.

Now Ker ( R - R ' ) = X by [4], 8.16. and [9], 2.6.d; (see also [11], 1.9.), and we have R = R' and J = J'.

REMARKS ON NORMAL OPERATORS ON BANACIt SPACES 323

COROLLARY 1.9. (Polar decompositin) Let T E t I (X) be normal and

0 E p(T)U at(T), that means T is injective and has dense range, then there

are commuting operators U and H with U n-unitary and H n-hermitian

such that:

T = H U = U H

Proof." Let V :=ark{o}, then X r ( V ) and Xr({0,oo})= {0} by our assumptions. Let h(z) := Izl and u(z) := z/[z I. h and u are continuous on V and by 1.6. H := h{T} and U := u { T } are normal and commuting. a(H) = h(a(T) N V) C [0, oo] and o(U) = u(a(T) N V) c 81. T = U H = H U = HU.

If = E L)(T), then U-~Tx = U - 1 H U x = H U - 1 U x = Hx. Hcnce, D(T) C P(H)

and U H = T.

Remark 1.10. This decomposition is unique, too. But we do not need this in the following and omit the proof which is similar to the proof of 1.9. The multiplication operator z on C~ L) := {z (~ ar : Iz I < 1) shows that such a polar decomposition, i.e. U, does not exist for every normal

operator T.

THEOREM 1.11. Let T E •(X) be normal and let T:=~{T} then:

a ) �9 --- R - iJ and o ( ~ = {z e ~ : -~ ~ a cr))

b) /)(7) = L)(~r') = L)(H) and IIT=II = IIT~II = IIH=II f o r every x E L)(T).

c) T T = T T = R 2 + J2= H 2

d) T is n-hermitian r T = T

e) T is n-unitary r T = T - l r H = I

Proof." a) Using 1.6. wa see that ~ is normal and 1.8. shows = Re{T} + ilm{~}. But {Rez} = r = R and Im(~} = r = - J .

b) We show L)(T) c L)(H) and IIH=II <- IIT=II for every w ~ L)(T). All other inclusions follow similarly. Using the definition of H we have V = r Xr(r = X, and r = 1 on U,(0) and r c U2,(0). Let ~ > 0 and r ~ C~

with 'b_._e, c U,(0) and r = 1 near zero.

Let x fi L)(T).

(hr = ( r 1 6 2 + ( r - r

Now, r and ( 1 - r are bounded. Therefore, (hr converges

to H x and x E L)(H). Moreover,

IIHxll <_ sup(ll(hr : - ~ ~ }

324 ERXCH ,~SCHALL

_< sup{Ix~,(z)z~.(z)l Ilxll : - ~ ~ ; z ~ )

+sup{l(1 - ~,(z)lzl/zxo.(z))l IIT~II : - ~ ~ ; z r

< ellzll + Ilrzll

c) Using [10], 1.1. B := [z[2{T} is an extension of each operator in c.

But D(B) = D(H 2) = D(T~) = D(~T) = D(R 2) f3 D(J 2) = D(T 2)

follows as in b.

d) o(T) c~q(J-)=(0}~J=0c~T=

e) If T is n-unitary, then T E [O~ and ~= z-' in G~

Hence T=~(T) =z-'(T) = 2 -I . T=T -I implies I = TT= Izl2(T}. Therefore, {I) = {Izl 2 : z 6 a(T) UdT) and o(T) c S,.

Remark 1.12. If T is normal and X is a Hilbert space, we have = T* and ~ has some additional properties being an adjoint operator. But

also in the general case 7 ~ is normal if T is normal and /{T'} = f{T} ' ([10], 1.1.). Using the State diagram of Taylor and Goldberg one obtains a lot of spectral properties of normal, i-hermitian, and /-unitary operators (cf. [11], 1.9. and 1.10.).

In the Hilbert space case Fuglede's theorem (l.3.a) shows that the sum and the product of commuting normal operators are normal, too. The example of Kakutani and McCarthy ([5], 9.) shows that this fails to be true even in reflexive Banach spaces.

This is surly a shortcomming, but besides 1.6. we have the following theorem of Vidav, Berkson, Glickfeld, and Palmer: An algebra V c / ) ( X ) is called a V*-algebra, if each A e V can be written as A = R + i J with R , J (~ V and i-hermitian. On V we have the Vidav-involution (R + iJ)* := R - iJ.

