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. Thus we obtained a T(H)-valued gramian [■, -]x on X. Clearly, it holds that (i) [x, x)x > 0, and [x, x]x = 0 iff x = 0; (ii) [x + y, z]x = [x, z]x + [y, z\x\ (iii) [ax,y]x iv
=
a[x,y]x;
( ) [x,y]*x = [y,x}x, for x,y,z € X and a € B(H), the algebra of all bounded linear operators on H. Since X is a left _B(#)-module under the module action (a,x) i-> a - x = ax for x € X and o e B(H), and is a Hilbert space with a T" (if)-valued gramian, we
1.3. MULTIDIMENSIONAL AND OTHER EXTENSIONS
13
call it a normal Hilbert B(H)-module. As we shall see later (Sections 2.1 and 2.3) Ll(il; H) and S(L\{Q), H) are isomorphic as normal Hilbert B(H)-modules by the isomorphism U : x H-> T* where Tx is defined by (3.3) (cf. (3.4) and (3.5)). We shall make full use of this space later in stochastic analysis. X = LQ(U ; H)-valued operator stationary processes are defined similarly as in the case of [Lg(fi)]"-valued processes. Now we want to define X-valued harmonizable processes. Let {x(t)} be an X-valued process with the operator covariance function T and suppose that F has an integral representation T(s,t)=
f[
2
el("u-tv)
M{du,dv),
(3.6)
s,t 6
JJR
for some T(H)-valued positive definite bimeasure M on 03 x 03. If M is of bounded semivariation, which is equivalent to sup {||M(>1, 5 ) | | r : A,B e 23} < oo, then the integral in (3.6) is a well-defined vector MT-integral developed by Ylinen [2]. In this case we can prove by an analog of RKHS theory for vector valued positive definite kernels that there exists a £ 6 ca(23,X) such that x{t)
s
itu
Z{du),
te
Hence {x(t)} may be termed "weakly harmonizable," but it does not have an oper ator stationary dilation in general. [See Chapter IV for more details.] Thus we need a stronger notion of harmonizability. In order to integrate oper ator valued functions with respect to a T(H)-valued bimeasure M or an X-valued measure f we introduce the following. The operator semivariation ||M|| 0 (-,-) of a T(.ff)-valued positive definite bimeasure M is defined by
||M|U(A,B)=sup
X ^ o j - M ^ - , £**)&;
A,BG23,
j=lfc=l
where the supremum is taken over all finite measurable partitions {Ax,. .. , Am} of A and {Bu... , Bn) of B, and a,j,bk € B{H) with ||aj||, \\bk\\ < 1 for 1 < j < m and 1 < k < n. If M is of bounded operator semivariation, i.e., ||M|| 0 (R,R) < oo, then the representing measure f of {x(t)} is also of bounded operator semivariation, denoted £ e 6ca(Q3,X). Here the operator semivariation ||£|| o (0 of £ is defined by \\„(A) = sup
Yla^(Ak)
A 6 23,
fc=l
the supremum being taken over all finite measurable partitions {Ai,... ,An} of A and ak € B(H) with ||afc|| < 1 for 1 < k < n. If this is the case, {x(t)} is said
14
I.
INTRODUCTION AND PRELIMINARIES
to be weakly operator harmomzable and we can show that {x(t)} has an operator stationary dilation, i.e., there exists a normal Hilbert B(H)-module Y = LQ(Q,;H) containing X as a closed submodule and a K-valued operator stationary process {y{t)} such that x(t) = Py{t), t e K, where P : Y -+ X is the (gramian) orthogonal projection. For a further investigation we need to develop integration theory of operator valued functions w.r.t. (= with respect to) an X-valued measure £ G ca(25,X) and a T(H)-valued bimeasure M. Let us add a remark here. Note that each x e X can be expanded as
X=^2{x,cj>a)H<j)a,
(3.7)
which converges in the norm of X, {4>a}aei being the CONS of H. If H is separable, then it has a CONS {k)kLi- For each X-valued process {x(t)} put n fc=i
Then, since \\xn(t) - x(t)\\x -* 0 for every £ 6 R by (3.7), we can regard as a finite dimensional approximation of {x(t)}.
{xn(t)}^'=1
There are other infinite dimensional extensions of second order stochastic pro cesses. Let X be a Banach space and consider the Banach space LQ(Q;X) of all X-valued strong random variables x on Q, with zero expectation such that fn\\x(uj)\\x ti(dbj) < co, where \\-\\x is the norm in X. Take an x e Lg(fi;X) and observe that x*[x()) 6 LQ(Q) for every x* e X*, the dual space of X, since / \x*{x{u))\\{du>)
< I b H . / ||x(uOlllM) < ° ° ,
|■ ||x- being the norm in X*. Hence each x 6 L Q ( 0 ; X ) defines a bounded linear op erator Tx : X* -> L Q ( O ) . Thus, instead of thinking about LQ(U ; X)-valued processes we can consider B(3£*,Lo(fl))-valued processes, or more abstractly, B(2), K)-valued processes, where 2) is a Bancah space and K is a Hilbert space. So far the index set of processes has been restricted to the real line M. When we consider Z, the set of all integers, then {x(n),n e 1} C LQ{Q) may be called a stochastic sequence. If we take Rk, then { x ( t ) , t e l ' } may be called a stochastic field. More generally, we can take an LCA (= locally compact abelian) group G as an index set and call {x(t),t e G} a stochastic process as before, and we shall mainly be concerned with this case. If G = Rk, then we can consider isotropy on processes, which concerns the rotational invariance about the origin. If G is a nonabelian locally compact group, some difficulties arise. To define and obtain
BIBLIOGRAPHICAL NOTES
15
integral representation of stationary or harmonizable processes, we have to define suitable Fourier transform by making use of C"*-algebra theory. A hypergroup was introduced by Jewett [1] and others, and has important applications. When G is a commutative hypergroup, we can consider various types of processes on G as well.
Bibliographical notes There are several books which deal with (weakly) stationary processes, e.g., Cramer and Leadbetter [1](1967), Doob [1](1953), Loeve [3] (1955), Rosenblatt [1](1985), Rozanov [2](1967), [3](1977) and Yaglom [2](1987). Those which treat nonstationary processes are very few. We refer to Priestley [1](1988) for time se ries and Yaglom [2]. Two Proceedings edited by Mandrekar and Salehi [4] (1983) and Miamee [4](1992) are good references for both stationary and nonstationary, or one-dimensional and multidimensional processes. We shall give a brief historical notes here. For one-dimensional second order stochastic processes, (weak) stationarity was first introduced by Khintchine [1](1934) as was mentioned in Section 1.1. Kolmogorov [1](1941) studied interpolation and extrapolation of stationary sequences. Cramer [1](1940) treated multidimensional stationary processes and gave a so called Cramer decomposition of the spectral measure of a process. An extensive study of multidimensional stationary processes was done by Wiener and Masani [1](1957) and [2](1958), Masani [1](1960), Helson and Lowdenslager [1](1958) and [2](1961). Masani [2] (1966) gives a survey for multidimensional stationary processes. Infinite dimensional (or Hilbert space valued) stationary processes appeared in Kallianpur and Mandrekar [l](1965). Later on Payen [l](1967), Nadkarni [1](1970), Mandrekar and Salehi [2](1970), Kallianpur and Mandrekar [2](1971) and Truong-van [2](1985) give several results in infinite dimensional stationary processes. Some of these au thors considered Hilbert-Schmidt class operator valued processes instead of con sidering L 2 ( f i ; _ff)-valued processes. Gangolli [1](1963) already treated B{H,K)valued stationary processes where H and K are Hilbert spaces. This idea appears in treating Banach space valued second order stochastic processes, which was seen in Section 1.3 (see also Miamee [1](1976)). Chobanyan and Weron [1](1975) stud ied B(2),i^)-valued stationary processes, where 2) is a Banach space and if is a Hilbert space. Loynes [3] (1965) considered stationary processes in V7/-spaces (cf. also Loynes [1, 2](1965)) and Saworotnow [4](1973) studied stationary processes with values in a Hilbert module over an //'-algebra. In this direction, we also refer to Rosenberg [3](1978) and Suciu and Valuses,cu [1](1979). As to a survey of infinite dimensional stationary processes one may refer to Salehi [l](1981) and Makagon and Salehi [2] (1989). It is interesting that classes of nonstationary stochastic processes considered so
16
I. INTRODUCTION AND PRELIMINARIES
far contain a class of stationary ones. Many authors have considered the extension of stationarity. Loeve [2] (1948) (see also references therein) defined harmonizability in the middle of 1940s and Rozanov [1](1959) introduced weaker harmonizability. Bochner [l](1956) defined V-boundedness which turned out to be equivalent to weak harmonizability if we consider weakly continuous processes. Karhunen [1](1947) and Cramer [2](1951) introduced new classes of processes which we termed as the Karhunen class and the (strong) Cramer class, respectively. Asymptotic stationarity was defined and studied by Kampe de Feriet and Frenkiel [1](1959) and [2] (1962) (see also Kampe de Feriet [1](1962)), Rozanov [1](1959) and Parzen [2](1962). All these notions were clarified, compared and classified by Rao [4] (1982), where he notified the importance of bimeasure integration theory developed by Morse and Transue in [2](1955) and [4](1956). An extensive study of bimeasure and nonstationary (espe cially harmonizable) processes was given in Chang and Rao [2] (1986) (see also Rao [7](1985) and [9, 11](1989)). When the parameter space is Kfc, Yadrenko [1](1983) gave an extensive study for stationary isotropic processes. Yaglom [1, 2] also treated such processes. Harmoniz able isotropic processes were introduced and studied by Rao [12] (1991) and Swift [1, 2](1994, 1995). If processes on a commutative hypergroup are considered, Lasser and Leitner [1, 2](1989, 1990) and Leitner [1, 2](1991, 1995) studied hyper station ary and hyper weakly harmonizable processes, where the definition of hyper weak harmonizability is due to Rao [10](1989). Study of processes on a nonabelian locally compact group is initiated by Ylinen [1](1975), where he introduced some kinds of stationarity. He also introduced harmonizability for such processes in [7] (1989). Rao [10] considered the case where the group is unimodular. As to infinite dimensional nonstationary processes Truong-van [1] (1981) defined (weakly) harmonizable (rather than F-bounded) processes and obtained (operator) stationary dilations under certain conditions. Kakihara defined weak and strong harmonizabilities and V-boundedness in [7](1985) and [9](1986). He also denned the Karhunen class in [11] (1988). A classification of various harmonizabilities is found in Kakihara [15] (1992) from which most terminologies are taken. Harmonizability and V-boundedness are also introduced to B(H, if)-valued processes and B(X, K)-val\ied processes by Makagon and Salehi [1](1987) and Miamee [2](1989), respectively, where H and K are Hilbert spaces and X is a Banach space. Rao [10] included a survey of harmonizable processes which are either of univariate, multivariate or infinite variate, and whose index sets are in varieties.
