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0 for any t, 0 < t ~ I. According to Lemma 5.4 there exist Qt E Lq(dp) and P, E LV (dp) such that P,Q, ~ 0, ! P, IP = = DP(S, 0", Ftdll) : Qt :q. P, belongs to the closure of {hF'IP, hE S,O"(h) = I} in LP (dp) and
Ch. S. HP -spaces
89
Moreover,
JhQ,PIP dp =
[DP(S, (/, Ftdp)],IP(/(h),
(h
E
S)
and dl = I Q, Iqdp = : Qt Iq F dm is a representing measure for cp with dl = dm as dl is absolutely continuous with respect to m. For any 0 < s ~ t ~ 1 we have [DP(S,
(5.2.5)
(1,
Pdp)]lIPq(h) =
~ hPfP (~ )"P Q.dp
(h
E
S).
We choose a sequence (h,,), h" E S, with (/(h,,) = 1 and h"PIP converging to P, in LP(dp). If we write (5.2.5) for every h", we obtain at limit (5.2.6)
[DP( S,
(1,
Pdp )]liP
= ~ P, ( ~
fP
Q.dp.
Applying Holder's inequality we get t-s
~P, ( ~ ) " Q.dp ~ = [J)l'(S,
nl ~ r' ]"P I P, jP (
u, P dp)]lIP
r'l I mr r
[~( ~
= [DP(S, u, Pdp)]l/P
J Q.:qdp =
dp
Q, qdp )"P =
~
s
dm
where we used the relation
JIQs lq dp =
1,
and the fact that dl = : Qt Iqdp is equal to m, and DP(S, (/, PdJi) I Qt Iq· Then, from (5.2.6) we obtain (5.2.7)
C( )I-S dm.
DP(S, (/, FS dp) ~ ~ DP(S, (/, FI dp) ) F
Hence, for 0 < s < t
~
I we have
I P, lP =
Function algebras
90
According to Lemma 5.1.2 there results 1
F.5dp) Jt-s lim sup [ DP(S, (1 '---t-+s DP(S, (1, Ftdp) for any 0
~
~
s
t
~
~
J ( -I , dm )
F
1. In this case it is known that
for any 0 ~ s < t < 1. If we write the last inequality for t - 1 and s = 0 there results J (F, dm) ~ DP(S,
Fdp) . DP(S, (1, dp) (1,
We now give up the restriction S C M(X). For this, let V E S with a(V) = 1. Since S C LP(dp) and S C LP(Fdp), we have
II V:PdJl
with the property that any representing measure for lP, absolutely continuous with respect to m, is identical with m. A linear closed subspace 5 of LP(dm) (norm-closed in LP(dm) for 1 ~ p < 00, and weakly closed in L ~(dm) if p = 00) will be called imJariant if for any f E A and 1z E 5 we havefh E S. From the duality LP(dm) - Lq(dm), we obtain that a subspace S C LP (dm) is invariant if and only if it is weakly closed and A 5 C 5. If S is an invariant subspace of LP(dm), we denote by 55 the weak closure, in L(dmt, of the set {hf: h E S, f
E
A,
Jfdm = OJ.
Function algebras
102
Obviously So C S. The invariant subspace S is said to be simply invariant if So :1= S. For a subset S C LP(dm), we write Sl = {g
E
LP(dm): Shg dm = 0,
hE
S}.
If S is a simply invariant subspace of LP(dm), then it is easy to see 1) that S& and Sl are invariant subspace in Lq(dm) and Sl C S6, 2) SJ. :1= SJ·. 3) A function g E Lq(dm) belongs to Sd- if and only if
-rgfh . dm =
°
for any hE Sand f E A with ((J(f) = 0, i.e. if and only if gh E K(dm) for any h E S. There follows at once that [S~]oC SJ., hence, if S is a simply invariant subspace of LP(dm) then S~ is a simply invariant subspace of Lq(dm). Theorem S.22. Let FE LfX(dm) with
IFI
1. The subspace S
=
=
= FHq(dm) of Lf(dm) is simply invariant. Then So
= {Fh, h E HP(dm), Shdm = o}
and
Proof. S is obviously an invariant subspace of LP(dm). The set {Fh:
hE
HP(dm), Shdm = O}
°
is weakly closed and, according to the definition of So, it contains So. Since FE S, there results that for any f E A with ((J(f) = we have Ef E So and then, as F is bounded and A is dense in HP(dm), we obtain Fh E So for any h E HP(dm) with
Shdm =
0.
Hence So
=
{Fh:
hE
HP(dm), S hdm
= O}
and, therefore, So :1= S, i.e. S is simply invariant.
Ch. 5. HP-spaces
103
If g = Fh belongs to So, then
Sg F dm
~
S Fh F dm = Shdm =
°
and FE S&. Since Sci- is invariant, there results that FHq(dm) C S&. Let now g E So. We then see that for any hE S we have gh E K (dm). Let hI E K(dm) with Fg = hI. Since FE L oc (dm) and g E Lq(dm), there results that hI E Lq(dm) and since hI E K(dm), we have hI E Hq(dm), hence g = FhI with hI E Hq(dm), that is S~ = FHq(dm}. This completes the proof. 0 Theorem 5.23. Let S be a simply inrariant subspace of L2(dm). Then there exists a function FE S such that IFI = 1 and S = FH2 ( dm) . The function F is uniquely determined up to a cOllstant factor of modulus 1. Proof. Let FE S be orthogonal on So and
Since f FE So for any f
E
A with qJU')
=
0, there results
(f E A, qJ(j)
=
0).
From Corollary 5.18 there results IFI2 E H2(dm) and, since it is real, according to Corollary 5.20, we have IFI = 1. Therefore, the multiplication by F is an isometry in L2(dm) and, since FE Sand S is invariant, we obtain FH2(dm)C S. Let now hE S orthogonal on FH2(dm). Then, for any f E A we have
(f E A), and, in particular, Fh gonal on So, we get
E
H2(dm). On the other hand, since F is ortho-
Sfh Fdm = 0
(f E A, qJ(f)
=
0)
that is hF E H2(dm). Then, according to Corollary 5.20, Fh is constant and, since
Jh Fdm =
0,
we have Fh = 0. But IFI = 1, hence h =
°
and, therefore, S = FH2(dm).
Function algebras
104
If S = FIH2(dm) = F2H2(dm), then FI = F'j,h2' F2 = Flhl with hi' h2 E H2(dm). There results that FIF2 E H2(dm), FIF2 E H2(dm) and from Corollary 5.20, FIF2 is constant. Therefore FI differs from F2 by a constant factor, whose modulus is obviously I. Theorem 5.24. Let
F be a function in L (dm) with IFI = 1. The
o.
0
5.6. The algebra HOC (dm) Let A be a function algebra on X and lfJ a nonzero multiplicative linear functional on A. Let m be a representing measure for lfJ with the property that any representing measure for lfJ, absolutely continuous with respect to m, is identical with m. The algebra LX (dm) is a commutative Banach algebra, with unit element, whose norm satisfies
IIf211
=
IIfll2
If we denote by Y the maximal ideal space of LX (dm), then it is known that L X (dm) is isometrically embedded into C(Y). Since L OC(dm) is obviously symmetric, from the Stone-Weierstrass Theorem we get LX(dm) = C(Y).
Ch. 5. HP-spaces
107
The algebra H-:L (dm) is a subalgebra of L -:L (dm), hence a subalgebra of C( Y), which contains the constant functions and is uniformly closed. According to Theorem 5.11, H-:L(dm) separates the points of Y. X H (dm) is therefore, a function algebra on Y.
Theorem 5.26. For any real function u E L -:L (dm) there exists E E H-:L(dm) such that E-1 E HX(dm) and eU = lEI. Proof. The functions F = eU and F1 = e- u belong to LX (dm) and we obviously have J(F, dm) > 0, J(Fl' dm) > O. According to Theorem 5.11 there exist the outer functions E and E1 in HX(dm) such that lEI = F, IEII = Fl' Since EEl is an outer function and I EEl 1 =- 1, there results EEl = 1 and, therefore, E-1 E HX(dm). 0
Notes The results contained in this chapter have their origin in classical problems of the theory of analytic functions of a complex variable and especially in the theory of Hardy HP-classes. A modern treatment of all these classical results can be found in K. HOFFMAN'S work [2]. H. HELSON and D. LoWDENSLAGER [1], [2] were the first to present a theory of HP-spaces under more general conditions than Hardy's theory. There followed a s~ries of works which proved in more and more general conditions, results similar to classical ones. We mention here some of them: P. R. AHERN, D. SARASON [1], I. GUCKSBERG [3], K. HOFFMAN [1], K. HOFFMAN, H. ROSSI [1], H. KONIG [1], [2], [3], [4], G. LUMER [1], [2], etc. The present chapter follows mainly the synthesis work of H. KONIG [4].
CHAPTER 6
Special classes of fu nction algebras 6.1. Dirichlet and logmodular algebras Let A be a function algebra on X. A is called a Dirichlet algebra on X if the space ReA is (uniformly) dense in CR(X). A is called a logmodular algebra on Xifthe set log lA-II = {u E CR(X): u = log If/,/EA, 1-1 E A} is dense in CR(X). Since for any I E A we have e l E A -1 and Rei = log lell, there results ReA Clog IA -11 and therefore any Dirichlet algebra on X is a logmodular algebra on X. Theorem 6.1. Let A be a logmodular algebra on X, 3'1t its maximal ideal space and qJ E 5JJl. There exists only one representing measure lor qJ with support in X. Proof. Let Ill, 112 be two representing measures for qJ, with support in X. If f E A-I, we have f(qJ) f-l(qJ) =
Since f( qJ )f- I ( qJ)
=
=
Jfdlll'
Jf-
I
d1l 2 ,
I f(qJ) I ~ JIf I dpl If-l(qJ) I ~
JIfl-
I
dp2·
1, there results 1~
JIII dill J1/1- 1 dill
(f E A -I).
But the set log lA-II is dense in CR(X), therefore for any function e U with u E CR(X) we have (6.1.1)
1 ~ JeUdll 1 Je- udJl2.
Function algebras
110
We now fix
UE
CR(X) and, for any real t, consider
pet) is a differentiable function and from (6.11) we get pet) any t. Since p(O) = I, there results p'(O) = O. Then
o=
p'(O) =
Judll
l -
J udll
~
1 for
2
hence
for any UECR(X) and, therefore, III = 11£. 0
Corollary 6.2. For any x E X, Gx is the only representing measure for x. Then, the Choquet boundary of A is equal to its Shi/ov boundary and both are identical with X. Due to the uniqueness of the representing measure for the elements q> of sm., the results obtained in the previous Chapter on HP-spaces hold for any q> E ~)ll and its representing measure m on X. As an application of these results, we shall determine the structure of the Gleason parts of the maximal ideal space of a logmodular algebra A on x. It is worth noting that the proofs hold also for a more general case, when any element q> E ~l admits a unique representing measure with support in X. An element q>1 E ~1l is called bounded in H2(dm) if there exists a constant c such that
(f E A). A functional in clll which is bounded in H2(dm) can be continuously extended to a linear multiplicative functional on H2(dm), Let us denote H~
The subspace
H~
= {h E H2(dm): Jhdm =
OJ.
of H2(dm) is obviously invariant.
III
Ch. 6. Special classes of function algebras
Proposition 6.3. Assume the subspace H;lI simply inl'ariant and let H;l = ZH2(dm) be its writing gil'en by Theorem 5.23. For hE H2(dm)
we write (n=0,1,2, ... ).
