Finally, as an application we shall give a new proof of the Wold decomposition theorem for discrete stationary processes in complete correlated ac- tions

ORTHOGONAL DECOMPOSITIONS IN HILBERT C^∗–MODULES AND STATIONARY PROCESSES

D. POPOVICI

Abstract. It is obtained a Wold-type decomposition for an adjointable isometry on a HilbertC^∗-module which is sequentially complete with respect to some locally convex topology, denoted bys. Particularly self-dual HilbertC^∗-modules satisfy this condition. Finally, as an application we shall give a new proof of the Wold decomposition theorem for discrete stationary processes in complete correlated ac- tions.

1. Introduction

Hilbert modules over aC^∗-algebra were introduced by I. Kaplansky in [2], the variety of applications, emphasized later by the papers of W. L. Paschke [6] and M. A. Rieffel [9], inciting the interest on these objects.

Today Hilbert C^∗-modules represent an important instrument of study in a general K-theory introduced by Kasparov (to see for example [3]) and called KK-theory, in theC^∗-algebraic approach to quantum group theory (to see [15]), but also in the study of various prediction problems in correlated actions.

In the development of the prediction theory, at the same time with factorization theorems by analytic functions (for example, using the Szeg¨o [12] factorization of a positive scalar function by an analytic scalar function, Kolmogorov [4] give an elegant solution for the univaried prediction problem), another result, of geometric type, namely the Wold decomposition theorem, in its various variants, played a very important role. This decomposition theorem for discrete stationary processes in complete correlated actions allow us to obtain (in some supplementary Harnack- type condition) the predictible part of a process, and also the prediction error operator. We cannot omit here the contribution of H. Wold in [14], and also of I. Suciu and I. Valu¸sescu for example in [10], [11] or [13].

In the following we shall approach this last subject in connection with Hilbert modules. The decomposition theorem for discrete stationary processes in complete correlated actions mentioned above can be obtained as an application of the main

Received July 6, 1997; revised February 26, 1998.

1980Mathematics Subject Classification(1991Revision). Primary 46L10; Secondary 60G15.

Key words and phrases. HilbertC^∗-modules, Wold decomposition, stationary processes in complete correlated actions.

result of this paper, a Wold-type decomposition for adjointable isometries on cer- tain Hilbert C^∗-modules (in a class which contains in particular self-dual Hilbert modules).

2. Notations and Preliminaries

LetAbe aC^∗-algebra. Apre-HilbertA-moduleis a rightA-moduleE(hav- ing a compatible vector space structure) equipped with a maph·,·i:E×E →A linear in the second variable and having the properties:

(i) hx, xi ≥0, x∈E; hx, xi= 0 if and only ifx= 0;

(ii) hx, yi^∗ =hy, xi, x, y∈E;

(iii) hx, yai=hx, yia, x∈E, a∈A.

E is said to be aHilbertA-moduleif verifies in addition (iv) E is complete with respect to the norm

kxk:=khx, xik^1/2, x∈E.

A pre-HilbertA-moduleEis said to beself-dualif every continuousA-module mapτ:E→A has the form

τ(x) =hy, xi, x∈E, y∈E being fixed.

If A is a von Neumann algebra, the s-topology on E is the locally convex topology onE generated by the family of seminorms (pφ)_φ∈A⁺∗,

pφ(x) =

φ(hx, xi)1/2

, x∈E

(we denoted byA⁺∗ the set of all positive linear functionals in the predual ofA).

IfE is self-dual thenE is the dual of a Banach spaceE^∗([6]), so its closed unit ball isσ(σ(E, E∗))-compact. Furthermorescontainsσand ifxα→σ xthen (1) φ(hxα, yi)→φ(hx, yi), φ∈A⁺∗, y∈E.