CHAPTER II
HILBERT MODULES A N D COVARIANCE
KERNELS
Special classes of Hilbert modules and operator valued positive definite kernels are considered. Let H be a Hilbert space and B(H) be the algebra of all bounded linear operators on H. Then a normal Hilbert B(i/)-module is defined to be a left B(H)-module with the trace class operator valued inner product as an abstraction of the space LQ(CI ; H) which was considered in Section 1.3. Fundamental properties of it will be studied. For infinite dimensional processes, covariance functions are positive definite operator valued kernels, and hence they will be treated in the frame work of the reproducing kernel space theory. Finally, a Stone and a Bochner type theorem is formulated for operator valued continuous positive definite functions on a locally compact abelian group. Also a Plancherel type theorem is established.
2.1.
N o r m a l Hilbert
B(H)-modules
Let H be a complex Hilbert space with norm \\-\\H and inner product (-,-)HB{H) denotes the algebra of all bounded linear operators on H with uniform norm ||-1|, and T(H) the space of all trace class operators on H with trace tr(-) and trace norm ||-|| T - For a left B(H)-modvle X we denote the module action of B(H) on X by B{H) x X B (a, a;) M> ax £X. In this section we define normal Hilbert £?(f/)-modules and give their elementary properties as well as examples. Definition 1. A normal pre-Hilbert B(H)-module is a left B(H)-module X with a mapping [•,■]: X xX —> T(H) which satisfies the following conditions: for x,y,z £ X and a € B{H) (a) [x, x] > 0, and [x, x] = 0 iff x = 0; (b)
[x + y,z]
(c) [a-x,y]
= {x,z} + =
[y,z};
a[x,y};
(d) [x,y]* = \y,x], 17
18
II.
HILBERT MODULES AND COVARIANCE KERNELS
where "*" stands for the adjoint of operators. The mapping [•, •] is called a gramian in X and we sometimes denote it explicitly by [,-}x- W e s e e t n a t bY ( c ) a n ^ (d) [x, ay] — [x, y]a* holds. In the above definition, if the gramian is B(#)-valued, then X has been termed a "pre-Hilbert B(//)-module." In our case, since the gramian is T(if)-valued and T(H) = B{H)m, the predual, we adopted the terminology of "normal" pre-Hilbert -B(i?)-module. The duality T(H)* = B{H) is realized in a way that, given p G T(H)*, then there is a unique ap e B{H) such that p(a) = tr(aa„) for a G T(H) with \\p\\ = ||a p ||, and conversely. This fact is frequently used hereafter. In a normal pre-Hilbert B(H)-moduie X we define ax = (al)-x,
(x,y)x
= tv[x1y],
\\x\\x = (x,x)x
= \\[x,x}\\^
for a e C and x,y G X, where 1 is the identity operator on H. Then we see that X is a vector space and (•, -)x is an inner product in X, so that X is a pre-Hilbert space. We need the following basic result. L e m m a 2. Let X be a normal pre-Hilbert B(H)-module. Then: (1) (a ■ x, y)x = (x, a* ■ y)x for a € B(H) and x,y G X. (2) ||a ■ XJ|JC < HalllNIx foraeB(H)
and x e
X.
(3) [y,x][x,y}< \\x\\x[y,y} for x,y e X. (4) \\{x,y}\\T)H for ' € H
as before. Then we see that tr [,ip] = ((/>, IP)H, the original inner product in H, and hence H is a normal Hilbert f?(.ff)-module. We shall see later that H has essentially a unique gramian in it which is the one given above (cf. Proposition 5 below). (2) Let q G N or q = oo and A" be a Hilbert space with the inner product (•, -)K and the norm || • \\K. Let H = C if q S N and H = £2 if q = oo. Consider the (inflated) Hilbert space Kq = {(xn)qn=1 For x = {xn)qn=v
y = {yn)qn=1 [x,y] = (ajk),
:xneK,l®ip) = a<j)®4>,
\®il),' ®ip'} =
{ip,i>')K(®¥)-
One can verify that H ® if is a normal Hilbert f?(ff )-module. By the way H®K and S(if, if) are isomorphic as Hilbert spaces, denoted H®K ~ S{K,H). To see this, fix a CONS j> Q } Q €Z of if. Then, every element in if ® if can be written in the form Yl 4>a®i>a with ^ ||0Q|IH < °°- D e m i e an operator U : i f ® i f -4 S(K,H)
by 1
V
V'c
a6Z
= 5Z$cr®^a,
^
®ip for every <j>,ip e H. Proof. Let 0 e H be such that \\4>\\H = 1 and observe that
[0,0] = [(0® 0)0,(0® 0)0] = (0®0)[0,0](0®0), which is a nonzero positive operator. Thus for each ip 6 H we have that [0,0]V>= ( 0 ® 0 ) [ 0 , 0 ] ( 0 ® 0 ) ^ = ( 0 ® 0 ) W > , 0 ) H [ 0 , 0 ] 0 = ([0,0]0,0)H(0,0)H0= ( [ 0 , 0 ] 0 , 0 ) H ( 0 ® 0 ) 0 .
Hence [0,0] = Q0 ® 0 for a = ([0,0]0, 0 ) H > 0. Then a = 1 since ||0O||H = 1 and 1 = t r [0o, 0o] = a. Thus for every 0 i , 02 € H it holds that [01,02] = [(01® 0 ) 0 , (02® 0 ) 0 ] = ( 0 1 ® 0 ) [ < M W > ® 0 2 ) = ^ 1 ® ^ 2 -
2.2. S u b m o d u l e s , operators a n d functionals In this section we consider submodules and gramian orthogonal complements in a normal Hilbert 5(i?)-module X. Among the bounded linear operators on X those which have gramian adjoints are of interest and they are characterized. Fi nally, T(i?)-valued functionals on X are considered and a Riesz type representation theorem is proved for bounded functionals. Definition 1. Let X be a normal Hilbert B(H)-modu\e. A subset of X is called a submodule if it is a left B{H)-mo&xAe. Every closed submodule is itself a normal Hilbert jB(#)-module. For a subset Y of X denote by &{Y) the closed submodule generated by Y, i.e., the closure of the set | ^TakVk *■
:ak € B{H),yk fc=i
eY,la ® Q for some <j>a e H of norm 1 say, for a e l Then, for each a £ I , Xa = { a i Q : a € B{H)} is a normal Hilbert B{H)module such that Xa = H since we have an isomorphism Ua : Xa -> H defined by Ua{a ■ xa) = aa for a e B{H). Now Theorem 4 (3) implies that X =* © Ha, Ha =
H{aeI).