For any ({J1 E c11I, bounded in H2(dm), and hE H2(dm) we hare I({J1(Z) I < 1 and 00
({J1(h)
=
~
an[(,Ol(Z)r·
n=O
The measure
is the representing measure of ({J1' Proof. We first show that I({J1(Z) I < 1. Indeed, since in H2(dm), there results
({J1
is bounded
for any natural n and therefore 1({J1(Z) 1 ~ 1. We suppose I({J1(Z) I = 1. Since Z is determined, but a constant factor of modulus 1, we can take (,01(Z) = 1. For a natural n let fn = 00
=
~ bkz k be a function of the standard algebra such that fn{l) = n k=l 00
and ~
00
Ib
k l2
k=l
~ 1 *). Then the function h"
=
~ bkZk belongs to H2(dm) k=l
and we have
which contradicts the fact that Since PI' P2 are positive measures of norm 1 ~ there results 0 ~ h ~ 1. We then have
2 = 11({J1 - ({J211 ~ IIPI - JL211 =
11(1 - h)JLlll
+ IIPsll ~ J (1
=
11(1 - h)JLI
- h) dpi
+ 1=
+ Psil = 2-
I hdpl'
Hence
and, since h is positive, we obtain h = 0 PI-almost everywhere, which implies P2 = Ps' 0 Theorem 6.6. Let A be a logmodular algebra on X and J1 E A ~ .
There exist: an at most countable set {cp,} of distinct elements in ollL any two elements belonging to two distinct Gleason parts, the func-
Function algebras
JI8
tions h; E Ht where A; are the representing measures, for lfJ;, and the measure (J E A.L, singular with respect to all representing measures for the elements of .)lll, such that 00
P =
~
+
h),;
(J,
;= 1
the series being convergent in norm. Proof. If P and A are two' measures on X, we denote by Pi. the absolutely continuous part of P relative to )., and by P;, the singular part of If with respect to l. Let ({J1."" ({In be a finite system of elements in j)1L, each two elements belonging to two distinct Gleason parts and )'1" .. , An their representing measures. According to Proposition 6.5, A; and )'j are mutually singular for i ¥= j. Let P E A.L and
P= P -
"
~ Pi-;' ;=
1
We have p
=
p;'j -
~ Pi.;'
(j
= 1, 2, ... , n)
;t::.j
hence p is singular with respect to every).;. Since P
" = p+ ~
Pi.;
;=1
there results
IIpll = Ilpll
+
t1
~
lip;)!
i= 1
which implies n
~ i=1
Ilpi.;1I
~
IlplI.
Ch. 6. Special classes of function algebras
119
This inequality holds for any system At, ... , An with the above properties, therefore there exists at most a countable set of Gleason parts which contain an clement lfJ with Pi, #- 0, A being the representing measure of lfJ. Let {~j} be this set. We fix cp; E ~j and let ;'j be the representing measure of lfJj. Since if;
~
Illli,; I
~
111l1I
i~l
J:;
'E
the senes i
=
Ji;.i
is convergent in norm. Let
1 if;
a
=
P-
~
p),;'
i = 1
Let A be the representing measure of an element cp E . .mL If there is a j such that lfJ E ~, then A and Aj are mutually absolutely continuous and, for i -=F j, A and Ai are mutually singular. There results (Ii).j);' = Ji). and (p;)). = 0 for i -=F j, hence a i. = Iii, - P). = O. If q> E ~i for any i, then, from the properties of the set {~i}' we get Ii;. = 0 and, according to Proposition 6.5, A and A; are mutually singular for any i, therefore (P;.);. = 0 which implies a). = O. Thus a is singular with respect to any representing measure of the elements of J)1L Following Theorem 5.6 we have Ii;'j E A.L for any i, hence a E Al.. Let h; E Ll(dA;) with dp).j = hidA;. Since Pi.j E Al., there results (fE A)
and, from Corollary 5.18, we get hi E Ht. This completes the proof of the theorem. (, The importance of this theore.m consists in the following: there are function algebras A with the property that the only measure orthogonal to A and singular with respect to any multiplicative measure on A, is the null measure; at the same time, there are orthogonal measures to A which are absolutely continuous with respect to no representing measure. In this sense give an example in § 6.4.
Function algebras
120
6.2. Algebras generated by inner functions Let A be a function algebra on X, its Shilov boundary. Let SI
lsI
= {s E A, s
#- 0: II Is II
m its maximal ideal space and r
=
11/11 for any I
We obviously have IIsll = 1 for any s E SI' Proposition 6.7. The function s of A belongs to =- 1 on the Shilov boundary, i.e.
SI = {s E A: Is(x) I = 1, X
Proof. If Is(x)1 = 1 for any x
II sIll
=
E
r, then
E
IIsll
E
A}.
if and only if
SI
r}. =
1 and
sup Is(x)1 (x) I = sup Is(x) I If(x) I = sup II/(x)1I xEr
xEr
=
11/11
xEr
for any I E A, that is s E SI' Conversely, assume there exist s E SI and Xo E r such that Is(xo)1 < 1. Let e > 0 and U be the neighbourhood of Xo on which
Is(x) I < 1 -
8
(x
E
U).
According to Theorem 2.16 there exists If I < e off U. We then have
Is(x)f(x) I = Is(x) I II(x) I
. Then x -+ ex applies X on m. The proof is complete. 0 We notice that, according to the maximum modulus principle X" is the union of X with the bounded connected components of the complement of X. Let us suppose that X belongs to the boundary of the unbounded connected component of its complement. In this case we shall prove that P(X) is a Dirichlet algebra on X. We first prove two helping lemmas (see CARLESON, [2]). Let dA be the Lebesgue measure in the plane. Lemma 6.25. Let X be a compact set of the complex plane and n the unbounded connected component of the complement of X. Let P be a real measure on X. Then u(z)
= ( log
J
x
1
Iz- xl
dp(x)
converges absolutely almost everywhere with respect to the Lebesgue measure dA. If u(z) = 0 in n, then u(z) = 0 in any point ZEn in which we have absolute convergence. Proof. Since the function log _1_ is locally integrable with respect
Izl
to dA and the measure d
Ipi is finite, there results
(
{ ( log
;zi(z). If ex ¢ X then (z - ex)-l E R(X), hence I = q>(l)
=
q>[(z - a) (z - a)-I] = (q>(z) - ex)q>(z - ex)-I = 0
which is impossible. Therefore a
e«(f)
=
f(ex)
X. It is then clear that
E =
f(q>(z) = q>(f)
for any rational function of R(X) , hence for any function of R(X). There results q> = ea and therefore the map x ~ ex is surjective. Lemma 6.29. Let Jl be a measure on X. Let
N(y)
dllli . J Ix-YI
= (
x
Ch. 6. Special classes of function algebras
137
Then N(y) is finite almost everywhere with respect to 1. If
F(y)
= (
dJl(x)
Jx-y
x
vanishes almost everywhere with respect to 1. then Jl = o. If Jl E R( X) 1. , then U = {y E C: N(y) < 00, F(Y)::J= O} C X and for any y E U, the measure Jl y defined by dJl y = f(y)
=
1 (z -y) -1 dJl(z), verifies the relation F(y)
Jfdpy
(f E R(X».
Proof. The first part of the lemma follows from the fact that
1
is an integrable function with respect to 1 on any compact in
Izl
plane, and IJlI is a finite measure on X. Assume now F(y) = 0 A-almost everywhere. Let g be a continuous function with compact support and indefinitely differentiable. It is then known that g(y) = _ _ I C(y _ 2n
J,
W)-l(~+ i~)g(W)dA(W). (}X
(}y
By integrating relative to dJl and applying Fubini's theorem we get
~ g(y)dp(y) = x
-
:
~ H(Y -
W)-l
~ ( o~ + i
:y)
g(w)dA(w) } x
x
= -
_I F(y) ( (~+ i ~)g(W)dA(W) = O. 2n J (}x (}y
Since any continuous function on X may be uniformly approximated on X with continuous functions with compact support and indefinitely differentiable, Jl = O.
Function algebras
138
Suppose now Jl
=
E
R(X)l. and let y
E
U.
If y; X then f(x) =
(x - y)-l E R(X), hence
F(y) =
J(x -
x
y)-ldJl = 0
which is impossible. Therefore U C X. Let y E U. Since F(y) =F 0 and (z- y)-l is integrable with respect to Jl on X, then Jl y defined by 1 F(y)
= - - (z - y)-ldJl
dp Y
is a measure on X. Since for any rational function of R(X), the function
hex) = f(x) - fey) x-y is also a rational function in R(X), there results
fey) - JfdJly x =
=
1
fey) - - - Jf(x) (x - y)-ldJl(x) F(y)
(( f(x) - fey) dll(X) ] F(y) J x - y
____ I
=
=
0
for any rational function in R(X) and, by continuity, we get
fey)
=
Jfdpy,
O. Proof. According to Theorem 2.23 there exists Il ~ ex such that Jl({x}} = o. Let v = p - ex. Then clearly v E R(X)l., v:l= O. Let F
Ch. 6. Special classes of function algebras
139
and U be as in Lemma 6.29. From Lemma 6.29, there results U C X and that F does not vanish everywhere with respect to l. Then l( U) = l{y E C: F(y) '1= O} > O.
We now show that UCA. Let y E U. According to Lemma 6.29, the measure 1
dv y =
Vy
defined by
(z - y)-ldv
F(y)
satisfies the relation
f(y) = Jfdvy
(f E R(X».
On the other hand, from Theorem 3.15 there follows a representing measure for y, absolutely continuous with respect to Vy , hence absolutely continuous with respect to v. This measure must obviously be of the form hdp, + C8y with hE Ll(dJL). Then JL y = (h + c)JL is a representing measure for y, absolutely continuous with respect to p,. Then, from Proposition 6.5 we get YEA. Hence UC A and therefore l(A) ~ l (U) >0. 0 Corollary 6.31. There exists at most a countable set of Gleason parts of X (with respect to R(X)), which do not reduce to one point. Proof. Since the points of ~ form point Gleason parts (Corollary 3.14), the proof follows immediately. 0 Corollary 6.32. A point x E X forms a point Gleason part of X with respect to R(X) if and only if it is a peak point for R(X). Hence the set P of all points x E X which form point Gleason parts is identical to 1:. Theorem 6.33. Let X be a compact set of the complex plane. 1: its Choquet boundary with respect to R( X) and P the set of all the points of X which form point Gleason parts of X (with respect to R( X)). The following assertions are equivalent. (a) R(X) = C(X) (b) X = P (c) X = 1:.
Function algebras
140
Proof. We already know that (a) ~ (c) and that (b) and (c) are equivalent. It remains to prove (c) ~ (a). Let I' E R(X)~. If I' =F 0 then, according to Lemma 6.29, there exists y E X such that F(y) =F 0 with F constructed as in the mentioned lemma, and the measure Jl. y defined by
satisfies the relation fey) =
Jfdp,y
(f E R(X».
Now, according to Theorem 3.15, there exists a representing measure for )" absolutely continuous with respect to /ly, hence absolutely continuous with respect to 1'. Since )' E 1:, By is the only representing measure for )" hence Gy is absolutely continuous with respect to p, that is p( {y}) =F O. Since y is a peak point for R(X), there exists f E R(X) such that fey) = I and If I < 1 on X - {y}. It is then clear that for sufficiently large n
which contradicts the fact that Jl E R(X).l. Then p = 0, i.e. R(X) = C(X). The theorem is proved.