We can prove now that (E, s) is quasi-complete and therefore sequentially com- plete. Indeed, if (xα)α is a boundeds-Cauchy net in E, it is alsoσ-Cauchy and thereforeσ-convergent to anx∈E. For eachφ∈A⁺∗, (xα)α being Cauchy, con- verges to an xφ in a Hilbert space Eφ, the completion ofE with respect to the sesquilinear formφ◦ h·,·i. Due to (1)xφ=x,φ∈A⁺∗ and so (xα)αiss-convergent tox, remark which complete the proof.

For a pre-Hilbert module collection (Eα)α over a von Neumann algebraA its ultraweak direct sum(according to [6]) is the pre-HilbertA-module

α+∈IEα=n

x= (xα)α∈I ∈ Y

α∈I

Eαsup

F

X

α∈F

hxα, xαi<∞o ,

F belonging to the setF of all finite parts ofI.

A submoduleE0of a Hilbert moduleEover aC^∗-algebraAis calledcomple- mentableif there exits a submoduleE1such thatE=E0+E1andhE0, E1i= 0.

We use the notationE=E0⊕E1.

If E and F are Hilbert A-modules a map T: E →F is called adjointable if there existsT^∗:F →E such that

hT x, yi=hx, T^∗yi, x∈E, y∈F.

Denote by L_A(E, F) (L_A(E) if E = F) the set of all these maps. For T in L_A(E, F) we shall use the notation [E, F, T] (respectively [E, T] ifE=F). Every adjointable operator is a boundedA-module map. In some additional conditions onE the converse is also true:

Proposition 2.1. If E is self-dual then every bounded A-module map is ad- jointable.

AnA-module mapU:E→F is calledunitaryif it is isometric and surjective.

Using the terminology from [7] an isometry [E, V] is called ashiftif E=

M∞ n=0

VⁿL

:=n x=

X∞

n=0

Vⁿlnln∈Land X

n

hln, lniconverges in norm inAo , whereL= kerV^∗.

The following result gives a necessary and sufficient condition on a Hilbert module adjointable isometry in order to admit a Wold-type decomposition.

Theorem 2.2([7]). An isometry [E, V] admits a unique decomposition of the formE=E0⊕E1 where:

• E0, E1 reducesV;

• V|E0 is a unitary operator;

• V|E1 is a shift if and only if

(hV^∗nx, V^∗nxi)n converges in norm in Afor allx∈E.

3. s-Shifts and Unitary Operators

For the sake of completeness we shall give now other proofs for the results obtained by E. C. Lance in [5] regarding the characterizations of unitary operators and adjointable isometries on HilbertC^∗-modules.

Lemma 3.1. Let A be a C^∗-algebra, E, F be two Hilbert A-modules and V:E→F be an isometricA-module map. Then

hV x, V yi_F =hx, yi_E, x, y∈E.

Proof. Firstly let us observe thatV E⊂F is a closed submodule. The operator V0:E→V E, V0x:=V x, x∈E

is bijective and, in addition,V₀⁻¹ is a boundedA-module map. Using the charac- terization in [6] of boundedA-module mapsT:E→F by

hT x, T xi_F ≤ kTk²hx, xi_E, x∈E, we can write

hx, xi_E =hV₀⁻¹V x, V₀⁻¹V xi_E≤ kV₀⁻¹k²hV x, V xi_{V E}=hV x, V xi_F. Applying the same inequality forT =V we obtain

hV x, V xi_F ≤ hx, xi_E

which leads, using the polarization identity, to the conclusion.

Starting with this lemma, result which represents, in fact, the main part in the proofs given by E. C. Lance we can enunciate:

Proposition 3.2. LetA, E, F be as in the previous lemma andU:E→F be anA-module map. ThenU is a unitary operator (that is isometric and surjective) if and only if U ∈ L_A(E, F),U^∗U =IE andUU^∗=IF.

Proposition 3.3. WithA, E, F as above we considerV: E→F a linear map.