Corollary 7. Every normal Hilbert B(H) -module X is isomorphic to the HilbertSchmidt class S{K,H) and hence to the tensor product Hilbert space H ® K for some Hilbert space K. Proof. By Theorem 6 we can find an index set 1 such that X = Y = 0
Ha,
Ha = H ( a e l ) . Let if be a Hilbert space which has a CONS {rf>a}aei- Define an operator V on Y by
Vy=^24>aipa, Since J2 II^QHH < ° ° ' ^V ^
a
y = (4>a)aax e Y.
well-defined Hilbert-Schmidt class operator from
Q£I
K into H for each y e Y, i.e., Vy e S(K,H). Clearly V gives an isomorphism Y = S{K, H). S{K, H) S H®K was shown implicitly in Example 1.4 (5). Therefore, we have X S 5(A", H)^H®K. F if Y of as
Let us make some remarks here. Let X be a normal Hilbert £ ( i i ) - m o d u l e and be a closed submodule. According to Corollary 7, we can find a Hilbert space such that X = H ® K. Hence there is a closed subspace Ki of K such that = H®K1. Moreover, B(X) = B(H ® K) = B(H) ® B(K), the tensor product C*-algebras. Then it follows from Proposition 2.4 (2) that we can identify A(X) the commutant of B{H) g) 1, which constitutes 1 ® B(K) = B{K).
Corollary 8. Let X be a normal Hilbert B(H)-module, and { X Q } Q € I and be two gramian bases of X. Then 1 and J have the same power.
{y0}
31
2.3. CHARACTERIZATION AND STRUCTURE
Proof. It follows from Corollary 7 that X = H ®K1 = H ®K2, where Kx and K2 are Hilbert spaces with dimensions | I | and \J\, respectively. Then it follows that K\ ~ K2, so that \1\ = \J\, i.e., I and J have the same power. D e f i n i t i o n 9. Let X be a normal Hilbert £?(#)-module. The power of any gramian basis of X is called the modular dimension of X, denoted Dim(X). X is said to be separable if Dim(X) < NoC o r o l l a r y 10. Let X and Y be two normal Hilbert B(H)-modules. Then they are isomorphic iff Dim(X) = Dim(Y). If X is a normal Hilbert 5(.ff)-module, then we have that dim(X) = Dim(X)dim(H), where dim(-) is the usual dimension of Hilbert spaces. Thus, if X = H ® K, then Dim(^C) = dim(i'C) and, hence, X is separable iff so is K. Let us give an example of a gramian basis of X = H ® K, K being a Hilbert space, as follows. P r o p o s i t i o n 1 1 . Let X = H ®K where K is a Hilbert psace, {ipa}a^z be a CONS in K and take £ H of norm 1. Then the family {a,4>g)K ® 4> = 0 if a / /3, [ ®ipa,a,® ipa]2 and | | 0 ® 4>a\\x = 1 for a g I . N let {<j>0}p£j be a CONS in H and observe that {0 ® ipa}aei,pej forms a CONS in H ® K. Take any x e H ® K and consider its Fourier expansion:
a,/3
Since the nonzero terms in the above sum are at most countable, we can choose an at most countable subset {0}i3£j (possibly N = oo) such that N
N
X = ^2^2{x,(j)0k k=laeX
®i>a)x4>0k
®4>a = Yl Yl (X> k ® 4'a)x4'a
£ K for 1 < k < N, w
TV
see that x = ^
4>k®i>k- Then for each a g I it holds that
fc=i r
[ x , $ ® ^ a ] ■ (® V"c
iV
^ ^ f c ® l/,fc,® V'a fc=l
®ipa-
i®1pa)
32
II.
HILBERT MODULES AND COVARIANCE
KERNELS
N fc=i N fc=l N
k=l
Consequently we have that N aelk=l
N
N
\
M=l
'
I
a£Xk=l
X
^s^a
a£Tk=\
= ^2{x,®il>a). Thus Theorem 4 (4) holds. Therefore {<j> Cg> V'ajaex is a gramian basis of X.
2.4. Positive definite kernels and reproducing kernel spaces C-valued positive definite kernels and their RKHS's were briefly discussed in Section 1.2. In this section we are concerned with T(//)-valued positive definite kernels and their reproducing kernel normal Hilbert B(i?)-modules consisting of T(H)-v&l\ied functions. Let 0 be a nonempty set. Definition 1. A function F : 9 x 0 —> T(H) (p.d.k.) if n
is called a positive definite
n
S X ^ r ^ . t k t e >o j=i
kernel
(4.i)
fc=i
for any n € N, O i , . . . , a„ € B(H) and tu ■ ■ • , tn € 0 , where the LHS (= left hand side) of (4.1) is in T+(H), the set of all nonnegative elements in T(H). [Note that
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
if T is B(if)-valued, above.]
33
then we can also define T to be a p.d.k. in the same manner as
P r o p o s i t i o n 2 . A function
F : 6 x © —» T(H) is a p.d.k. iff it satisfies that n
n
^^(r(i
f c
,^)^,^)
H
>0
(4.2)
J=lfc=l
for any n € N, 4>i,... , 4>n 6 H and * i , . . . ,tn 6 8 . Proo/. Suppose that (4.2) holds and let n € N, fli,... ,an € -B(#) and £ l 7 . . . ,£„ 6 9 be arbitrary. Then for any <j> 6 H it holds that
VV
;,fc
7
/H
j , k
Therefore F is a p.d.k. Conversely, suppose that T is a p.d.k. and let n € N and 4>\, ■.. , 4>n 6 H be arbitrary. We can choose a nonzero $ 6 H and Oi>... , a n g B(H) such that a*4> = 4>j, 1 < j < n. Then it holds that
]T (r(ttl tj)4>iAk)H = £ (r(tt,*i)o^, a^) H (j2akr(tk,tj)a;)'P,A
>0.
J.fc
Thus (4.2) is satisfied. Elementary properties of a p.d.k. are noted next. P r o p o s i t i o n 3 . Let F : © x © -> T(H) be a p.d.k.
Then:
(1) T(t,t) > 0 for every te ©. (2) T(i, *)* = T(s,t) for every t, s € ©. (3) r ( t , s)F(s, t) < | | r ( s , s) || T F(i, t) /or euen/ t, s G 0 . Proof. (1) is clear from the definition. (2) Let t, s € 0 be fixed. Then, for every , T/J e if it holds that ( r ( M ) ^ V ) H + {r{t,t),4>)H + {T{s,s)rl>,il>)H + (T(s,t)^,cP)H It follows from (4.1) that
(r(t,*)^)„ + (v>,r(*,t)'*)H
> 0.
34
II. HILBERT MODULES AND COVARIANCE KERNELS
is real. Putting ip = 4> and ip = i(f> (i = \f—l),
(r(t,s))H + (0,r(s,t)V) H ,
we have that
i{{r(t,8)4,4>)a
are real, so that (r(t,s), )H = (T(s,t)*^,)H. we conclude that T(t, s)* = F(s,t). (3) follows from Lemma 1.2 (3).
- (,r(s,t)*)H}
Since this holds for every <j> e H,
An analogy of the RKHS theory can be obtained by introducing reproducing kernel normal Hilbert B{H)-modules. Definition 4. Let r : 6 x 0 -> T(H) be a p.d.k. and X be a normal Hilbert B(if)-module with a gramian [•, •] consisting of T(H)-valued functions on 0 . Then r is said to be a reproducing kernel (r.k.) for X if (a) r ( t , ■) eX for each t e 9 ; (b) x(t) = [x(-),T(t, •)] for each t € 9 and x e X. The property (b) is called the reproducing property. In this case X is said to be a reproducing kernel (r.k.) normal Hilbert B(H)-module of T. In the case where X is a normal pre-Hilbert i?(//)-module, we can also say that T is a r.k. for X if (a) and (b) above hold. Note that, if H = C, the above definition reduces to the usual RKHS. Propositions 5-11 below give basic results on r.k. normal Hilbert B(.ff)-modules. P r o p o s i t i o n 5. / / a normal Hilbert B(H)-module is unique. Proof. If T, T' : 9 x 9 -> Tin) that, for t, s 6 0
X admits a r.k., then the kernel
are r.k.'s for X, then it holds by Proposition 3 (2)
r(s,t) = [r( s ,-),r'(t,-)] = [r'(*, •),!>, •)]* =r'(t,sy
= r'(s,t).