00
Now let h E H have the form h
=
~ Ucp/li' ;=1
In
F or any k Nb by
=
1, 2, ... we define the function g~ on X, with values
if there exists J such that x
g~(X) =
E
E,?, where
o for the other x For k = 00, J = (j1,j2,···),A < j2 < .... We now show that g~ E L2(Nk; Xwkdll). We have
J (!h(x)lqJ jl(x)1 2 + ih(x)lqJ jlx) I2 + ... + !ik(X) IqJik(X) I2d.u(x) ~
~
card .1 = k E,? oc
~ ~
JIqJi(X)12!i(X)d/l(x) =
;=1
~ (U: fPi 2h;, hi) ;=1
11 - c, 437
~
JIqJ;(x)1 2d /l
i
=
;=1 X
oc
=
00
oc
=
oc
~ (UfPjlzi' U fPi hj ) = ~ ;=1
;=1
II UfPj hi ll 2 =
IIh!12.
Function algebras
J62
ffi
Let gh be the element of
L2(Nk' XcolcdJl) defined by
k = 1, 2, .. ,00
:xl
For hE H of form h
~ U~ihj, ;=1
=
we put
One easily verifies that U is a linear map from the subspace of elect)
ments ~ U~ihj of H, which is dense in H, into i
=
ffi k = I, 2, ... ,
1
L2(Nh XcokdJl). 'J.)
Moreover
I Uhl1 2 = IIghl1 2=
Ilg~112 =
~ k = 1, 2, ... ,
k
~
~
k = 1, 2, ... ,
card ,I = k
:xl
~
JIlg~(x)112d,u =
~ k
:xl
=
1, 2, ... ,
00
Wk
J1q>j;(x)12fj;d,u =
i = 1 E,I ct)
~ k
=
~
1,2, ... ,
card ,I = k
ct)
~ i
=
J lq>i(X)12fi(x)dp =
1 E,I
00
=~ i
=
:xl
=
~ ; =
~
1 k = 1,2, ... ,
~
~
00
Jlq>j(x)12fi(X)dp =
1 k = 1, 2, ..• , x
~ i
=
Jlq>lx)1 2dJLi =
;=1
.'Y.)
~ (Uq;;q;jh j , hi)
=
;=1
~ i =1
~ (U~ih;, U~ih;) ;=1
"1)
=
J1q>;(x)12f;(x)dJL =
1
"1)
.'Y.)
=
'J.)
IIh;ll2 = Ilh11 2 •
=
Ch. 7. Operator representations of function algebras
163
At a certain step we used the fact that for x E E" we have fi(X) = 0 for any i ¢ f. The map U can therefore be extended to an isometric operator ® L2(Nk; XWkdJL). from H into k = 1.2, .. .• 'Xl
We now prove that the image of H by U is dense in
®
k= 1,2, ... ,00
®
L2(Nk; XWk dJl). LetgE
L2(Nk; lWk dJl) of the form g=(gl, g2, ... , gOO)
k=1.2, •..• 'Xl
with gk = (g}, ... , gt), gk
E
L2(Nk; lWk dp), be orthogonal to any Uh co
with h E H. For h E H of form h =
o= ~
=
k=l.~ •.
(UIz, g)
(g~, gk) = ..• Xl
UfPi hi, we have
= (gh, g) ~
k=1.2 ..... 'Xl
~ k ~=1.2 ..... Xl
~
~
=
J(gk(x), g~(x)) dp =
k
~ JJ fjlx)
qJii(X)
g7(x) dJl.
card.1 =k i = 1
Let now
Theorem 8.6. Let f -+ Tf be a representation of A H has a unique decomposition of the form H
=
011
H. The space
Hs EB He
such that Hs and He are doubly invariant subspaces, the representation f -+ TflHs is spectral and the representation f -+ TflHe is completely non-spectral. Hs is the largest spectral subspace of H. Proof. Let {M j }iE.1 be the set of all spectral subspaces of H. The subspace {OJ is, obviously, spectral, hence this set is non-void. {Mih E,? is increasingly directed, according to Proposition 8.4. Let Hs be the closure of U A(. Then, from Proposition 8.5, Hs is spectral and, iE.1
clearly, it is the largest spectral subspace of H. We write H = Hs EB He· B s is doubly invariant, hence He is doubly invariant too. Hs being spectral, f -+ TflHs is also spectral and, as Hs is maxin1al, f -+ TflHc
is completely non-spectral.
171
Ch. 8. Elements of spectral theory
Consider another decomposition of H
with HI, H2 doubly invariant, / -+ TjlHl spectral and / -+ TjlH2 completely non-spectral. Hence, HI is a spectral subspace and, as Hs is maximal, HI C Hs. Let
Since Hs and HI are doubly invariant, M is also doubly invariant and, according to Corollary 8.3, M is spectral. But M is orthogonal to Hh hence M C H2 and, as f -+ TjlH2 is completely non-spectral, there results M = {o}. Therefore
and the proof is complete. 0 The representation / -+ 1jIHs will be called the spectral part of the representation f -+ 1j, and f -+ TjlHc the completely non-spectral part.
8.2. The spectral dilation and attached spectral measures A representation q> -+ U
Ig21, and
E
=
rtp,*(g).
I , C2 be two constants, with CI > IgII,
CR(X) and
C
We used the fact that p,* is a positive functional, and (8.2.12). Hence (8.2.17)
for any gI' g2 E CR(X). From (8.2.14) there follows (8.2.18) and therefore (8.2.19)
JI*(g) ~ IIgll IIhl1 2
-
p,*(llgll - Igl) ~ Ilgll IIhl1 2
Ch. 8. Elements of spectral theory
177
From (8.2.15), (8.2.16), (8.2.17) and (8.2.18), p* is a seminorm on CR(X) and, according to Hahn-Banach theorem and (8.2.19), there exists a real measure Ph on X such that Ph(1) = IIhl1 2 and (8.2.20) Then, obviously
hence Ph is a positive measure on X. We have
for g
~
0 and c
~ g,
and therefore
for any g ~ 0 and c ~ g. Since any positive function in CR(X) may be written under the form c - g with g ~ 0, c ~ g, then for any g E CR(X), g ~ 0, we have (8.2.21 ) Then
that is
and the proposition is completely proved. 0 12 - c. 437
178
Function a1gebras
Proposition 8.9. Let ilia be a positive measure on X such that PIa(I) = IIhl1 2 and
(8.2.22) Then Ilh is a spectral measure attached to h by f
(/, g E A). -+
Tf .
Proof. It remains to prove only that (8.2.23) Let A be an arbitrary complex number and / there results
E
A. From (8.2.22)
hence
or (8.2.24) Since (8.2.24) holds for any complex number A, we get
and the proof is complete. 0 From Proposition 8.8 and 8.9 there results the following Theorem 8.10. Let / -+ Tf be a representation of the algebra A on Hand h E H for which (8.2.7) holds. Then there exists a spectral measure attached to h by f -+ Tf . In the case of logmodular algebras we shall prove the uniqueness of attached spectral measures together with the existence and unique-
Ch. 8. Elements of spectral theory
179
ness of the spectral dilation when there are spectral measures attached to every point h of H. Theorem 8.11. If A is logmodular on X and f --+ Tf is a representation of A on H, then the spectral measure attached to 11 E H by f --+ Tf is unique. Proof. Let Ilb /12 be two spectral measures attached to h E H by f --+ Tf · For f, f- 1 E A we have
Since A is a logmodular algebra, we have
for any u E CR(X). Hence, the 'eal function
has an extreme point in 0 for any u E CR(X) and therefore
o=
qJ'(O)
for any u E CR(X), i.e. J11
= 2( Jud III -
= J12.
Jud /12)
0
Theorem 8.12. Let A be a logmodular algebra on X and f --+ Tf a representation of A on H such that for any hE H there exists a spectral measure attached to It by f --+ Tf . Theil there exists a unique spectral dilation of f --+ Tf . Proof. Let Ph be the spectral measure attached to h by f --+ Tf . We first prove the following relation (8.2.25)
180
Function algebras
Let /,j-I EA. We have 2(lIh 1 1l 2 = (1f(h l
~ I Tf(h l ~
+h
2 ),
+ IIh2112) = Ilhl + h2112 + IIhl Tj-l(h l
+ h » + (Tf(h 2
l -
1z2112 =
h2 ), TJ-l(hl - h2» ~
+ h2) I I Tj-l(hl + h2)11 + II Tj~-1(hl-h2)11 I Tf (h l -h 2) I
[II Tf(h l
+ h2)11 2 + 111/(h l
-
h2)112]1I2[11 Tj-l(h l
~
+ h2)11 2 +
Since A is a logmodular algebra we get
for any U E CR(X). Then, the standard argument used also in the proof of the preceding theorem, yields (8.2.25). If we put
we get a semispectral family (/1h 1 • h2)h1 • h2 eH on X such that
Using Theorem 7.5 we obtain a representation ({J --+ U", of C(X) on a Hilbert space K~ which is a spectral dilation of j --+ Tf .
Ch. 8. Elements of spectral theory
Let cp
--+
U; be another spectral dilation of f
181
--+
Tf . Writing
for hI, h2 E H, then, as we have seen, the measure mil = mh,h is a spectral measure attached to h by f --+ Tf . From Theorem 8.11 there follows mh= Ilh hence, the semispectral families (mhl> hz)h1 , h2 EH and {J1h1 • h2 )h1 , h2 EH coincide. Since the spectral dilation for semi spectral families is unique (Theorem 7.6), the spectral dilations qJ --+ UqJ and cp --+ U; of f --+ Tf coincide (in the sense given above). The theorem is proved. 0 Theorem 8.13. Let A be a logmodular algebra on X and f --+ Tf a representation of A on H. The representation f --+ Tf has a spectral dilation if and only if (8.2.7) holds for any h E H. If f --+ Tf has a spectral dilation then it is unique. Proof. If (8.2.7) holds for any h E H then, from Theorem 8.10 there results that f --+ Tf has a spectral dilation. The uniqueness of this spectral dilation follows from Theorem 8.12. Let qJ --+ UqJ be a spectral dilation off --+ Tf and qJI,"" qJn a finite system of elements in A + A, of the form cp = fi + gi' with l: IqJil 2 ~ 1 and h E H. We have
~IITlpih112
=
~11(Tfi
+ TiJ hll
~ ~ I Ulpjhll2
(ii) is obvious. So we can suppose that condition (ii) holds. Firstly, we shall prove (by induction) that n
n
~
(8.2.27)
I/il
2
~ 1 => ~
;=1
I Tf ,hll 2
~
IIhl1 2
;= 1
for any finite system {/b ... ,I,,} of elements in A and hE H. For n = 2 (8.2.27) is contained in (8.2.26). Let thus n > 2, e > 0 and n
lb'" ,I" E A such that ~ I/il 2 < I. Since A is logmodular algebra on ;=1
X, we can find g
E
A such that g-1 n-l
~
(8.2.28)
E
A and
I/il 2 < Igl2 < 1 -1/,,\2 + e
;=1 n-l
Let gi =/jg-l, i
=
1,2, ... , n - 1. Then, by (8.2.28) ~ Igil2~ 1 ;=1
thus by the induction hypothesis n-l
~
II Tg ,kll 2
~
IIkl1 2
;= 1
for all k
E
H. Particularly, for k
= Tgh we obtain
n-l
~ II Tfi h II 2 ~
II TJzll 2
;=1
But, from (8.2.28) we have also
//,,/2 + Igl2
~
1+e
so that using (8.2.26) we infer
that is n
~
II Tfihll2
~ (I
+ e) IIhl1 2
;= 1
Letting e -+ 0, we obtain finally (8.2.27).