THEOREM 1.13. (cf. [121, 2.8. -2.10. and [13])

a) Let Vo c B(JO be a commutative V*-algebra, then Vo, the sequence closure in the strong or weak operator topology of the algebra generated by I and Vo is a commutative V'-algebra and with the Vidav-involution a C*-algebra.

b) Each A E V~ is normal.

COROLLARY 1.14. Let T 6 A(X), then T is normal, if and only if T is normal in the sense of Palmer.

REMARKS ON NORMAL OPERATORS ON BANACH SPACES 325

Proof." " =~ " Let V := {f{T) : I is continuous and bounded on aT), then 1.6. and 1.8. show that V is a commutative V'-algebra.

Now T = R + i Y with R, J even n-hermitian, and the group generated by R, i.e; UR(t)= expit{R} = (expitRe){T}, is contained in V. Similar we have Uj(t) E V Yt E R and T is normal in the sence of Palmer [12], Def. 4.1.

"~=" Let KCL-ar and

X~s(lO := {x ~ X : an(X) c K; ~j(x) c K} = XR(/O n X~(/O,

then X r is a closed subspace which is invariant for every A E B(X) which commutes with tit and J. Let V be the V'-algebra generated by the groups

UR(t) and U~(t), t ~ R.

Let Vr := {AK := A I X r : A ~ V}, then Vx is a commutative V'-algebra.

But RK := RIXK E B(Xr) and RK = lira t-~(UR(t)IXK -- I lXx) . t - .o .

Hence R~c E Vr, and similarly Jr ~ Vr. Therefore, we obtain TK E VK,

and TK is normal by 1.13.b. Hence, IIf(rr)xll <_ Ilflloll=ll for every x ~ X k and f E 6'~ If K : LCE~, then Y(TL)IXK = f(T~) because the calculus is unique. Therefore, f(T) is well defined on Xo := LJ x r = xe~.,(~), and

Kazar the proof is complete if X0 is dense. Since R, J are i-hermitian, they are of class F ~ in the sense of [11]. Hence, Xn(I~')=X by [11], 1.6

and it in enough to show that XR,J(r Xn(r Let {~b,,},~N C F ~ be an approximative unit as in [11], 1.6. and let x E XR(ar), then x = lira ~bn(d-):c.

�9 11 -"*OO

Since R commutes with Uj(t), it commutes with ~,,,(J'), and using [9], 2.6. we obtain c,j(~b,,(J-)z) c ~b.__E, CE~ and oR(~,(J-)-~) C on(x).

Hence, Tb,(J-)~ C XR,j(r for every n ~ IV.

Now T.W. Palmer ([121, 4.3. 4.5.) obtained useful characterizations of bounded or unbounded normal operators which, of course, characterize scalar type prespectral, resp. spectral operators, if X is a dual space resp. X is w.s.c. If T is hermitian, we have:

THEOREM 1.15. (Palmer) Let Y E ~I(X) and densely defined, then the

following are equivalent:

(i) T /s n-hermitian

(ii) ~r~ is i-hermitian Yn E IV

(iii) IIR(r",~)ll <_ ~Vn ~ IV and t~ E ~ with I~1 >- Iv(.) l U I

(iv) T is i-hermitian and C(t) := ~(U(t)+ U(-t)) and S(t) := l ( u ( t ) - U(-t))

are i-hermitian Yt E

326 ERICH ~SCaALL

(v) T is i-hermitian and 11/'(TDll _< sup{If(z)l : z ~ _~) v f ~ Ao c ~ := { f : l is the Fourier tranform of ] E Ll(~q)} and Ao is dense in C~ ( ~ [111 for the definition o f F~

An other characterization with ordertheoretic methods will be proved in a following paper.

2. Normal operator semi-groups and cosine operator functions.

If X is a Hilbert-space, the generator of a strongly continuous semi-group {T(t ) : t > 0} is normal, iff T(t) is normal for every t > 0 ([7], 22). A. R. Sourour [13] generalized this result to scalar type operators in w.s.c. Banach spaces. But since the sum and product of scalar operators are not always scalar, he could not use the C'-methods of the proof of R.S. Phillips [7]. Sourour used the polar decomposition. But this proof does not generalize to cosine operator functions. Now the mathods prepared in the first part clarifies the situation even in the Hilbert space case and yield a simple proof which generalizes to cosine operator functions.