P r o p o s i t i o n 6. Let X be a normal Hilbert B(H)-module consisting of'T(H)-valued functions on 0 . Then, X admits a r.k. iff, for each t 6 0 , i(x) = x(t) (x 6 X) is a bounded functional on X. Proof. Suppose that X has a r.k. I \ For a fixed t € 0 we have that i(x) = x(t) = [&(•), r ( i , - ) ] for x e X. Hence | | i » | | T < | | i | U I | r ( t , - ) I U for x € X, so that i is bounded. Clearly, t is a functional on X. Conversely suppose that, for each t G 0 , i is a bounded functional on X. Then, by Proposition 2.6 there exists a unique xt g X such that i(x) = x(t) = [x,xt] for x 6 X. Putting T(t,s) — xt(s) for t,s 6 0 , we can check that T is a r.k. for X.
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
P r o p o s i t i o n 7. Suppose that F : 9 x 9 - > T(H) B(H)-module X.
is the r.k. for a normal HUbert
(1) If a sequence {i„}™ =1 C X converges strongly to x £ X,
||x n (t)-x(£)|| T ->0,
35
then
tee.
(2) / / a sequence {xn}^=i C X converges weakly to x € X, then {x„(t)}'^=1 T(H) converges weakly to x(t) E T(H) for every t 6 0 .
C
Proof. (1) For each t 6 0 observe that
\\Xn{t) - x(t)\\T = ||[xn(-) - x ( . ) , r ( t , -)]|| T < ||*n - x\\x\\r{t,-)||.v
-+ o.
(2) For each t € 0 and a G B ( i / ) we have that tr((xn(t)
~ x(t))a)
= tr [ » „ ( ■ ) - i ( - ) , a T ( * , - ) ] = ( i „ - i , a T ( ( , - ) ) x -> 0.
P r o p o s i t i o n 8. Let { x Q } a g z 6e a gramian orthonormal subset of X. Then it is a gramian basis of X iff T admits a representation r(s,t) = ^ x a ( s ) * x Q ( t ) ,
s,t€B.
Proof. Suppose that { X Q } Q 6 X is a gramian basis of X. 3.4 (4) and the reproducing property that for s e 6
(4.3)
It follows from Theorem
so that (4.3) hold. Conversely, suppose that (4.3) holds. If x € X and [x,x a ] = 0 for a 6 I , then we have that x(t) = [x(-),r(t,-)] =
s ( 0 . 5 3 *„(*)* ■««,(•) = ^ 3 Q £ I
[x(-),x Q (-)]x Q (i) = 0
QgZ
for i 6 0 , which implies that x = 0. Thus Theorem 3.4 concludes that {xa}aeJ a gramian basis of X.
is
P r o p o s i t i o n 9. Suppose that X(T) is a normal Hilbert B(H)-module admitting a p.d.k. r : 0 x 0 —► T(H) as a r.k. Let X be a larger normal Hilbert B(H)-module containing X as a closed submodule and P : X —> X be the gramian orthogonal
36
II. HILBERT MODULES AND COVARIANCE KERNELS
projection. x = Px.
For each x 6 X define x(t) = [x,F{t, ■)] for t € 0 . Then it holds that
Proof. For each x 6 X write x = xi + x2 where x\ £ X and X2 £ X* Then we have that
[*,r(t,-)] = [xi}r(t,-)] =xiW, Hence, xi(t) = (Px)(t)
= [x,T(t,-)],
= X Q X.
fee.
t e 0 , i.e., x = Px.
P r o p o s i t i o n 10. Let T : © x © -> T(if) fee tfte r.k. for a normal Hilbert B(H)module X. (1) Every closed submodule of X admits a r.k. (2) If X = Xi © X 2 and Tj : 0 x © —> T ( # ) is £/ie r.fc. /or a closed submodule
Xj {j = i, 2), then r = r 1 + r 2 . Proof. (1) Let Y be a closed submodule of X and P : X —> Y be the gramian orthogonal projection, and define r'(£, ■) = PT(t,) for t € 0 . Note that r" : © x © -> T(fl') is a p.d.k. Then, for each y e Y it holds that y(t)=
[y(-),r(tt-)]
= [Py,r(t,-)]=
[y,PT{t,-)}
= [y,V(t,-)},
tee.
Thus V is the r.k. for Y. (2) follows from (1). P r o p o s i t i o n 1 1 . Let T : 0 x 0 —> T ( / / ) 6e a p.d.fc. and X be a r.k. normal Hilbert B{H)-module of V. (1) .Eac/i bounded functional
I on X can be expressed as
£(x) = [x,X(\,
x £ X,
where xe(t) = e(T(t, ■))* for t 6 0 . (2) Each T 6 A(X) is expressible as
(TaO(t)=[*(-),rr(t,-)],
tee,
wftere T T ( i , ■) = T*T{t, ■) /or t e 0 . Proo/. (1) By Proposition 2.6 there is an xe 6 X such that £(x) = [ar,ar^] for x e X . Then we see that for t G 0 Xe(t)=
[xe,r(t,-)}
= [r(t,-),xe]'
=
e(r(t,-))'.
(2) If we define a kernel r T : 0 x © -> T ( H ) by rr(t,aJ=(TT(t,.))(a),
«,t€0,
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
37
then we see that for x 6 X and t 6 0
(Tx)(t)=[Tx,r(t,-)} = [i,rr(vl] = [*,rT(t,0]. Now given a p.d.k. r : 0 x 0 —>• T(H), we want to construct a normal Hilbert B(i7)-module Xp admitting T as the r.k. To do this we introduce the notion of "functional completion," which is different from the usual norm completion of preHilbert spaces. Definition 12. Let X 0 be a normal pre-Hilbert B(H)-module consisting of T(H)valued functions on 0 . Then the functional completion of Xo is to obtain a normal Hilbert B(H)-module X by adding T(H)-valued functions on 0 in such a way that the value of a function x e X at t e 0 depends continuously o n i e X and that X0 is a dense submodule of X. In this case the resulting space X itself is called the functional completion of XoT h e o r e m 13. Let T : 0 x 0 —> T(H) be a p.d.k. Hilbert B(H)-module X r admitting T as the r.k. Proof. Let us define a left B(H)-module 0 as follows:
Then there exists a unique normal
X0 consisting of T(ff)-valued functions on
Xo = I ^ a j r ( t i , - ) : oj e B(H), tj e 0 , 1 <j < n,n 6 N I ,
where the module action of B(H) is defined by n
n
a ■ ^2 ajV(tj,•)
= ]T)aa,-r(tj,0,
a e S(H).
If we define a mapping [■, -]o : Xo x Xo —¥ T{H) by
n
for x = XI a j r ( * j ' ' ) j=i
m an
d 2/ = S W£sfc,') 6 -X'o. t Q e n
we see
that [-, -] 0 satisfies all
fc=i
the properties of a gramian except that [x, x]o = 0 implies x = 0 (i.e., x(t) = 0 for f 6 0 ) . We shall show later that [-,-]o has this property and, hence, is actually a gramian on X 0 . P u t ||a:||o = |j[a;,x]o||*
for x
£ Xo-
38
II.
HILBERT MODULES AND COVARIANCE KERNELS
Now we can show that (a) and (b) of Definition 4 hold. In fact, (a) is obvious 71
and (b) is seen as follows: for x = J2 aj?{tj,
■) € %o a n d
t e
©^
nolds
tnat
,=1
[x(-),r(i,-
52 0,1% •), r(t, •)
= E ^'rfe-*) = XWj = i
i=i
Suppose that x 6 X 0 and [x,x] 0 = 0. [x(-),T(i, •)] implies that
Then the reproducing property x(i) =
l|xW|| T 0
as n, 77i —> oo for t € 0 . Hence there exists a T(if)-valued function x on 0 such that ||x„(t) — x(t)|| T —> 0 for t 6 0 . Denote by X r the set of all such functions x obtained in this way. It is clear that Xp is a left /3(i/)-module and X0 C Xp. Define for x, y € ^ r [x,j/].Yr = lim [xn,yn]o
(in the trace norm),
n—too
where {x n }^L x and {j/nj^Li Q Xo are the determining sequences of x and y, respec tively. [■, -].Yr is well-defined. For, if {x^l}^°=1 and {y'n}^=1 C X 0 are other such deter mining sequences of x and y, respectively, then | | x J l ( t ) - x n ( i ) | | T , \\y'n(t)-yn(t)\\T —> 0 for f e 6 . Since {x'n - xn}^=1 and {y'n - y„}'^L1 are Cauchy in X0 and the limit functions are zero, we conclude that \\x'n - xn\\0, \\y'n - yn\\0 —i 0. Hence ||K.2/n]o - [a?„,j/n]o||T < | | K - xn,y'n]0\\T < \K
+ \\[xn,y'n -
yn]o\\T
~ S n | | o | | l & H o + | | x n | | o | | j / ; - 2/nHo - > 0.