Ch. 8. Elements of spectral theory
183
In a similar way we infer that n
n
~ Ifil2 ~ 1 => ~ II TJjhl12 ~
(8.2.29)
;=1
IIhl1 2
;=1
for any finite system {/h" .,I,.} in A and h E H. Now define for gEeR(X), g ~ 0 Il*(g)
sup ~ 1\ T/ hll 2
=
j
where the supremum is taken for all finite systems {/h ... ,f,.} C A such that 1:/fiI2 ~ g. Then reproducing the proof of Proposition 8.8 we obtain (using (8.2.2i) instead of (8.2.i») a positive measure ilion X such that 111(1) = IIhl\2 and 1\ Tf hl1 2
(8.2.30)
~
J1/1 2d1l
(f E A)
1,
Analogously, (8.2.29) will lead (again by reproducing the proof of Proposition 8.8) to the existence of a positive measure 112 on X such that 112(1) = IIhll 2 and
(fEA)
(8.2.31 ) In virtue of the proof of the Proposition 8.9 we will have
(fEA)
(8.2.32) Actually III
J.l2 since for f,/-l
=
IIhl14
E
A we have
~ (Tfh, T'f-lh)2 ~
~
IITfh!\21IT7-1hI12
~
J1/1 2dll J1/1- 2dIl2• 1
The fact that III = 112 is easily obtained as in the proof of Theorem 8.11. Let us put Ilh = 111 = 112' Finally if qJ; = fi + Ki' with 1:lqJ il 2 ~ 1 and f1,'" ,f,., gh"" g,.EA then n
~ IIT"jh112 ;=1
n
=
~ IIT/jh
+ T~jhll2
n
~ ~ (II Tfih II 2
;=1
+ 2Re (Tfigjh, h) + "TgihI12) =
;=1 n
~ ;=1
(II Tfihll2
+
+
Function algebras
184
+ 2Re Sfigi d J1h + II T~/1112) ~
n
~
(S IJ;\2dJ1h +
;=1
+ 2Re Jfigidp. + JIgiI 2dp.) = J( ~1
Irp.!2 d P.) .;;; 111111 2,
The proof of the proposition is complete. 0
8.3. Szego measures and natural representations Let fl be a positive measure on X. We recall that H2(dJ1) is the Hilbert space obtained by closing A in L2(dJ1). The measure J1 is called a Szego measure (relative to A) if, for any Borel set E of X for which
XEL2(dJ1) C H2(dJ1), we have J1(E) = 0 (1.E = the characteristic function of E). The following theorem asserts the existence of Szego measure for any algebra A '# C(X): Theorem 8.15. (i) Any representing measure of A is either a Szego measure or a point measure. (ii) if v is a non-zero complex measure on X, orthogonal to A, then Ivl is a Szego measure. (iii) If A i= C(X) then there exist Szego measures on X (relative to A). Proof. (i) Let J1 be a representing measure of A. One easily verifies that J1 is multiplicative on H2(dJ1.). Assume that for a Borel set E we have (8.3.1) Then XE
E
H2( dJ1) and
S XEdJ1 =
J X1dJ1 = [J XE dll]2
Hence J1(E) = 0 or /1(E) = 1. If /1(E) = 1 then from (8.3.1), there results L2(dJ1) = H2(dll), hence /1 is multiplicative on L2(dJ1) and, in particular, on C(X). It is known that such measures are point measures. (ii) Let v be a complex measure, orthogonal to A. Then v is orthogonal to H2(dll) and, writing J1 = lvi, we have (8.3.2)
dv = FdJ1,
IFI
=
1
Ch. 8. Elements of spectral theory
Suppose XEL2(d/1) C H2(d/1). following (8.3.2), we have
/1(E) =
There results
185 --
XEF E H2(d/1) and
JX d/1 = JXEPFd/1 = JXEFdv = O. E
(iii) is a consequence of (ii) and of the Hahn-Banach theorem. The proof is therefore complete. 0 A positive measure /1 on X be called Szego - singular (relative to A) if L2(d/1) = H2(dJi.). We shall now establish the relation between the Szeg6 measure and a completely nonspectral representation. Theorem 8.16. Let f -+ Tf be a representation of A on Hand Ji.h a spectral measure attached to the element h E H. Let M be a closed subspace of H such that hEM and TfM C M for any f E A. If the representation f -+ TflM of A on M is completely non-spectral, then Ji.h is a Szego measure. Proof. Let V, = TfIM; we define the operator V on A + A, with values in M, by (f, g
Since Ji.h is a spectral measure attached to h by f
I V(f + g)112
=
II Vfh ~
+ V:h1l 2 ' then, according to (8.3.3) and (8.3.4), the M-c1osure of VXEL2(dJlh) is a doubly invariant subspace Nand (8.3.5)
V(Uf IXEL2(d'Lh»
=
Vf (Vlx EL2(dJlh»
V(UjIXEL2(dJlh» = vj(VIXE L2(d,Lh».
Following (8.3.5), we have V:VfVe = V:VUfe = VU:Ufe = VUgfe = VUfge = = VUfU:e
= VgV:Ve
for any e E X EL2(dJlh)' and by continuity (8.3.6) For cP
(/, g E
C(X) of the form
(8.3.7) with/i' gi (8.3.8)
E
A, and n EN, we put
E
A; n EN).
Ch. 8. Elements of spectral theory
187
Then, using (8.3.5), we obtain V~ Ve
(8.3.9)
=
~
vj;vgi Ve = V(lpe)
for e E XEL2(dp,,). Hence, if lp = 0 then V~Ve = 0 for any e E XEL2(dp,,) and, by continuity, V; = o. Therefore lp --+ V~ is a well-defined linear map on the elements lp of the form (8.3.7). Also from (8.3.9) we get (8.3.10) and, following (8.3.9), (8.3.10), there results
!! v;mVe!! = !! v;mVell = II v(~/)me)!! ~ JIlpl21el 2mdph = Illpllmllell
~
where IIlpll resp. lIell are the norms in C(X) resp. L2(dp,,). Therefore
II v;mVell
(8.3.11) Since
~
IIlpllmli e ll.
V; are normal operators, we have II V; Veil = (V; Ve, V; Ve)1/2 = {V;*V; Ve, Ve)1/2
~
II V;*V;VeIl 1/2 Veil 1/2 = (V;V;Ve, 11
=
~
V;*V~Ve)1/2111 Veil 112 =
(V:'2 V;2 Ve, Ve) 1/21 11 Vell1!2 ~ 1
1
~ II V;*2 V;2 Veil II Veil 2 + 22. Taking into account (8.3.11) we obtain 111
(8.3.12) and, for m
I V; Veil -+ 00,
~
-+-+ ... +--
II Veil 2
1
2mlllplillell2m
21
therefore results
hence (8.3.13)
I V~II
~
IIlpll.
Since the set of all functions of the form (8.3.7) is dense in C(X) then, according to (8.3.13), we may extend the map lp -+ V; to a representation of C(X) on N. As VI = V; for f E A, the subspace N results spectral for the representation f --+ VI'
Function algebras
188
But
I
Vf is a completely non spectral representation, hence N = {O}. Therefore Ve = 0 for any e E XEL2(dll h). Since XE L2(dll h ) C C H2(dph), for any e E XEL2(dll h ) there exists a sequence In of functions in A, such that e = limL2 In. We have --+
o=
(Ve, h) = lim (Vln' h) = lim (Vfnh, 11)
=
= lim (Tfnh, h) = lim SInd/lh = Sedph for any e E XEL2(dll h). In particular
o = SXEdll h =
Illl E).
Therefore Ph is a Szego measure. The theorem is proved. 0 In the" following we shall study a class of representations of A on H for which the converse assertion is also true. The representation I --+ Tf of A on H will be called subspectral if there exists a spectral dilation qJ --+ Utp of I --+ Tf such that Ufh E H for any I E A and h E H. We obviously have
(I E A; hE H)
(8.3.14)
One easily verifies that any other spectral dilation of the subspectral representation I --+ Tf , which satisfies (8.3.14), coincides with q> --+ Ucp. In the following by spectral dilation of the subspectral representation I --+ 1/, we mean the (unique) spectral dilation qJ --+ Ucp of I --+ Tf which satisfies (8.3.14). Proposition 8.17. Let 1--+ Tf be a subspectral representation of A on Hand M C H a doubly invariant subspace to I --+ Tf . The representation I --+ TflM 01 A on M is subspectral. Prool. Let qJ --+ Ucp be the spectral dilation ofI --+ Tf and K the dilation space. If we put K(M)
= clm {Ucpm; mE M,
qJ E
C(X)}
then q> --+ UcpIK(M) is clearly a spectral dilation of I satisfies (8.3.14) for I E A and m E M. 0
--+
TfIM, which
Ch. 8. Elements of spectral theory
189
An important class of subspectral representations is the class of natural representations. Let p, be a positive measure on X, p(X) = 1 and N a Hilbert space. We have already seen in paragraph 7.3 how the natural representation q> ~ UqJ of C(X) on L2(N; dp) is constructed: (q> E C(X); k E L2 (N;
dp»
If we consider the elements of N as functions with constant values from X to N, we can consider N as a subspace of L2(N; dll). Let H't.(N; dp)
=
H't.(N; A; dp,) = elm [{ Ufn, f
It is clear that, for any f
E
E
A, n E N}J.
A, we have
If we write Tf = Uf IH2(N; dp), one easily verifies that f ~ Tf is a subspectral representation of A on H2(N; dp,), q> ~ UqJ being its spectral dilation. We call this representation the natural representation of A on H2(N; dJl). If N is one-dimensional we shall write H2(dJl) instead of H'l(N; dp,), and it is easy to see that this notation is in agreement with that of the previous chapters. N ow let f ~ Tf be a subspectral representation of A on Hand q> ~ UqJ its spectral. dilation. Let (P,k l • k 2 )kl • k2 EK be the spectral family attached to the representation q> ~ UqJ of C(X) on K(K is the dilation space) and Ph = Ph,h the spectral measure attached to h by f ~ Tf , for any hE H. We have
that is (8.3.15) For h E H, we denote by Mh the closed subspace of H spanned by Tfh, f EA. Using (8.3.15) we may uniquely define the operator Von M h , with values in L2(dJlh), such that
Function algebras
190
The operator V is an isometry from M" onto V(Mh). It is clear that V(M h) is invariant to the multiplication by functions of A and (8.3.16) Proposition 8.18. The closed subspace M of M" is spectral with respect to f ~ TfIM" if and only if there exists a Borel subset E of X such that VM = XEL2(dph). E is Ph-essentially unique determinedby M. Proof. Let VI = TIIMh. Assume VM = XEL2(dph). For mE M and qJ E C( X) we define U•tpm --
1.'-1 J'
cP If J' m.
Since XEL2( dph) is an invariant subspace to the multiplication by functions of C(X) , Utp are well defined and linear on M. Obviously
At the same time
Hence qJ ~ Utp is a representation of C(X) on M. From (8.3.16) there results
(fE A; For m
E
M and k
(V;m, k) =
E
=
M).
M" we also have
= (m, Vfk)
(Vm,fVk)
mE
=
(fVm, Vk)
(m, Tfk) =
=
(m, V-IfVk)
(V-I fVm, k)
=
= (U;m, k).