THEOREM 2.1. Let (T(t) : t >_ 0} be a strongly continuous semi-group and let A be the infinitesimal generator, then A is normal iff T(t) is normal for each t > O. In this case we have T(t)= expt{A} and r =expt(c,(A) nO) for every t > O.

Proof." "~= " Let V c 8(X) be the algebra generated by TO/n) and T0/n ) , n E iN. V is a commutative V*-algebra. For, let W ~ V, then there exists no ~ ~ ' such that:

./-1 L-~t jffil

N = f { Y ( l / n o ) ) with f ( z ) := ~ajzn ' -~ m/ E C~

jffil

Hence, W is normal by 1.6. and W = R - i J with R and J n-hermitian and commuting. But R , J E V because W = f{T(1/no)} E V. Now, V' the sequence closure of V in the strong-operator-topology consists only of normal operators. But if ,~ > too we have (cf. [4]., 2.8.)

REMARKS ON NORMAL OPERATORS ON BANACH S P A O ~ 327

o o

R(A, ~)x = f e-xtT(t)~.dt for every ~: E X, and R(A, A) E ~ ' ff we assume 0

the integral as the strong limit of Riemann sums with rational supporting-points

tj. Now A is normal by 1.6. or [9], 3.10.

"=~ " Let ,.q(t):= expt{A},t >__ O. Since o(A) Ntg is contained in a left halfplane, expt(.) is bounded on tr(A) N~. Therefore, 8 ( t ) E B(X) and #(S(t)) = expt(a(A) Nag') and $(t) is normal. It is clear that {8(t) : t > 0 ) is a

semi-group and the methods used in [9], 4.13. or [11], 1.7. show that S(.) is

strongly continuous and A is the generator of {S(t) : t > 0}. Since the semi-group generated by an operator is unique, we have 8(t)= T(t) for every t >__ 0.

Remarks 2.2. This proof only needs that {TOn -n) : n E _a/'} is

a normal-equivalent set for some rn E / N , that 'means each T(m -'~) is

normal-equivalent, but with common bound II,t, ro~-.(.)ll <_ M < ~o.

If { T ( t ) : t E !q} is a periodic group of operators and, for instance,

T(1) = I = T(q) for every q E 2 , then using [5], 10.10. T(p/q) is spectral of

scalar type, hence normal-equivalent for every p/q E t~. But the translation group

on the continuous periodic functions on ~q shows that in this case the generator needs not be normal-equivalent, too. Instead of defining the whole matter of

cosine operator functions we refer the reader to [8] and the papers cited there,

and only remark that using equations (0,3) and (.0,8) of [8] the theorem 2.1. and the proof of it holds for cosine operator functions, too. This generalizes [8] and

is the main reason to reprove 2.1.

COROLLARY 2.3. (Stone, Berkson) Let {UCt):t E lg~} be a strongly continuous group of n-unitary operators, then there is one and only one n-hermitian operator It such that U(t)= exp it (H} for every t E ~ .

Proof" The generator iH of {U(t) : t > 0} is

a(U(t)) = expt(a(iH)N~) C 81. Hence, a(H) C ~q

Moreover, U(t)= exp it {H} for every t E lq.

normal by 2.1. and and H is n-hermitian.

COROLLARY 2.4. If (H(t) : t >_ 0} is a strongly continuous semigroup of i-hermitian operators, then the generator A is normal, a ( A ) C [-oo, too], and {II(z) := expz{A} : Rez >_ 0} is a strongly continuous extension of {H(t) : t >__ 0} which is even holomorphic on gt := {z E ~7 : Rez > 0}.