Thus, [x, y]xr is independent of the choice of determining sequences of x and y, and is well-defined. Now it is easily seen that [■, -}Xr is a gramian on Xr. Moreover, A'0 is dense in Xr- For, let x € Xr be arbitrary. First we note that the equality \\y\\xr = \\y\\o holds for y 6 XQ- If {xn}^Li £ XQ is a determining sequence of x S Xr, then lim \\xn-x\\Xr= n—too
lim n—yoo
lim ||x n - xm\\0 = 0. m—too
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
39
Furthermore, to see that X-p is complete, let {x n }£° =1 C X r be a Cauchy sequence. Since X0 is dense in X r , there exists a Cauchy sequence {x'n}'^'=1 Q Xo such that ||x'n — x n | | x r —^ 0. This sequence determines an element x g Xr such that ||x'„(£) — x{t)\\T -> 0 for t € 6 , and \\x'n - x\\Xr -> 0. Therefore ||x„ - x\\Xr -> 0. Finally, we shall verify the uniqueness of X ^ Let X be a normal Hilbert B(H)module with the gramian [■,■] which consists of T(i?)-valued functions on 0 and admits T as a r.k. Since F(t, ■) E X for t g 0 by definition, we have X 0 C X . Moreover, it holds that [z>2/]o = [x,y],
x,y e X0-
(4.4)
Furthermore, XQ is dense in X since {T(t, ■) : t g 0 } is complete in X . Thus we must have Xp = X . For any x,y g X we can choose sequences {xnj^-^^, {yn}'^'-i C Xo which converge to a: and y in X , respectively. Then we have by (4.4) [x,y}=
\im[xn,y„]= n—>oo
lim [x„, yn]0 =
[x,y]xr-
n—foo
Therefore X and Xp are identical normal Hilbert B(i?)-modules. The above proof readily shows the following characterization of functional com pletion for a r.k. space. Corollary 14. Let X0 be a normal pre-Hilbert B(H)-module consisting ofT(H)valued functions on 0 . Then there exists a functional completion X of XQ iff the following conditions hold: (1) For each t g 0 the functional t on Xo defined by t(x) = x(t), x g Xo is bounded; (2) / / {xnj-^Lj C Xo is a Cauchy sequence such that \\xn(t)\\T —> 0 fort g 0 , then \\xn\\0 —> 0, where | | | | 0 is the norm in X0. If a r.k. is restricted to a subset 0 O x 0 O of 0 x 0 , then it is again a r.k. for some normal Hilbert B(H)-modu\e Xo, which is characterized by the following: T h e o r e m 15. Let ©o be a nonempty subset of 0 , Y : 0 x 0 —> T(H) be the r.k. for a normal Hilbert B(H)-module X, and T0 = r|e 0 xe 0 > the restriction of F to ©o x 0 O . Then, To is the r.k. for the normal Hilbert B(H)-module X0 = {x\&0 : x g X } and the equality [x0,x0} = miii{[x,x] : x g X, x | e 0 = x0} holds for x0 g X 0 , i.e., the operator inequality [X0,XQ] < \x,x] holds for x g X such that X\Q0 = XQ, the restriction of x to ©o, and for some x g X the equality holds. Proof. P u t X i = {x g X : x | e 0 = 0}. Then Xi is a closed submodule of X . To see this, let {x„}™=1 C X i be a Cauchy sequence. There exists an x g X such that
40
II.
HILBERT MODULES AND COVARIANCE KERNELS
\\xn ~ x\\x -» 0. By Proposition 7(1) it holds that \\xn(t) - x(t)\\T -> 0 for t E 9 . Since xn\e0 = 0 for n £ N, we have that x|e 0 = 0, i.e., x & X\. Let Ti and r f be r.k.'s for X i and X * (the gramian orthogonal complement of Xi), respectively, which exist by Proposition 10, and P : X -t Xf be the gramian orthogonal projection. Then F = Fi + r f again by Proposition 10. Note that V(t, ■) = rf{t, ■) for t E 9 0 . For x0 £ X0 we define [XQ] = {x e X : z | e 0 =
Note that PX\Q0
x0}.
= XQ, i.e., Px E [XQ] for x E [XQ] because xo = x\e0 = Px\e0 + {x - Px)\e0
=
Px\e0-
Moreover, Px = Py for x,y £ [x0] because Px = Py -^ Px\e0 = Py\e0can define x'0 = Px for x E [xo] independent of x. We also note that [x'0,x'0] < [x,x],
Hence we
x £ [x0]
because [x,x] = [x'0,x'0] + [x - x'0,x - x'Q] > [x'0,x'0]. Define [ZQ.^OIO = [^'o^ol for x0 E Xo- Then, the operator U : (X 0 ,[-,-]o) -> (Xf, [•,•]) defined by t/xo = x'0, xo E Xo gives an isomorphism between these two normal Hilbert B(.H")-modules. Note that (Uxo)(t) = xo{t) for XQ E Xo and t E OoWe now prove that T 0 = F | e 0 x e 0 *s the r.k. for (X 0 , [-, -]o)- Take any XQ E X 0 and t 6 0 o . Then it holds that x0(t)
= (Ux0){t)
= [Cteo, i f («,-)] = [ i o , r 0 ( t , - ) ] 0
since r f (t, •) £ X* and UT0{t, ■) = r f (*, ■) for t € 0 . Thus T 0 is the r.k. The sums and differences of r.k.'s are considered below. T h e o r e m 16. Let Yj : 9 x 9 —> T{H) be a r.k. for a normal Hilbert B{H)-module (Xj, [-, •];, | H | j ) , j = 1,2. Tftera F = r x + r 2 W a r.k. for the normal Hilbert B{H)module X = {x\ + x2 : x\ £ X i , 2:2 6 X2}, where the gramian and the norm are respectively given by [x,x] = min{[a;i,a;i]i + [x 2 ,x 2 ] 2 : a; = Xi + x2, 11 £ Xi, x2 € X 2 } ,
(4.5)
||x|| 2 = min{||x 1 ||2 + || a ;2||i : x = xx + x2, xx £ Xlt x2 £ X 2 } .
(4.6)
Proof. Observe that X = {(x%,x2) ■ Xi € X i , x2 £ X 2 } is a normal Hilbert module, where the module action, the gramian and the norm are defined by a- ( i i , i 2 ) = {a-xua-x2),
[{xi,x2),{yuy2)]x
= [x1,yi}1
+
[x2,y2}2,
B{H)-
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
41
ll(zi,x2)|i;r = iKiii + i M i , respectively. That is, X is the direct sum of Xi and X%. Define X0 = Xlf)X2)
X0 =
{{X>~x):xeX0}.
Then it is easily seen that X0 is a closed submodule of X. Define U : X —> X by (f/(xi,x 2 ))(-) = a;i(-) + x2(-)>
(xi,x2)
e X.
Then f/ is jB(#)-linear. Moreover, U\y# is one-to-one and onto, i.e., X = \x' + x" : (x',x") e X^}.
Define for x 6 A" with x = x' + x", (x',x") € X^ [x,x] = [{x',x"),(x',x")]x,
\\x\\ =
\\(x',x")\\x.
Now we prove (4.5). For x = xi+x2 there correspond (x\,x2) € X and (x',x") £ X*, so that a; = xi+x2 = x'+x". Hence, x2—x" = —(xi—x1) and (xi—x1, x2—x") E X0. Moreover, it holds that [xi,Xl]l +
[x2,X2]2
=
[(xi,x2),(x1,x2)]x
= [(x',x"), > =
(x',x")]x
+ [ ( m - x',x2 - x"), (Xl - x',x2 -
x"))x
{{x\x"),{x',x")}x \x,x).
Thus (4.5) is proved. (4.6) is clear from (4.5). Finally we show that V = r x + T 2 is the r.k. for X. Let x € X and t € 8 . Then we have that x(t) = x'{t) + x"(t),
= = =
where x (x',x") €
Xf,
[x'(),r1(t,-)]1+[x"(-),r2(t,-)]2 [(x',x"),(rl(t,-),r2{t,-))]x [(x',x"),(F'(t,-)X'(t,))]x
+ [(x',x"), (Tiff, •) - r'(i, ),r 2 («, •) - r"(t, ■))]x, where T(t,-) =
^ {r'(t,-),T"(t,-))
e X*,
[(x',x"),(T'(t,-),T"(t,-))]x, since (Vi(t, ■) - r'(t, •), r 2 ( t , ■) - r " ( t , ■)) e * 0 and ( x ' , i " ) e * * ,
= [z(.),r(t,-)].