We used (8.3.) 6) and the fact that Vand V-I are isometries. Therefore M is doubly invariant to f ~ Vf and qJ ~ UqJ is an extension to C(X) of f -+ VI' i.e. the subspace M is X-spectral relative to f -+ Vf·
Ch. 8. Elements of spectral theory
191
Conversely, assume M is X-spectral relative to f -+ Vf and let cp -+ Ucp be the representation of C(X) on M, which extends / -+ V,IM. According to (8.3.16), V M is invariant to multiplication by functions of A. Let P be the projection of L2(dflh) on VM. For m, n E M we have
(vjm, n) = (m, Yin) = (m, Tin) = (m, V-I/Vn)
= (Vm,JVn)
=
=
(fVm, Vn) = (P/Vm, Vn) =
Hence (8.3.17)
V;m = V-IP/Vm.
Since V, are normal operators, we have
= I V-l/Vmll = IIJVml1
-
=
II/Vmll
for any mE M. Therefore fVM C VM for any f EA. Then VM is invariant to multiplication by any function of C(X) of the form cp = r.figi and, by continuity, to multiplication by any function cP E C(X). From Theorem 7.14 -there results VM = XEL2(dflh)' with E a Borel set, JLh-essentially unique. The proof is therefore complete. 0 Theorem 8.19. Let f -+ Tf be a subspectral representation of A on Hand flh the spectral measure attached to h by f -+ Ti . flh is a Szego measure if and only if the representation f -+ TflMh oj A on Mh is completely nonspectral. Proof. Since, obviously, V(Mh} C H2(dJLh) then, according toProposition 8. I 8, f -+ TflMh has no X-spectral subspaces different from zero, if and only if for any Borel set E with XEL2(dJLh) C H2(dJLh) we have JLh(E) = 0, that is, if and only if JLh is a Szeg6 measure.
192
Function algebras
Let Jl be a positive measure on X with Jl(X) = 1 and f -+ Tf the natural representation of A on H2(dJl). Taking as h E H2(dlt) the function identically equal to 1 on X, we have
hence Jl is the spectral measure attached, by the natural representation, to the element 1 of H2( dJl). Furthermore MI = H2(dJl) and the operator V defined above is the embedding operator of H2(dJl) in L2(dJl). Theorem 8.20. Let p. be a positil'e measure on X and f -+ Tf the natural representation of A on H2(dp.). (i) The subspace M of H2(dJl) is X-spectral if and only if M = = XEL2( dJl), where E is a Borel set of x. E is Jl-essentially unique determined by M. (ii) The natural representation of A on H2( dJl) is completely nonspectral if and only if Jl is a Szego measure. (iii) The representation f -+ Tf is spectral if and only if p. is Szego-singular. (iv) The measure Jl has a unique decomposition under the form p, = JlI + Jl2 where JlI is a Szego-singular measure and Jl2 a Szego measure. M oreOl'er
,
where the subspaces H2( dJlI) , H2(dJl2) are doubly invariant,f-+ TfIH2( dP.I) is the spectral part of f -+ Tf,f -+ TfIH2( dJl2) is the completely non-spectral part of f -+ Tf . The measure Jl2 is the supremum of the Szego measures v, such that v ~ Jl. Proof. (i), (ii) and (iii) are direct consequence of Proposition 8.18 and Theorem 8.19. It remains to prove (iv). Let
the canonical decomposition of the representation f -+ Tfo Since Hs is an X-spectral subspace, then, according to (i) it has the form
Ch. 8. Elements of spectral theory
193
with E a Borel set in X. Let JlI = XEJl. We have
Since any function of L2(dJl) which belongs to H2(dJl), belongs also to H2(dJ1l)' there results L2(dJll) = Hs = H2(dJll)' hence Jl is a Szego·singular meas ure. Let J1.2 = (l - xd Ji. Since XE E H2(dJl), there follows
Then
The representation f -+ TjlHc is therefore the natural representation of A on H2( dJ1.2); as it is completely non-spectral, Jl2 results a Szego measure. Hence (8.3.18)
Jl = J1.l
+ 112
with 111 a Szego-singular measure and 112 a Szego measure. Let v be a Szego measure, v ~ J1.. We have
and therefore veE) = 0, i.e. v ~ Jl2. Thus, Jl2 is the supremum of the Szego measures v, v ~ Ji. This proves also the uniqueness of the decomposition (8.3.18). The Theorem is proved. 0 The measure Jll is called the Szego singular part of Jl, and Jl2 the Szego part of Jl.
8.4. The Wold decomposition· Let A be a function algebra on X and Jl a representing measure for A. Recall we denoted A Jl 13 - c. 437
=
J
{f E A : f dl1 = O}.
Function algebras
194
In the following we suppose that Jl has the uniqueness property, i.e. any representing measure which coincides with J1. on A is equal to J1.. The subspectral representation f ~ Tf of A on the Hilbert space H is said to be J1.-spectral if elm [U TfH] I
= H.
f::. A,l
Proposition 8.21. Let f ~ Tf be a J1.-spectral representation of A on Hand M C H a doubly invariant subspace. The representation f - 4 Tjl-I\{ is a It-Spectral representation of A on M. Proof. According to Proposition 8.16, f ~ TflM is subspectral. Let m E M, orthogonal to [U TJ-A-/]. For any f E A JL and h E H of the form h = hI
+ h2'
lEA"
with hI E M, h2 E MJ., we have
Hence m is orthogonal to
which is equal to H, therefore m = O. 0 The subspectral representation f ~ Tf is called /l-completely non spectral if for any doubly invariant subspace M, for which f -+ TflM is jl-spectral, we have M = {OJ. Theorem 8.22. Let f ~ Tf be a subspectral representation of A on H and Jl a representing measure of A with the uniqueness property. The space H has a unique decomposition of the form
such that HI' H2 are doubly invariant subspaces for f ~ Tf , f ~ TflHI is a J1.-spectral representation andf ~ TflH2 is a J1.-completely Ilonspectral representation. HI is the largest doubly im'ariant subspace for which f~ TflM is a J1.-spectral representation.
Ch. 8. Elements of spectral theory
195
Proof. Let us write N
= [U
TfA]-L.
ftAp.
Let qJ -+ Utp be the spectral dilation off -+ Tf and (J1k h kzh 1 • k2 E K the spectral family attached to qJ -+ Utp (K is the space of spectral dilation). For any n E N andf E All we have:
according to the definition of N. Since
using the uniqueness property of j1, there results (n EN).
8.4. 1) Let us put HI
elm [U TfN].
=
lEA
It is clear that TfHI CHI for any fE A. If n E Nand g for any hE H we have:
E
All' then
hence
T*n g
(8.4.2)
=
0
(n EN, g E All)'
It results that Tt N C N for any g E A. Let now h E HI be of the form h = Tfn with n E Nand f EA. For any g E A we can choose a gk with fk' gk E A such that sequence fk
+
lim Slfk k -+ x"
+ Kk -
fil 2dJl
=
0,
Function algebras
196
since A
+ A is dense in L2(dJl) (Corollary 5.19). Since
we obtain lim (Tfk n
k .... oo
+ T:kn) =
T:Tfn.
Hence T:h
=
Tg*1jn = lim (Tfkn k .... oo
+ Tg~n) E HI
thus the subspace HI is doubly invariant. Let us write
where HI and H2 are doubly invariant subspace. We have
Indeed if m E HI is orthogonal to
then for any f E All and h2 E H2 we have
H of the form h
hE
therefore m is orthogonal to elm
[U
TfH]
fEAt~
which contains HI_ i.e. m = O.
= hI
+ h2
with
hI E
HI'
Ch. 8. Elements of spectra] theory
197
It results that I -+ T,IHI is Jl-spectral. Let M be a doubly invariant subspace of H such that
clm
[U
T,M]
=
M.
fEA/.1
I
E
Then it is easy to see that M is orthogonal to N. Then for any A, m E M and n E N we have (m, T,n)
= (T,*m, n) = O.
Hence M is orthogonal to H2 i.e. M C HI. Therefore HI is the maximal doubly invariant subspace M for which I -+ T,IM is Jl-spectral. It is now clear that I -+ T,IH2 is Jl-completely non-spectral. Let H= H{ €a H~
be another decomposition of H as required by the theorem. By the maximality of H2 we have H; C HI. Let us put
I t is then clear that M is a doubly invariant subspace and, since M C HI, from Proposition 8.21 there results thatl -+ T,IM is Jl-spectral. Since M C H~ we conclude that M={O}. Thus H; = HI, H~ = H 2 • 0
PropositioD 8.23. II Jl is not a point measure and M is a spectral subspace of I -+ T, then f -+ Til M is Jl-spectral. Proof. Let
n
and assume there is an hEM N, II h II = 1. Let Mil be the closed subspace generated by the elements of the form Tfh, f EA. As in the case of H2 from the previous theorem, we can prove that Mil is doubly invariant to f -+ Tj • As Mil C M and M is spectral, Mil is also spectral
Function algebras
198
and, according to Proposition 8.18, there exists a Borel set E, J1.h = = J1. - essentially unique, such that
where V is the operator defined on M h , with values in L2(dJ1.), by
But, clearly, VMh C H2(dJ1.), and therefore XE L2(dJ1.) C H2(dJ1.). According to Theorem 8.14, J1 is a Szeg6 measure, hence J1.(E) = O. There results Mh = 0 which contradicts the fact that hE Mh and IIhll = 1. Therefore
Now, let m
E
M be orthogonal to
For any h E H of the form h andl E AJl we have
This means that
mE
[U
=
hI
+ h2'
TfH~.l = N,
with hI
hence
mE
E
M, h2
M
nN=
E
Ml.
{O}.
fEAl1
I.e. m = O. 0
Corollary 8.24. Let 1 -+ Tf be a subspecfral representation 01 A on Hand J1. a measure which is not a point measure. If 1 -+ Tf is spectral, then it is also Ii-spectral. If.f -+ Tf is Ii-completely non-spectral, then it is also completely nonspectral. A subspectral representation 1 -+ Tf of A on H, J1.-spectral and completely non-spectral, will be called J1.-singular. Theorem 8.25. (Wold-Helson-Lawdenslager). Let f -+ Tf be a subspectral representation 01 A all Hand Jl a representing measure (l1,'hich
Ch. 8. Elements of spectral theory
199
is not a point measure) for A with the uniqueness property. H admits a unique decomposition under the form
such that H t , H 2 , H3 are doubly inl'ariant subspaces to f -+ ~,f --+ TflHt is spectral, f -+ TflH2 is p-singular and f --+ TflH3 is p-completely non-spectral. Proof. The proof follows immediately by applying successively Theorem 8.6, Corollary 8.24 and Theorem 8.22. 0 The following theorem furnishes some characterizations of natural representations and gives the reason for which Theorem 8.24 was called the Wold Theorem.
Theorem 8.26. Let f --+ Tf be a subspectral representation of A on Hand p a non-point mass representing measure of A with the uniqueness property. The following assertions are equivalent. a) There exists a Hi/bert space N such that f -+ Tf is unitarily equivalent to the natural representation of A on H2(N; dp). b) The representation f -+ Tf is p-completely non-spectral c) If M is a closed subspace of H such that Tt M C M for any fEA and -
M C elm [ U ~H], lEAp
then M = {OJ. d) If M is a doubly invariant subspace of H with M C elm
[U TfH] lEAp
then M = {OJ. e) If M is a doubly invariant subspace such that M
n [U
fEA/1
then M = {OJ.
TfH]J. = {OJ
Function algebras
200
Proof. Let us prove the implications a) -+ c). For this, let f -+ T, be the natural representation of A on H2(N; dJl). First we shall show that
[U
N =
(8.4.3)
T,H2(N; dJl)]l.
JEAJl
Indeed if n E N,f E AI" g
(T,gm, n)
E
A and mEN, we have:
= (fgm, n) = I (f(x) g(x) m, n) dJl(x) =
=
(m, n) Ijg dp = O.