Proof." Since H(t)'~ = H(nt) is i-hermitian for every t > 0 and n EN, 1.15. (ii) implies that each H(t) is normal. Now A is normal by 2.1. Since

328 ERICrX U A R S ~

expt(a(A) Nlff) = a (H( t ) )CE~, there exists too E ~ with o(A) C I -co , to0]. Hence, H(z) := expz{A} E B(X), z E ~ , is well defined and H(.) is a semi-group. Again as in [11], 1.7. for x in the dense subset XA((g),x(z):=H(z)x is an

entire function. Since H(.) is locally-bounded, H(.) is holomorphic on H and

strongly continuous on H-/.

We now list some more results on normal semi-groups.

(1) [9], 3.12. implies that co is not an isolated point of a(A), hence,

~r (A) N d7 ~r r and the type

too := lira t -1 log IlT(t)ll = sup{Rez : z E , (A) I"1~'} > - c o t---*OO

(2) IlT(t)ll = r(T(t)) = exptto0 for every t > 0.

(3) T(t) is injective and has dense range. This follows by [10], 1.3.e. and f .

(4) Therefore, T(t) has an unique polar decomposition (1.9.) T(t)= HtUt.

But since A is normal, we have A = R + i J with R , J n-hermitian and r c [ -co , to0]. Now, by 2.4. R generates a semigroup H(.), holomorphic on

the right halfplane, and i J generates a n-unitary group U(t), t E ~ . But

It(t) = expt{R} -- (exptRe){a} = (h o expt){A} = h{T(t)} = Ht,

and similarly U(t) = Ut for t > O.

(5) T(.) is periodic, i.e. T ( a ) = I for some c~ > 0, if and only if a(A)Neff C C 2~riz.

C[

Proof" u(T(c~)) = e x p a ( v ( A ) N ~ ) by 1.7. and the assertion follows if

a(T(c~)) = {I} implies T(ot)= I. But, since T ( r , ) - I is normal, 1.4. shows

l i T ( a ) - Z l l = r (T ( (~ ) - / ) = O.

The last result hold for i-hermitian operators, too, but not for general

generators. In fact, let 0 # N E ~(X) with N 2 = 0, then a(N) = {0}, but

e a /v= I+ i tN ~ I for every t ~0 . and T(.) is not periodic. Similar corollaries

hold for cosine operator functions.

REFERENCES

[1] Bcrkson E., Semi-groups of scalar type operators and a theorem of Stone, Illlonois J. Math. 10 (1966), 345-352.

[2] Braueli O., Robinson D.W., Operator algebras and quantum statistical mechanics L New York-Heidelberg-Berlin, Springer 1979.

REMARKS ON NORMAL OPERATORS ON BANACI-I SPACES 329

[3] Colojoara I., Foias C., Theory of generalized spectral operators, New York-London-Paris, Gordon and Breach 1968.

[4] Davies E.B. One-parameter semigroups, London-New York-San Francisco, Accademic Press 1978.

[5] Dowson H.R., Spectral theory of linear operators, London-New York-San Francisco, Acxademic Press 1978.

[6] Dunford N., Schwartz J., Linear operators III, New York-l-x)ndon-Sidney-Toronto, Wiley-Intersience 1971.

[7j Hille E., Phillips R.S., Functional analysis and semigroups, 3. edition, Rhode Island, Amer. Math. Soc. Coll. Publ. X'XXI 1968.

[8] Lutz D., (_]ber operatorwertige l_,~sungen der Funlctional-gleichung des Cosinus, Math. Z. 171, (1980), 233-245.

[9] Marschall E., Funktionalkalkiile J~r abgeschlossene lineare Operatoren in Banachriiumen, Manuscripta Math. 35 (1981), 277-310.

[10] Marschall E., A spectral mapping theorem for local multipliers, Math. Ann. 260 (1982), 143-150.

[11] Marsehall E., On the functional-calculus of non-quasianalytic groups of operators and cosine functions, Rend. Circ. Mat. Palermo (2) 35 (1986), 58-81.

[12] Palmer T.W., Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc.133 (1968) 385-414.

[13] Sourour A.R., Semigroups of scalar type operators on Banach spaces, Trans. Amer. Math. Soc. 200 (1974) 207-232.

[14] Weidmann J., Lineare Operatoren in Hilbertrdumen, Stuttgart, Teubner 1976.

Pervemuto il 1 apriie 1985

Erich Marschall Mathematisches Institut

Universt~ M iinater EinsteinstraBe, 64

D-4400 Miinster