42
II. HILBERT MODULES AND COVARIANCE KERNELS
Definition 17. Let T ( 9 ) be the set of all p.d.k.'s r : 9 x 0 -> T{H). Tx, T 2 e T ( 9 ) define
For
Ti r,-rier(e). Then ( T ( 0 ) , < ) is a partially ordered set. P r o p o s i t i o n 18. Let Fx, F2 € T(0) and Xx, X2 be the corresponding r.k. normal Hilbert B(H)-modules, respectively. If Fx < F2, then Xx C X 2 and [zi,Xi] 2 < [xi,%i]i for Xi £ Xx. Proof. By assumption T = T 2 - Ti e T ( 9 ) . Let X = Xr be the r.k. normal Hilbert B ( # ) - m o d u l e of F. Since T 2 = Ti + F, we have X 2 - {xx + x : xx € Xx, x £ X) by Theorem 16. Hence ^ C I 2 - Again by Theorem 16 it holds that for x2 6 X 2 [ x 2 , i 2 ] 2 = min{[xi,a:i]i + [ar.ar]: x2 = xx + x, xx € Xx, x € X } . Thus [xi,xx]2 5* [^li^ili f° r ^l £ X i . To consider the converse of Proposition 18 let F € T ( 9 ) be the r.k. for a normal Hilbert S(//)-module A" and X j C X be a normal Hilbert £?(#)-module such that [xi,a;i]i > [ x i , i i ] ,
i i 6 Xi,
(4.7)
where [•, ]i is the gramian in X\. L e m m a 19. With the above notation and assumption which is the r.k. for Xx.
there exists a Fx g T ( 9 )
Proof. By Proposition 6 for each t € 9 there is an at > 0 such that ||a;(t)|| T < o;(||xj| for x e X. Hence ||i 1 (t)|| T < a*II*Hi for xx e X i by (4.7). Thus, again by Proposition 6, X i admits a r.k. Pi e T ( 9 ) . To show that the above obtained Pi satisfies the relation Fx < F we shall construct a normal Hilbert B{H)-modu\e (X 2 , [•, ]2) admitting r 2 = F - Fx as the r.k. Define an operator L : X —> Xx by [xi,x] = [xuLx]i,
xi 6 X i , x € X.
To see that L is well-defined, observe that, for each x 6 X , l{xx) = [xi,x] (xx e Xx) is a bounded functional on X and hence on Xx by (4.7). Thus by Proposition 2.6 there exists a unique Lx € X^ such that l(xx)
= \xx,x\
= \xx,Lx\^
xx e Xx.
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
Now L € B(X) x<E X \\Lxf
n B{X,Xi),
Xx C X, with ||L|| < 1 (cf. Definition 2.3) since for
< \\Lx\\\ = WlLx^xhl
= \\[Lxtx]\\r
< \\Lx\\\\x\\
0,
(cf.
x 6 X.
is also gramian positive because for x € X
[(I — L)x,x]
= [x, x] — [Lx,x] = [x,x] — [Lx, Lx}\ > 0
by ||L|| < 1 and by Proposition 2.4 (4). Thus there exists a gramian positive operator V e A(X) such that I - L = L'2. Here we define X2 = {L'x:xeX},
X'=
{xeX
:L'x = 0},
X" = (X')*
=xex'.
Then we see that L' : X" —> X2 is a gramian unitary operator, where the gramian in X2 is defined by [x2,a;2]2 = [Px,Px]
for a;2 = L'a; with x 6 X".
Here P : X —> X" is the gramian orthogonal projection. Consequently (X", [•, ■]) = (^2,[-,-]2).
We shall show that r 2 = F - Ti is the r.k. for X2. For t e Q
r2(i,■) = r(t,■) - r,(t,■) = (/-£)r(t,•) = L'(L'F(i,■)) e x 2 . Moreover, for t g 0 and x 2 £ -X'2, letting x 6 X be such that L'a; = x2, the reproducing property is seen as follows: x2(t) = [ar a ,r(i,-)] = [ £ ' x , r ( t , - ) ] = [x,LT(t,-)] = [Px,PLT(t,-)] = [L'i,L'LT(t,-)]2
= [x 2 ,r(i,-)-r 1 (t,-)] 2 = h,r 2 (t,-)] 2 . We summarize the discussion into the follwoing. L e m m a 20. Under the assumptions
of Lemma 19 the kernel V\ satisfies Ti < F.
II.
44
HILBERT MODULES AND COVARIANCE KERNELS
From Lemmas 19 and 20 we deduce the following. T h e o r e m 21. Let F € T(O) be the r.k. for a normal Hilbert B(H)-module X. If F = Fi + T 2 , r i , r 2 e T ( 0 ) is a decomposition of F, there is a decomposition I = Li + L*2 of I into two gramian positive operators in A(X) given by (LlX)(t)
= [i(-), T1%-)],
(L2x)(t)
=
[x{-),F2(t,-)]
for ( 6 6 and x 6 X. Conversely, if I ~ Li + L% is a decomposition of 1 into two gramian operators in A(X), then there correspond p.d.k. 's Ti, F 2 G T ( 0 ) such that
Fj(t,-) = L3F{t,-){tee,j
= i,2)
and
positive
r = r i + r2.
As seen from Theorem 13, every F € T ( 6 ) admits a r.k. normal Hilbert / ? ( / / ) module Xp. According to Corollary 3.7 there exists a Hilbert space K such that Xr = H ® K. How can this Hilbert space K be realized? Can it be obtained as a RKHS? To answer this question we begin with the following definition. Definition 22. Let F 6 T(6) and S) be a Hilbert space consisting of //-valued functions on ©. Then Sj is said to be a reproducing kernel Hilbert space (RKHS) of F if the following conditions are satisfied: (a) L(-, t)(f> 6 f> for each t € 6 and e H, (b) (p{t),4>)H = (p(-),r(',«)^) i 5 for each t e 0 ,
j 6 H, 1 < j < n, n € N | . Then f>0 becomes a pre-Hilbert space if we define an inner product (■, ) 0 by (P.9)o = X !
(r(sk,tj)4>j,4'k)H
j,k
n
for p(-) = J ] r (->*i)^j j=i
m and
"(■) = E r(-,sfc)Vfc € £o- That (p,p) 0 = 0 implies fc=i
p = 0 can be seen after checking that F)0 satisfies (a) and (b) of Definition 22. Thus we can obtain the RKHS 9)r by the functional completion of Sjo-
2.4. POSITIVE DEFINITE KERNELS AND REPRODUCING KERNEL SPACES
45
Note that Proposition 23 also holds for B(H)-valued p.d.k.'s. Now the RKHS S)r obtained above will play the role of K of Corollary 3.7. T h e o r e m 24. Let T £ T ( 6 ) , and Xr and Sjr be the normal Hilbert and the Hilbert space which admit T as a r.k., respectively. Xr^
B(H)-module
Then it holds that
H® Sir-
Proof. Let {pa}a£i
be a CONS in f j r . Then we claim that r(s,t)
= Y^Pa(s)®P~Jt),
s,teQ.
(4.8)
In fact, for <j> £ H and t 6 0 we have that T{-,t)4> 6 fjr by Definition 22 (a) and
r(-,i)tf = £(r(-,t)*.p*(0) ar P«(-) aex = Y^ (*,Pa(*)) H Pa(-).
by Definition 22(b),
= X](pa(-)®M*)) £ H define a T(.ff )-valued function 4> ® P on 6 by
(0®p)(-) = ^®pTTThen we claim that, for 4>,ip £ H and p, g G f>r, [0®P,V , ®g] = ( 9 1 p ) $ r ( 0 ® ^ ) . where [•, •] is the gramian in Xroo
(4-9)
In fact, since p and q can be written as p(-) =
oo
53 ^{-,tj)4>j and g(-) = 53 r(-,sfcfc)i/'fc Er(-,tj)0,-and «,(■)= Er(-,s )v for some sequences {tj}^lt
jJ == li
{skj^Li
£ @
fc=i fc=l
{ ^ } ~ x , {fc}~=i C i / , we see that
J2 {® ^)r(i„ ■)> £ > ® ^)r( ajb ,•
[0 ® p, V> ® q]
i ^
* ( ® ^ ) [r(tj, •), r(s f c , •)] (V'fc ® ? ) .