Conversely, if hE H2(N; dp) is orthogonal to Nand
[U
T,H2(N; dJl)],
fEAJl
then for any n EN and g
E
A, if we put
f
= g -
I gdp,
we obtain (h, gn) = (h,jn) =
(h, 1Jn)
+ (h, I gdf.l n) =
+ I gdp(h, n) =
O.
Let now M be a closed subspace of H2(N; dJl) as in c). From (8.4.3) M results orthogonal to N. Then for any m E M, f E A and n EN we have
(m, fn) = (m, T,n) = (T/m, n) = 0 i.e. m = O. Hence M = {O} and implication a) -+ c) is proved. c) -+ d) and d) -+ b) are obvious. b) -+ e). Let M be a doubly invariant subspace as in e). Then we have (8.4.4)
elm
[U 1jM] = fEAJl
M.
Ch. 8. Elements of spectral theory
201
Indeed, if m E M is orthogonal to
then for any hE H of the form h = hI andfE A we have
+ h2'
with hI EM, h2 E Ml,.
Thus m
E
[U
TfH]l
f.:. A Il
nM
i.e. m = O. Hence M verifies (8.4.4) and from b) we obtain M = {O}. This completes the proof b) -+ e). e) -+ d) is obvious. To complete the proof of the theorem it remains to prove b) -+ a). ]f b) is true then from Theoreln 8.22 there results that
where
Let (Ilk l • k2 )k l • k2 eX be the spectral family attached to the spectral dilation qJ -+ U" off -+ Tf . Using (7.1.3) and (8.4.1), for any n, mEN, We obtain
Function algebras
202
Then for any
hE
H of the form ~ Tfin j • with
Ii E A
and ni
E
N,
i=1
we have:
p
I;
-
p
IliljdPninj =
p
-
~ Ililini' nj) d/l = I ~ lilini' n) dp = i, j=l
i,j=l
-
i,j=l
Now it is clear that the map TIn ~ In may be extended to an isometric operator U from H on H2(N; dp) such that ({EA, hEH).
This completes the proof of the theorem. Hk(with H" = H for any k). k=l .......
n
If h = (h k ) is a vector in EE> Hk, then k=l
""-
hence (9.1.2) is equivalent to the positivity of the operator T. We show that T = W*DW, where and 15 are operators on
tv
n
""-
EE> Hk and D is positive. k=l
Ch. 9. Elements of prediction theory
215
We put Wij
={
T gj-1 gj.. if i
o
~
j
if i > j
and
Du = I D·· 'J
then
W=
= 0 if i:l=}'
(Wij),
D=
(Dij).
Since Tg are contractions, jj is, clearly, a positive operator. We now show that T = As T and W* DWare self-adjoint operators, it is sufficient to prove that Tij = (W* D W)ij for i ~ j. At the same time (W*)ij = ~~ = 0 for j > i, hence
w*nw.
........,
-. . . . . .
(w*nW);j
=
i.................. .........
~ (W*)iiDW)kj. k=1
For i
=
1 we have
and for i > 1 ........ . . . . . --..
i
.....
(W*nW)ij = ~ (W*)UDllWIj 1=1
=
Function algebras
216
But from (9.1.3) and (9.1.4) there results
T*
~-1
,
g i
T* g-1
- T* 8-1 g
g -
1-1'
1-1
i
and therefore
This is nothing else than
which yields
(t'h, h) = (w*n wh, h) = (DWh, Wh) ~
o.
Thus, T is a positive operator, hence g --+ Tg satisfies (9.1.2). The theorem is completely proved. Let G, S, H be as above. A semigroup of contractions on H is a map s --+ Ts of S in L(H), such that Tl = I, TSI TS2 = T S1S2 ' for any SI' S2 E S and II Tsil ~ 1 for any s E S. A unitary representation g --+ Ug of G on a Hilbert space K, IS called a unitary dilation of the semigroup {Tsh eS, if H C K and (s E S, hE H),
where P is the orthogonal projection of K on H. Corollary 9.6. If G is totally ordered by S, i.e. if G = SU S-l , then allY semigroup {Ts}s eS of contractions on H has a unitary dilation.
Ch. 9. Elements of prediction theory
217
Proof. Indeed, jf we write
g=
SE
S
then we obtain a map g -+ Tg of G in L(H) which, obviously, satisfies the conditions of Theorem 9.5. Therefore g -+ Tg verifies (9.1.2), hence admits a unitary dilation. Theorem 9.7. Let G, S, H be as written above, with G - SS-l. Let {Ts}sES be a semigroup of contractions such that
(9.1.5) and
T*T =STG T* G S
(9.1.6)
for
SO'-14q;; S U S-1 •
Then there exists a unitary dilation of the semigroup {Ts} S ES' Proof. Let us write
T:TS { T= g
T* G
if g =
0'-1, O'E
S.
According to (9.1.5), the mapping g -+ Tg is well defined and Tl = I, Tg-l=T:, IITgll ~ 1. We now show that it satisfies also (9.1.3). Letgb g2 EG be such that gl' g2, glg~ ~ S-I. We put gl = slO'l\ g2 = SO'il. Since gh g2' g1g2 ¢ S-1 we have
If gh g2
E
S, then (9.1.3) is clearly satisfied. Suppose g2 ~ S. Then
Function algebras
218
and, using (9.1.6), we obtain (9.1. 7) Now taking into account (9.1.7) we get
Similarly, we have
Therefore (9.1.3) is verified by the map g 4> Tg • Now by Theorem 9.5, here exists a unitary dilation for the semigroup {Ts} s ES' 0 Let G, S, H be as written above and {Ts}sES a semigroup of contractions on H. A closed subspace M of H is called invariant (to {Ts}s ES) if TsMCM for any s E S. The subspace M is said to be doubly invariant if TsMCM, T;MCM for any s E S. If M is invariant, we denote by TslM the restriction of Ts to M and by {TsIM} s ES the corresponding semigroup of contractions. The semigroup of contractions {TsL ES on H is called unitary if the operator Ts is unitary on H for any s E S. The semigroup {Ts}sES is called completely non-unitary if for any doubly invariant subspace M of H for which {TsIMLES is unitary we have M = {OJ. Theorem 9.8. Let {Ts} s ES be a semigroup of contractions on H. The space H has a unique decomposition under the form
such that Hu and He are doubly invariant, {TsIHJsES is unitary and {TsIHcJ s ES is completely-non-unitary. Proof. Let.A be the family of doubly invariant subs paces M of H, with the property that {TsIM}sES is unitary.
Ch. 9. Elements of prediction theory
219
U M.
One easily verifies
Let Hu be the subspace generated by
MEV«
that Hu is doubly invariant and {TsIH,,} s ES is unitary. Hence Hu is the largest doubly invariant subspace M, for which {TsIM}sES is unitary. Let
He results doubly invariant. Let M C He be double invariant, with {TsIM}sES unitary. Then MCH" and, since it is orthogonal to H", we have M = {OJ. Therefore {TsIHe} s ES is a completely non-unitary semlgroup. Let
be another decomposition of H, with HI, H2 doubly invariant {TsIH1}sES unitary and {Ts IH2 } s ES completely non-unitary. Hu being maximal there results HI C H u • Let us write
Then M is doubly invariant and, as MC H u , {TsIM}SES results unitary. But M is orthogonal to HI' hence Me H2 and, as {Ts IH2 }sES is completely non-unitary, M = {O}. Thus Hu = HI and Hs = H 2 , i.e. the decomposition is unique. The theorem is therefore completely proved. 0, such that XEL2(dJl) C H2(dJl). Therefore lEL2(dJl)nH2(dJl) = ZEL2(dfl) #- O. Since
Function algebras
236
J1 is a Szego-total measure, we have J1(E) = 1, hence L2(dJl) = = XEL(dJl) C H2(dJl) C L2(dJ1), that is L2(dJl) = H2(dp). This yields XEL2(dJl) C H2(dJ1) for any Borel set E, hence peE) = 0 or Jl(E) = I for any Borel set E of X; that means Jl is a point measure which contradicts the fact that J1 is a Szego-total measure. 0 Theorem 9.24. Let J1 be a positive measure on X, Jl (X) = 1, and f -+ Tf be the natural representation of A on H2( dJ1). Then (i) If Jl is a Szegii-singular measure, f -+ Tf is a spectral representation. (ii) If Jl is a Szegii measure and N = [ U T:Ts H2(dJl)]J- = O,f-+ Tf a-1sffS- 1
is m-singular. (iii) If Jl is a Szego-total measure and N :f= 0, f -+ Tf is m-completely nonspectral. Proof. Points (i) and (ii) result from Theorem 8.19. Point (iii) follows from Corollary 9.17 and Proposition 9.18. We know that, for (i) and (ii) the converse assertions are also true. As concerns point (iii), we have the following statement: Proposition 9.25. If f -+ Tf is m-completely nonspectral, then for any Borel set E of X, with XE E H2(dJ1), we have Jl(E) = 0 or J1(E) = 1. Proof. Let E be a Borel set with XE E H2(dJl). Then M = XEH2(dJ1) is a doubly invariant subspace of H2(dJl). Indeed, TgM eM for any g E A. Let hEM and gh = hI + h2 with hI E M and h2 E XE L2(dJl) orthogonal to M. We have
Jkh dJl = JXEkh2 dJl = 2
0
for any k E H2(dJl), hence h2 E H2(dp)l.. There results Tg*h = Pgh = = hi E M, and therefore Tg* M C M. Thus M is a doubly invariant subspace and, since f -+ Tf is totally nonspectral, from Corollary 9.17 we get M = {O} or M = H2(dJl), i.e. Jl(E) = 0 or J1(E) = 1. 0 We now show that Theorem 9.24 may be interpreted as a prediction theorem. Let K be a Hilbert space (K may be, for instance, t~e space L 2(dP) of square integrable functions on a probability space)~-- G an abelian group and S eGa subsemigroup of G such that G = SUS-I, SnS- 1 = {l}. . We say that the map g -+ Fg of G in K is a G-stationary process, if (9.5.1 )
237
Ch. 9. Elements of prediction theory
The prediction problem relative to the future S, for such a process,
is that of approximating an element Fg by linear combinations of elements Fy with g-l yeS. More precisely, let us suppose, without any loss in generality, that the space :f( is the closure of the set spanned by the elements Fg , gE G. For any gE G we put Jig
If gl Jl gt
C
=
clm [{Fy; Y E G, g-l}, e S}].
~ g2'
Jig!,
i.e. if g2gl -1 E S, then gil yeS yields gil yeS, hence Thus, {..II g } is an increasing family of subs paces of .it.
Let us write :fl
=
n .A
g•
KEG
Let Eg be the orthogonal projection of Fg on the orthogonal complement of "lt g. For any g E G, Eg is orthogonal to:f l and {Eg; g E G} is an orthogonal set. It spans the closed subspace :f3 of :ff, orthogonal to :fl' We put (9.5.2) We say that g~ Fg is a deterministic process if % = %1' If % = %3 the process is called innovation process and if :f = :f 2 the process is called evanescent. The prediction problem is, then, nothing else than the characterization of the deterministic, evanescent and innovation parts of a process. Let us define on G the real function p, by (g
E
G).
p is a positive definite function on G. Indeed, according to (9.5.1)
we have ~ CjC;p(g;l gi)
=
~ cjc; (F
-1
gi
• gj
Fl )
=
Function agebras
238
for any finite system cl , ... , Cn of complex numbers and any finite system gl' ... ' g,. of elements in G. Following the Herglotz-Bochner-Weil theorem, there exists a positive measure lIon the dual X of the discrete group G, such that
peg)
=
SgdJ-l
(g E G),
where on the right handside g is viewed as character on X. We now define the operator U on :f{, with values in L2(dJl), by (g E G).