^(^®?j)r(t,-,*fc)(v»fc®?)
easily justified,
and
46
II. HILBERT MODULES AND COVARIANCE KERNELS
= ^
( ® p)(-) e Ay for $ 6 # and p € fjrNow take €// of norm 1. Then {0 ® p Q }aeJ forms a gramian basis of XrIn fact, by (4.9) we see that [
® 4> = [4>®Pa,4>® Pa}2 for a € l Moreover, we have that by (4.8)
J2(®pa){sy(®pa)(t) = ]C p «( s )®p°W = r(s,i) for s , ( 6 0 . Thus Proposition 8 implies that {<j> ® pa}aex Therefore Xr £* H ® Sjr.
is
a
gramian basis of Xp-
2.5. Harmonic analysis for normal Hilbert B(.rY)-modules In this section we consider a T(i/)-valued continuous positive definite function denned on a topological group and its relation to a gramian unitary representation on some normal Hilbert B(H)-module. If the group is LCA (= locally compact abelian), we can prove a Stone type and a Bochner type theorem for such a positive definite function. Fourier transform is defined for a normal Hilbert B(i7)-module valued function on an LCA group, and a Plancherel type theorem and an inversion formula are established. Definition 1. Let G be a topological group and X be a normal Hilbert B(H)module. A T(ff)-valued function r on G is said to be positive definite (p-d.) if YJajr{t]tk1)al ^ ° f o r a n y n e N , tti,... ,o„ € B(H) and t z , . . . , tn e G. F : G ->• i,k T(H) is said to be weakly continuous if tr(aF(-)) is continuous for a € B(H). A gramian unitary representaion (g.u.r.) of G on X is a mapping [/(■) from G into A(X) for which U(s) is gramian unitary for every s e G and satisfies that U(e) = I and U(st) = U{s)U(t) for s,t 6 G, where e is the identity of G. A g.u.r. [/(•) of G on X is said to be weakly continuous if (f/(-)ar,y) is continuous for x,y E X. A vector i 0 6 I is said to be cyclic for a g.u.r. [/(■) if T{H), we put F(s,t) = Fist'1) for s,t € G. r : G x G —> T(H) is a p.d.k. in the sense of Definition 4.1.
47
Then F is p.d. iff
P r o p o s i t i o n 2. Let G be a topological group and F : G —> T(H) be p.d. Then there exist a normal Hilbert B(H)-module Xp with the gramian [•,], a g.u.r. [/(■) of G on Xp and a cyclic vector xo G Xp such that F(s) = [U(s)xo,x0],
s E G.
It also holds that for s,t £ G
r( 5 - 1 ) = r( s )*,
||r( s )|| T < ||r(e)||T,
||r( s )-r(i)||? X 0 is onto and gramian preserving, indeed, [U(s)x,U(s)y] = [x,7/] for x, y € Xo- Hence f/(s) can be uniquely extended to a gramian unitary operator on X r , still denoted U(s). It is easily verified that U{st) = U{s)U{t) and (/(e) = / for s, t e G. Thus (/(■) is a g.u.r. of G on X r . If we put x 0 = f (e, ■) G X r , then we see from (5.1) that F(s) = [U{s)x0,x0],
seG.
The equality r ( s - x ) = F{s)* for s £ G follows from r ( s ~ 1 ) = f (e,s) = F{s,e}* = F{s)*. The inequality ||r(s)|| T < ||r(e)|| T for s G G results from | | r ( S ) | | T = ||f( S ,e)|| T = |lff(. S ,-),r(e,-)]|| T
48
II.
HILBERT MODULES AND COVARIANCE KERNELS
L2(G, Q) is a unitary operator, where T is defined by
(^/)(X)= [ (t,x)f(t)s(dt), JG
feL2(G,g)
and j | / | | 2 = H-77II2 for / G L2(G, g). We may write L 2 (G) = L2(G, g) and L2(G) L2(G, g) if g and g are fixed.
=
2.5. HARMONIC ANALYSIS FOR NORMAL HILBERT B(tf)-MODULES
51
Let X be a normal Hilbert 5(H)-module and L2(G;X) be the Hilbert space of all X-valued strongly measurable functions $ on G such that
I l * l l 2 = ( j f ||*(OIL*0( L2(G ; X) is a gramian unitary operator given by 2
(^- 1 *)(t)= / (tTx)*(x)e(dx), JG
teG,*eL2(G;X).
Thus we have the following inversion formula:
*(*)= [Wx){F*)(x)e{dx), JG
teG,^eL2(G;X).
:
52
II. HILBERT MODULES AND COVARIANCE KERNELS
Bibliographical n o t e s For a general theory of inner product modules we refer to Istra^escu [1, Chapter 12](1987) and Lance [1](1995). Inner product modules over an operator algebra (C*-algebra and ylW-algebra) with values in the same algebra was first considered by Kaplansky [1](1953). Later on Saworotnow [1](1968) studied Hilbert modules over an # "-algebra, and Paschke [1](1973) and Rieffel [1](1974) treated modules over C*-algebras. For more infor mation relevant to this chapter we refer to Ambrose [1](1945), Giellis [l](1972), Kakihara [4](1983), Saworotnow [5](1976) and Smith [1](1974). 2.1. Normal Hilbert B(H)-modules. A (normal) Hilbert 5(if)-module was intro duced by Kakihara and Terasaki [l](1979) to treat Hilbert space valued stochastic processes. As was seen in Section 1.3, a normal Hilbert 5(.ff)-module is a natural abstraction of Lg(fi; H). Lemma 1.2 is esssentially due to Kaplansky [1] and Pashke [1]. Proposition 1.5 is due to Ozawa [1](1980). 2.2. Submodules, operators and functionals. Lemma 2.2 was noted by Kakihara and Terasaki [1]. Proposition 2.4 is the collection of results in Kakihara [l](1980), Ozawa [1] and Paschke [1]. Proposition 2.6 was proved in Kakihara and Terasaki [!]■ 2.3. Characterization and structure. The characterization theorem (Theorem 3.1) is in Kakihara [1]. Theorem 3.4, the structure theorem (Theorem 3.6) and Corollary 3.7 are due to Ozawa [1]. Kakihara [17](1994) gave an alternative proof to Theorem 3.6 using Corollary 3.5. Corollaries 3.8 and 3.10 are also given by Ozawa [1]. Proposition 3.11 is in Kakihara [9](1986). 2-4- Positive definite kernels and reproducing kernel spaces. The original idea of RKHS goes back to Moore [1](1916). The entire idea of Section 2.4 is taken from Aronszajn [1](1950); see also Burbea and Masani [1](1984), and Pedric [1](1957). Definition 4.1 through Theorem 4.21 are formulated in Kakihara [10](1987). Propo sition 4.23 is in Kakihara [ll](1988) and Theorem 4.24 is in Kakihara [17]. Some related topics can be seen in Miamee and Salehi [1](1977) and Umegaki [1](1955). 2.5. Harmonic analysis for normal Hilbert B(H)-modules. Propositions 5.2, 5.4 and 5.5 were formulated in a more general setting in Kakihara [4]. Proposition 5.6 is noted by Kallianpur and Mandrekar [2](1971) in the particular case G = Z and B and X = S{K,H). We also refer to Saworotnow [2](1970) and [3](1971).
CHAPTER III
STOCHASTIC MEASURES OPERATOR VALUED
AND
BIMEASURES
The contents of this chapter is fundamental and extensively used in the follow ing chapters since we consider those processes which are representable as integrals w.r.t. X = LQ(Q ; #)-valued measures and their properties are totally determined by representing measures. Thus, in this chapter, we first discuss some technical proper ties involving semivariations and variations of X-valued measures and T(H)-vaiued bimeasures. These variations and semivariations are essential for integral represen tations of various processes and their classifications since the latter depends on the behavior of these different variations and semivariations. Orthogonally scattered dilation and gramian orthogonally scattered dilation are then considered, which are then used to obtain stationary dilations of harmonizable processes. For T(//)-valued measures and bimeasures we can construct L 1 -spaces and L 2 -spaces. Some prop erties of these spaces are studied. When the measurable space is locally compact, we can consider Riesz type representation theorems for operators on (vector val ued) continuous functions. Finally, convergence and approximation properties are obtained for X-valued measures.
3.1.
Semivarations and variations
Let X = Ll(Q;H) be as in Chapter I. That is, the Hilbert space of all Hvalued strong random variables x(-) on the probability space (f2, J, fi) such that E{x) = Jn X(UJ) ii(dui) = 0 and Jn \\X(LU)\\2H fi(dio) < oo, where H is a complex Hilbert space with the inner product (•,■)// a n d the norm j|-||#. Throughout this chapter H is assumed to be separable. Recall that X is a normal Hilbert S(i?)-module with the T(H)-valued gramian [x,y]=
/ x{ui) ® y(u) n(dw), in 53
54
III. STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES
the inner product {x,y)x = tv[x,y] and the norm \\z\\x = [x,x)x = \\[x, z]\\* for x,y e X (cf. Section 2.1). In this section we consider various semivariations and variations of X-valued mea sures and T(H)-valued bimeasures. ^"-valued measures are sometimes referred to as stochastic measures. Interrelations among them are obtained mainly as inequalities. Several examples are presented to distinguish them. Let ( 0 , 21) be a measurable space where 0 is a set and 21 is a cr-algebra of subsets of 0 . We begin with general definitions of Banach space valued measure theory. The reader should refer to Diestel and Uhl [1], Dinculeanu [1] and Dunford and Schwartz [1, IV]. Definition 1. Let X be a Banach space with the norm | | - | | j . ca(2t,3t) denotes the set of all ^-valued c.a. (in the norm) measures on (0,21). For each A 6 21, 11(A) denotes the set of all finite measurable partitions of A. Let £ £ ca(2l, X). The semivariation \\£\\(A) of £ at A 6 2t is defined by
ueii(A)=Sup
Ea^A'
■■aAeC,
|QA| < l , A G w e n ( A n .
A6vr
The variation |f |(A) of £ at A 6 21 is denned by
|e|(A) = suP
Ell^ A )Hx^en(^)
Denote by -uca(2l,3£) the set of all £ 6 ca(2t,3E) of bounded variation, i.e., |£|(0) < oo. R e m a r k 2. Let X be a Banach space and X* be its dual. (1) Let £ e ca(%X). Then ||£||(-) is monotone, i.e., A C B implies ||?||(A) < CO
°°
HfH(B) and countably subadditive, i.e., ||£||( U i „ ) < E "-
1
UClPn) for {An}™=1 C
n=l
21, while |f|(-) is c.a. iff £ e nca(2t, 3£). For A e 21 the following relations hold: U{A)\\x
< U\\(A) < |£|(A),
sup ||£(A n B)ll* < UW(A) < 4 sup U(A n 5 ) | | 2 , sea Bea IKH(A) = sup {|z'£|(A) : * * € £ * , ||x*|U- < 1 } , where (i*£)(-) = a* (£(•)) e ca(2t,C) and |af£|(-) is its variation, | | - | | j . being the norm in X* (cf. Diestel and Uhl [1, pp. 2-5]). Since £ is denned on a cr-algebra, it is bounded, i.e., sup ||£(A)|| 2 < oo by the Nikodym Uniform Boundedness Theorem A£• 0 if An | 0. A(A)->0
A can be chosen so that X(A) < ||£||(A) for A g 21 (cf. Dunford and Schwartz [1, IV.10.5]). (3) (co(2t, 3£), | | | | ( 0 ) ) and (uca(a,3t), |-|(6)) are Banach spaces and the latter is a subspace of the former. The proof is similar to that of Proposition 7 below. We list three classical theorems in vector measure theory in the following since they will be used frequently hereafter. O r l i c z - P e t t i s T h e o r e m (cf. Diestel and Uhl [1, p.22]). / / £ : 21 -> X is weakly c.a., i.e., x*£ € ca(2l, C) for every x* g X*, then £ g ca(2l,X). V i t a l i - H a h n - S a k s - N i k o d y m T h e o r e m (cf. Diestel and Uhl [1, p. 23-25]). Let {^n}^Li C ca(2l,X) fee a sequence such that lim £B(-A) = £(A) exists m norm for n—foo
euen/ J4 6 21, tften £ 6 ca(2l, X) and the countable additivity of {^n}^Li is uniform, i.e., Ak 4,0 implies that lim (,n{Ak) = 0 uniformly in n. k—► oo
N i k o d y m Uniform B o u n d e d n e s s T h e o r e m (cf. Diestel and Uhl [1, p. 14]). Let {£,a}aex Q ca(%L,X) be a family such that sup||£a(4)||* eH.
(1.2)
Hence, if we define V : X ->■ S ( # , L § ( Q ) ) by (7x = T£ for a; G X, then [/ gives the isomorphism in (1.1) (cf. (1.3.4), (1.3.5) and p. 12). We shall frequently use this identification later. The semivariation and the variation for X-valued measures are defined as in Definition 1. We introduce several other notions in the following. Definition 4. Let £ E ca(%X). (1) The operator semivariation ||£|| 0 (A) of £ at A e 2t is defined by
ll£llo(A)=sup
£ «AC(A)
aA e £(#}, ||aA|| < 1, A e n e U{A) } .
A£TT
6 c a ( a , X ) denotes the set of all £ € ca(2l, X) of bounded operator semivariation, i.e., ||€|U(e) < 00. (2) The strong semivariation ||£|| S (A) of £ at A £ 21 is defined by
na(A) = suP.
Yl Te(A)*A
feefl, II0AIIH)H = % ) ? € ca(a,Lg(fi)) and | | ^ | | ( A ) is the semivariation of £4, at A.
3.1. SEMIVARATIONS AND VARIATIONS
57
(4) The weak variation \£.\W{A) of £ at A G 21 is defined by |fU(A) = sup {|&|(A) : 4, e H, | H | „ < l } , where | ^ | ( A ) is the variation of ^ at A. If X is a Banach space and is also a left 5(.ff)-module, then for £ € ca(2l,£) we can define the operator semivariation ||£||0(") as in Definition 4 (1). If, in particular, F G ca(2t, T(H)), then we see that for A G 21 |F|| 0 (A) = |F|(A) = sup
(1.3) A6TT
where the supremum is taken over 6A G B ( # ) with ||6A|| < 1 for A 6 TT e 11(A). This can be seen from the proof of Theorem 5 (2) below. T h e o r e m 5. (1) Let £ G ca(2t,X) and A G 21. Tfeen ii holds that \\aA)\\x<sup{U(AnB)\\x:Be%}
< U\U) =
sup{\(a-),x)x\(A):xeX,\\x\\xl,B) = sup
Y, £
aA&yM(A,A';
AgTT A'GTr'
||m||(A,B)=sup
£
^
' A€TT A'STT'
aA£A,m(A,A') '
where the suprema are taken over 0 4 , /3/y 6 C with |QA|, |/?A' | < 1 for A g 7r g 11(A) and A' g n' g 11(B). The terminology of Vita/z variation is sometimes used to stand for variation of bimeasures. The terminology of "semivariation" has usually been used for a vector measure and not for a bimeasure, and Frechet variation is used for bimeasures. However, we adopt the terminology of semivariation for both vector measures and bimeasures. Note that for M G OH (21 x 21; X), ||M||(-, ■) is separately monotone and countably subadditive, and \M\(-, ■) is separately c.a. iff M g 9K„(2t x 21; X).
3.1. SEMIVARATIONS AND VARIATIONS
63
Immediate consequences of the Orlicz-Pettis Theorem and the Vitali-Hahn-SaksNikodym Theorem are: T h e o r e m 10 (Orlicz-Pettis T h e o r e m for B i m e a s u r e s ) . Let X be a Banach space and X* be its dual. Then M : 2t x 21 —> X is an X-valued bimeasure iff M is separately weakly c.a., i.e., x* M is a scalar bimeasure for every x* € X*. T h e o r e m 11 ( V i t a l i - H a h n - S a k s - N i k o d y m T h e o r e m for B i m e a s u r e s ) . Let {Mn}^L1 C 9Jl(2l x 21; X) be a sequence of X-valued bimeasures such that lim Mn{A,B) = M(A,B) exists for every A,B €21. Then, M e 3Jt(2l x 2t; 3£) and n—yoo
are separately uniformly {M n }™ =1 In Theorem 11 it holds that
c.a.
||Af„||(i4,B) a) is clear from Lemma 19 (5). (2) follows from (1) by taking H = C. (3) Observe that the following equality holds for every A G 21: |e|(A)=sup
E
5A
^A)
(1.14)
Ag7T
where the supremum is taken over 5 A € B{X) with ||5A|| < 1 for A g -K e n(A). If we identify X with S(X, C), then the gramian in this case is [x, y}0 = x ® y, i.e., [x,y]02 = (z,y)xx for x,y,2 G X. Note that [x,y]Q > 0 iff (a:,y)x = tr[x,y] 0 = ||rc ® y||T = ||x||x||y||x iff there exists some a > 0 such that x = ay oi y = ax.
70
III.
STOCHASTIC MEASURES AND OPERATOR VALUED BIMEASURES
Also note that the variation |£|(-) of £ £ ca(2t,X) equals the operator semivariation HCIU-) of £ e ca(Sl, S(X,€)), which is seen from (1.14). Now since h) => g) is obvious we prove g) => h). Without loss of generality we may assume that there is some B £ 21 such that £(B) ^ 0. Put x = £(B). By (1) we have that [£(A), £(B)]o = £(;4) ®x" > 0 for yl £ 21. Hence there exists some I/(J4) > 0 such that £(A) = v{A)x for A £ 21. It is immediate that v 6 ca(2l,R+). Therefore h) holds. R e m a r k 23. (1) Theorem 5(3) and (4) follow from Theorem 22(1) and (2), re spectively. (2) It follows from the proof of Theorem 22(1) that, for x,y £ X, [x,y] > 0 iff \\x + y\\x > ||u ■ x + y\\x for every unitary u £ B(H). Hence, [x,y] = 0 iff ||x ± y\\x > ||w • x + y\\x for every unitary u £ B(H). Moreover, if H = C, we have that, for f,g £ Lg(O), (/,ff) 3 > 0 iff ||/ +