(9.5.3) Since
U may be defined by (9.5.3) and is a unitary operator from :f{ to L2(dJl) U realises an equivalence, from the point of view of the prediction theory, between the G-process g -+ Fg on :f{ and the G-process g -+ F~ = = g, on L2(dJl). Indeed, if we write
then
Let us remark that }Jg = elm [{y: y E G, y = gs, S
E
S, s #- 1}].
Let A be the algebra generated by S in C(X) and f -+ Tf be the natural representation of A on H2(dJl). We denote by m the Haar measure of X, viewed as the dual group of the group G.
Theorem 9.26. The process g -+ Fg is deterministic (evanescent, innovation) if and only if the natural representation f -+ Tf of A on H2(dJl) is m-spectral (m-singular, m-completely non-spectral).
Ch. 9. Elements of prediction theory
Proof. Let K
239
= L2(dJI) and H = H2(dJI). If we denote by Ug the
operator of multiplication by g in L2(dJl), then g ~ Ug is the unitary dilation of the semigroup {Ts} sES which corresponds to the natural representation f --+ Tf of A on H2(dJI). Let and
H
=
HI (±) H2 (±) H3
be the Wold decomposition of the semigroup of isometries {Ts} sESe Since MI
=
elm [{ S:
S
E S, s i= I}]
there results MI C H, hence KI C H. By the definition of M g we get that, for any S E S, S i= 1 and h E H, we have gsh E Mg. Therefore, UgKI CKI for any g E G and, from (9.1.8) we obtain KI CHI. Conversely, since HI reduces any operator U g to a unitary operator, for any hE HI and g E G there exists hI E HI such that h = gShI' with EE sS, s i= 1, i.e. h E Mg. Then HI C K I , hence KI = HI. Since the natural representation f --+ Tf is spectral if and only if HI = H2(dJI) = L2(dJI), then f --+ Tf is spectral if and only if KI = = L2(dJI), that is if and only if the process g --+ Fer is deterministic. Let now e g be the projection of g on the orthogonal complement of Mg in K. As Ug is a unitary operator on K, there results (9.5.4) Assume now that f
--+
Tf is m-singular. Then
1 E elm [ U T s H2(dJI)] = MI. s+=1
Hence el = 0 and, from (9.5.4) we get eg = 0 for any g E G. Therefore Ka = O. At the same time from HI = 0 there results KI = 0, that is g --+ Fg is an evancescent process. Conversely, if the process is evanescent, then KI = K3 = {Ole From K3 = 0 there results el = 0 and therefore 1 E Ml = elm [ U TsH]. S1=1
Function algebras
240
f
-+
On the other hand, as KI = 0 implies HI = 0, we obtain that T, is m-singular. Let M be a subspace of H, doubly invariant to f -+ T" such that elm [ U TsM]
=
M.
s+1
According to a previous remark, M C MI' Then el is orthogonal to M. Since I E H2(dJl) and Ml C H2(,dJ1.), there results el E H2(dJ1.). We have
(m, e s ) = (m, sel )
= (sm, el ) =
(Tsm, e l ) = 0
for any s E S, where P is the projection of L2(dJ1.) on H2(dll). Therefore M is orthogonal to K 3 • If the process is innovation, then M = {OJ, that is f -+ T, is completely non-spectral. If f -+ T, is completely non-spectral then H2(dJ1.)
=
(±) TsN sES
where N = Mr is one-dimensional. Then, clearly, L2(d/l)
=
tt> UgN. gEG
Let k E L2(d/l) be orthogonal to eg for any g E G. As e1 EN, el i= 0 and N is one-dimensional, we have ng = cgel for any ng E N, with cg constant, and (k, gn g)
=
(k, Cggel) = cik, gel)
=
cik, eg} = O.
Hence k is orthogonal to UgN for any g and, from (9.5.4), we obtain k = O. Thus, we have L2(d/l) = K 3 , i.e. g -+ Fg is an innovation process. The theorem is proved. 0 In the case G = Z is the additive group of integers and S = Z+ the semigroup of positive integers, Theorem 9.24 gives exactly the
Ch. 9. Elements of prediction theory
241
classical prediction theorem for stationary discrete processes. The following two theorems are precising this fact. In this case we know that X is the unit circle {Izl = I} of the complex plane, m is the normalized Lebesgue measure on X and A the standard algebra on X. Theorem 9.27. Let A be the standard algebra on X = {Izl = I} and m the normalized Lebesgue measure on X. A positil'e measure Jl. on X is a Szeg5 measure if and only if it is a Szego-total measure. This happens if and only if dJl = hdm with h ELI ( dm) and (9.5.5)
Jlog h dm >
-
00.
Proof. First observe that, in this case, the function 1 belongs to the L2(dJl.)-closure of A~ if and only if L2(dJl). H'I(dJ.l). Let dJ.l = hdm + dJl.s be the Lebesgue decomposition of J.l with respect to m. If E is the support of Jl.s then, from the Szego theorem and the above remark, there results (9.5.6) Suppose Jl. is a Szego measure. Then (9.5.6) yields J.l(E) = 0, that is Jl.s = O. Hence dJ.l = hdm. As L2(dJl) + H2(dJ,l), once more from the Szego theorem and the above remark, we get
Jlog hdm >
-
00.
Now let dJ,l = hdm where h satisfies (9.5.5). The Szego theorem yields H2(dJ.l) ~ L2(dJ,l). Let E be a Borel set and
Then h- 1h1l. E E H2(dm) and we get m(E) = 0 or m(X - £) = 0, hence p, (E) = 0 or p,(X - £) = 0, i.e. J,l is a Szego-total measure. Since a total Szego measure is a Szego measure, the theorem is completely proved. 0 Theorem 9.28. Let f ....... Tf be the natural representation of the standard algebra A on H2(dJ1}. f . . . . Tf is an m-completely non-spectral representation if and only if J,l is a Szego measure. 16 - c. 431
Function algebras
242
Proof. There remains to prove that if Jl is a Szeg6 measure, then N
= [
U
Tf H2(dJI)1l. # O.
lEAm
Indeed, if N = 0 then I belongs to the L2(dp)-closure of Am and therefore the Szeg6 theorem yields H2(dp) = L2(dp); but this contradicts the fact that Jl is a Szeg6 measure. The theorem is proved. 0 Summing up all these results we now obtain the prediction theorem for discrete stationary processes.
Theorem 9.29. Let n -+ Fn be a discrete stationary process given by a positive Ineasure p on the unit circle X = {Izl = 1} of the complex plane, and m the Lebesgue measure on X. Let dp = hdm + dps' hELl(dp) be the Lebesgue decomposition of p with respect to m. 11 -+ Fn is a deterministic process if and only if
Slog It dm n
-+
Fn is an innovation process
=
-
00.
if and only if dp
Slog h dm > -
=
hdm and
00.
In the last case the prediction is made with a non-zero error el given by Szego formula
el = exp
[~
jIoghdm
l
Notes The first proof to Theorem 9.1 has been given by M. A. NAIMARK [1]. Other proofs as well as essential generalizations to non-commutative case and to semigroups are due to B. SZ.-NAGY [1], [2]. Theorem 9.5 has first been proved in the case of the additive group of integers and of the semigroups of positive integers by B. SZ.-NAGY [l], [2], by means of complex functions methods. The proof given here has been first done by I. HALPERIN, also in the case of integers group. W. MLAK [3] has rewritten the same proof for the case when G is totally ordered by S. The general form appears in I. SUCIU [7].
Ch. 9. Elements of prediction theory
243
The theorems of canonical decomposition for semigroups of contractions have their origin in the works of B. SZ.-NAGY and C. FOIA~ [I], [2], [6], where they are proved for the semigroup of positive integers or strongly continuous semigroups of real parameters. Important contributions in this direction have appeared in W. MLAK [1], [2], [3]. The characterization of the unitary space given by Theorem 9.9, in the general case, is given in 1. SUCIU [4 J. Dilation theorem for semigroups of isometries has been given by T. ITO [1 ]. Wold type decomposition theorems for semigroups of isometries have their origin in H. WOLD [1]. A first Wold type decomposition in three terms has been proved by H. HELSON and D. LOWDENSLAGER [2]. Theorems 9.11, 9.12 as well as the characterization of the semigroups of unilateral translations given by Theorems 9.13 and 9.14 are expounded in 1. SUCIU [9]. The results of paragraph 9.4 appear in I. SUCIU llO] together with Theorem 9.24. The formulation of the prediction problem of paragraph 9.5 follows H. HELSON and D. LOWDENSLAGER [1]. Theorem 9.26 which establishes the connection between Helson and Lowdenslager prediction theorem and Theorem 9.24 is unpublished. Theorem 9.7 appears in C. FOIA~ and 1. SUCIU [1]. Theorem 9.29 is the classical prediction theorem for discrete stationary processes (cf. J. L. DOOB [1 Concerning prediction theorems see also G. LUMER [2].
n.
CHAPTER 10
Some examples in the spectral theory of non-normal operators
10.1. The case of a single contraction Let X = {z E c: Izl = I} be the unit circle in the complex plane and A the standard algebra on X, that is the algebra of continuous functions on X which can be analytically extended in the interior of the unit disk D = {z: Iz\ < l}. It is known that A is the uniform closure P(X) of the polynomials in z and that A is a Dirichlet algebra on X. At the same time, S = {l, Z, Z2, ••• }, where z is the function f E A defined by f (x) = x, X E X, is a closed mUltiplicative system of inner functions in A, analytic free, which generates A. If we put G = SS, then X is the dual of the discrete group G, the normalized Lebesgue measure is the Haar measure m of X, and A
= {IE C(X): Jz"fdnl = 0, n
=
I, 2, ... }.
Let H be a Hilbert space and T a contraction on H, i.e. T E L(H), "Til ~ I. We write Ts = Tn for S E S, S = zn. We thus obtain the semigroup of contractions {Ts} sES on H. Since G = SUS we obtain the map g -. Tg of G in L(H) by Ts = T:. According to Proposition 9.5, g -. Tg is a positive definite extension of the semigroup {Ts}sES to G. Then from Corollary 9.3 there results that for any linear combination ~CiSi of elements in S,
Function algebras
246
we have
that is for any polynomial p(z) !!p(T)!1 ~ sup !p(z) I
(10.1.1)
!z: =1
which is von Neumann formula for a contraction. Since A = P(X). f(T) can be constructed for any f E A by using (10.1.1). If we put Tf = J(T) then it is easy to see that J ~ Tf is a representation of A on H such that T z = T. It is clear that any representation J ~ Tf of A on H may be obtained in the same way starting from the contraction T = Tz of L(H). T is called completely non-unitary if the only subspace which reduces T to a unitary operator is {O}. One easily verifies that T is unitary (completely non-unitary) if and only if {TsL ES is a unitary (completely non-unitary) semigroup, hence if and only if f ~ Tf is a spectral (completely non-spectral) representation. From Theorem 8.6 or 9.8 and Theorem 9.9 there results Theorem 10.1. Let T be a contraction a unique decomposition of the form
011
H. The space H has
where Hu reduces T to a unitary operator and He reduces T to a completely non-unitar.v contraction. We have
Hu = {h
E
H: IITnhll = IIT*nhll
=
Ilhll,
11 =
1, 2, ... }.
F olowing Proposition 9.2 there exists a uniquely determined semispectral measure (F(U))aE B(X) such that (n
= 0, 1, ... ; h, k
E
H).
Ch. 10. Examples of spectral theory
247
Let (Ph ,k)h ,k e H be the semispectral family attached to the semispectral measure (F(O'))aEB(X) and J1h = J1h,h' hE H. From Theorem 8.16 and Theorem 9.7 there results Theorem 10.2. Let T be a completely non-unitary contraction on Hand (Ilh ,k) h,k E H the semispectral family on X attached to T. For any h E H the measure Ilh is absolutely continuous with respect to the Lebesgue measure m on X and
According to Corollary 9.5 the semigroup {TsLES generated by T admits a unitary dilation g -+ U g. The unitary operator U = Uz is called the unitary dilation of T. Therefore Tnh = PUnh
(h
E
H, n
=
1,2, ... )
p.
is the orthogonal projection of K (the dilation space) on H. It is immediate that T is an isometry if and only if {TslsES is an isometric semigroup or if and only if f -+ Tf is a subspectral representation. In this case we have:
where
elm [
U T:TsH] = clm [ U TfH] = TH.
so-lEES- 1
fe Am
Therefore {Tslses (f -+ Tf ) is a quasy-unitary semigroup (m-spectral representation) jf and only if it is unitary (spectral). At the same time {Ts} sES (f -+ Tf ) is totally-non unitary (m-completely non-spectral) if and only if it is completely nonunitary (completely non-spectral) and the singular semigroups (m-singular representations) do not appear. The decomposition theorems of Wold Type (Theorem 8.25, Theorem 9.12) reduce in this case to the classical Wold theorem. Theorem 10.3. (Wold). Let T be an isometry on H. Then H admits a unique decomposition under the form
Function a1gebras
248
where Hureduces T to a unitary operator and the restriction of T to H t is unitarily equivalent to the shift (translation) operator in H2( N) with N = [TH).l .
. Then Theorem 8.21 and Theorem 9.19 give Theorem 10.4. The isometry T on H is unitarily equivalent to the shift (translation) operator on H2(N). with N one--dimensional, if Qnd only if T is not unitary and has no proper doubly invariant subspaces.
10.2. Operators having spectral sets with connected complement Let X be a compact set of the complex plane contained in the boundary of the unbounded connected component of its complement. Let A = P(X) be the function algebra on X which is the uniform closure in C(X) of the set of all polynomials in z (z is the function f E E C(X) defined by f(x) = x for x E X). We recall that A is a Dirichlet algebra on X (Theorem 6.26). We also know that the maximal ideal space of X may be identified too with the polynomial convex hull X of X. We have A
A
X =
XU
G1
U
G2 U···
where G1 , G2 ,. •• , are the bounded connected components of the complement of X. One easily verifies that G1 , G2 ,. •• are the non point Gleason parts of X (relative to A) and any x E X forms a point Gleason part. A
Theorem 10.5. Let f
Tf be a representation of A on the Hilbert space H. Then H admits a decomposition under the form -+
where H s ' H", h = 1,2, ... are doubly invariant subspaces, f is spectral and for any k = 1, 2, .. . ,f -+ 1JIH" is G,,-continuous.
-+
TfIH.
Ch. 10. Examples of spectral theory
249
Prool. Let (G(I.)(l.E.1 be the family of all Gleason parts of X" relative to A, and H
= ES HfI. ES Ho (I.E.!
be the decomposition of H relative to I -+ T, as follows from Theorem 8.28. Therefore Ho,H«, (l E J are doubly invariant subspaces, 1-+ .T,IH« is Gtz-continuous, (l E J, andl -+ T,IHo is singular. If G!I={x} we denote Htz by Hx and write
Hs = EB Hx ES Ho· xeX
We now show that I -+ T,IHs is spectral. It is. sufficient, for this, to prove that I -+ TflHo and I -+ T,IHx, x E X, are spectral. Let (Ph ,k)h ,k E H be the semi-spectral family attached to I -+ Tf · If h, k E Hx , then Ph ,k is absolutely continuous with respect to ex and, since Ph,k(1) = (h, k), there results (l 0.2.1)
Ph,k(CP)
=
ip(x) (h, k)
Hence it is immediate that (ilia ,k)1a ,k E H:Ie generates a representation of C(X) on Hx which is an extension of I -+ T,IHx, that is I -+ 1/IHx is spectral. Now let h, k E Ho. We then know that I1h,k is singular with respect to any representing measure for A. For any I, g E A we have
Therefore the measure gdjl" ,k - dl1TgIl ,k is orthogonal to A. Since, obviously, it is singular with respect to any representing measure for A, from Proposition 6.34 we get
(f E C(X), g
(10.2.2)
. Let now g
E
A).
Tg be the positive definite map of C(X) in L(H0) given by the semi spectral family iP",l)",lEHe' Clearly g -+ Tg is an exten-+
Function algebras
250
Slon of I
-+
Tf . From (10.2.2) there follows
for any I E A and g E C(X). Therefore dph.T;k - g dph,k IS an orthogonal measure to A and, as it is singular with respect to any representing measure of A, it is eq ual to zero. Hence (10.2.3)
(h, k E Ho,
I, g E C(X».
From (l0.2.3) there results
for any h, k E Ho, i.e. g -+ Tg is multiplicative. Then I -+ TflHo is a . spectral representation. Since a direct sum of spectral representations is a spectral representation, I -+ TflHs is also spectral. The theorem is pr.oved. 0 Let T be a bounded operator on H. Assume that for any polynomial p in z we have (10.2.4)
/lp(T)/I ~ sup Ip(x)!. XEX
Then it is clear that for any I E A we can construct I(T) and, writing Tf = I(T), I E A, we obtain a representation 1-+ Tf of A on H, with Tz = T. One easily verifies that I -+ Tf is a spectral representation if and only it T is a normal operator and its spectrum is contained in X.
Ch. 10. Examples of spectral theory
In this case the spectral measure attached to f measure of T.
---+
251
Tf is just the spectral
Corollary 10.6. Let T be a bounded linear operator on H which verifies (10.2.4). For any k = 1, 2, ... , let mk be a representing measure with support in X, of a point in Gk • The space H admits a decomposition under the form
where Ho reduces T to a normal operator To with spectrum contained in X and its spectral measure singular with respect to m k , k = 1, 2, ... , H k, k = 1, 2, ... , reduces T to an operator which admits an absolutely continuous semispectral measure with respect to mk'
10.3. Finite system of commuting contractions
n {z n
Let X =
E
e: Izl =
I} be the n-dimensional torus in
en.
1
We denote by x = (Z1, Z2"",Zn) the points of X and by Zi the coordinate functions on X, i.e. Zi(X) = Zi' i = I, 2, ... , n. Let S be the closed multiplicative system of inner functions on X generated by Z1, Z2,"" Zn and G = SSe G is an abelian group, S is a sub-semigroup of G with S S-1 = {I} and X is the dual of the discrete group G. Let A = A(S) be the function algebra on X generated by S. Therefore A is the closure in e(X) of the algebra of polynomials in the variables
n
Z}, Z2,""
Zn'
Let T}, T2 , ••• , Tn be a system of n commuting contractions on a Hilbert space H. If for any s E S, S = Z~l Z~2 ••. z~n, we put Ts = = T~l T;2 ... T!n we obtain a semigroup of contractions {Ts}sES on H. Relative to the semigroup {Ts} sES it is natural to ask the following questions: I) Does a unitary dilation g ---+ Ug of {Ts} sES exist? 2) Does a representation f ---+ Tf of A on H, which extends the . '? semIgroup 0 and, since V is an isometry on K+, there results Vnl L -L V n2 L for any n1 ¥= n 2 From the obvious relation
+ ... -+-
vn -1 (V - T) h
there follows (10.3.3) and from (10.3.3) one has (10.3.4) Theorem 10.7. Let Z be a bounded linear operator on H with ZT = TZ. There exists a bounded linear operator Y on K+ with YV = V Y and such that
=
Function algebras
254
(i)
Tnzm h = P + V"ymh
(ii)
Y(K
e
H) C H +
(iii)
I YII
=
I Zll·
Proof. We may assume
e
IIZII
(/z
E
H,
11,
m
=
0, /,2 .. ) -.
H
=
I and look for an Y of the form;
(10.3.5) where Bo, BI , B 2 , ... are bounded linear operators from K+ into L. We shall determine Bo, BI , B 2 ,. .. such that Y satisfies (i), (ii), and (iii). Let us define the operator B acting on K + with values in L, by
We have IIBkll2 = II(V -
=
«V -
T) ZP +k1l2 = «V -
T) ZP +k, VZP +k) =
T) ZP +k, (V -
I VZP +k1l2
T) ZP +k) =
- (TZP +k, VZP +k)
=
where we used (10.3.2) and (10.3.4). Hence (10.3.6) Since IIZI1 = I, the operator IK + - P +Z*ZP + is positive on K+. Let Do be the positive square root of this operator. Then
255
Ch. 10. Examples of spectral theory
and by (] 0.3.6) we have
that is (10.3.7) The inequality (l0.3.7) allows us to define a contraction Co from K+ inio L such that
B
= CoDoV
as follows: we write CoDoVk = Bk on the subspace DoVK+ of K+, then extend by continuity to the closure of Do VK+ and, for k in the orthogonal of DoVK+ in K+ we write Cok = O. Let
Bo = CoDo·
Bo is an operator defined on K + with values in L. We have Bo V = B and .~~.
that is (I0.3.8) We now construct by induction the sequence B o, B1 , B2 , ••• of operators on K + with values in L such that (l0.3.9)
(n
= ], 2, ... )
and (10.3.10) UZP +kI12+ ~ IIBnkl12 ~ IIkl1 2 O~n (dm). Math. Zeitschr. 106 (1968), 261- 266
MERRILL,
MLAK,
W.,
[l] Representations of some algebras of generalized analytic functions. Bull. A cad. Polan. Sci. (ser. math. astr., phys.), 13 (1965), 211-214
[2] Unitaty dilations of contraction operators, Rozprawy mat. 46 (1965), 1-88 [3] Unitary dilations in case of ordered groups. Ann. Pol. Math. 17 (1965), 321-328 [4] Positive definite contraction valued functions. Bull. A cad. Polan. Sci, (ser. math., astr., phys.), 15 (1967), 509-512 [5] A note on Szego type properties of semi-spectral measures. Studia Math. 31 (1968), 241-251 [6] Decompositions and extensions of operator valued representations of function algebras. Acta Sci. Math. 30 (1969)" 181-193 MOCHIZUKI, N., [l] A characterization of the algebra of generalized analytic functions. Toh. Math. J. 16 (1964), 313-319 [2] Isometry between HP(dm) and the Hardy class HP. Toh. Math. J. 18 (1966) 311-315
270
Function algebras
[3] Correction to "Isometry between HP(dm) and the Hardy class HP·'. Math. J. 19 (1967), 373
To"~
P. S., [1] A structure theory for isometric representations of a class of semigroups,. J. Reine Angew. Math. 255 (1972), 135-154
MUHLY,
M. A., [1] Positive definite operator functions on a commutative group. Izvestia Akad~ Nauk SSSR, 7 (1943), 237-244 (Russian) [2] On a representation of additive operator set functions. Doklady Akad. Nauk SSSR, 41 (1943), 359-361 (Russian) [3] Normed Rings. GITTL., Moscow, 1956
NAIMARK,
M., [1] Mathematical Analysis, I, II, III. Ed. tehnica, (Romanian)
NICOLESCU,
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1957, 1958, 196{)
PARROTT, S.,
[1] Unitary dilations for commuting contractions. Pacific J. Math. 34 (1970',. 481-490
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