Then are equivalent:

(i) V is an isometricA-module map with complentable range;

(ii) V ∈ L_A(E, F)andV^∗V =IE.

In the following we suppose that Ais a von Neumann algebra if it is not oth- erwise specified.

Definition 3.4. Let E be a Hilbert A-module. A map S: E → E is called an s-shiftif there exists a Hilbert A-moduleF such thatS andSF are unitary equivalent (we denoted bySF the operatorSF: +^∞

n=0F → +^∞

n=0F, SF(x0, x1, . . .) = (0, x0, x1, . . .)).

We deduce some remarks from this definition.

Remark 3.5.

• Since S and SF are unitary equivalent (that is S = U^∗SFU, [E, +^∞

n=0F, U]

being unitary),S is an adjointable isometry onE.

•Identifying F with {(x,0, . . .) | x∈F}we can considerL =U^∗F, relation which permits to observe thatSⁿL=U^∗S_FⁿF,n∈N. Because

hSⁿL, Li=hU^∗Sⁿ_FF, U^∗Fi=hS_FⁿF, Fi= 0, n >0 Lis wandering forS (that ishSⁿL, S^mLi= 0,m, n∈N,m6=n).

Lemma 3.6. LetE be a pre-Hilbert A-module,(Eα)α∈I a parwise orthogonal family of submodules inE and (xα)α∈I ∈ Q

α∈IEα. If (xα)α∈I iss-summable then

Fsup∈F

k P

α∈F

hxα, xαik<∞. Proof. Denote byx=s- lim

F∈F

P

α∈Fxα. Because for eachφ∈A⁺∗, φ(P

α∈F

hxα, xαi)

F∈F

−→ φ(hx, xi) the uniform boundedness principle applyied to the family (TF)F∈F, TF:A^∗→C, TF(φ) = X

α∈F

φ(hxα, xαi), φ∈A^∗

shows that the net (kTFk)F∈F is bounded. The conclusion follows because kTFk²=k P

α∈F

hxα, xαik,F ∈ F.

Definition 3.7. Let E be a pre-Hilbert A-module and (Eα)α∈I a family of parwise orthogonal submodules ofE. We call thedirects-sumof the submodules Eα,α∈Ithe set

α+∈IEα:=n

x=s- lim

F∈F

X

α∈F

xαxα∈Eα(α∈I) and (xα)α∈I iss-summableo .

Remark 3.8.

•Because the map (E, s)3x7→xa∈(E, s),a∈A fixed, is continuous it can be easily proved that +

α∈IEαis anA-submodule ofE.Furthermore, if x∈ +

α∈IEα

andφ∈A⁺∗ then h

φ(X

α∈F

hxα, xαi)i1/2

−[φ(hx, xi)]^1/2=pφ(X

α∈F

xα)−pφ(x)

≤pφ(X

α∈F

xα−x)^F−→^∈F0

and consequently P

α∈F

hxα, xαi

F∈F ultraweakly converges to hx, xi. Using the polarization identity we can assert the same about the convergence of the net P

α∈F

hxα, yαi

F∈F tohx, yifor allx, y∈ +

α∈IEα.

•(xn)n∈ +^∞

m=0F if and only if (xn)n =s- lim

m→∞

Pm

k=0(0, . . . , xk,0, . . .).

Indeed, for allφ∈A⁺∗ andk∈N,

φ(h(x0, . . . , xk,0, . . .)−(xn)n,(x0, . . . , xk,0, . . .)−(xn)ni)

=φX^k

p=0

hxp, xpi − h(xn)n,(xn)ni_k→∞

−→ 0 ([6]).

We used here that

h(x0, . . . , xk,0, . . .),(xn)ni= Xk p=0

hxp, xpi,

which is a true relation because (hx0, x0i+· · ·+hxk, xki+h0, xk+1i+· · ·+h0, xmi)m

ultraweakly converges to (h(x0, . . . , xk,0, . . .),(xn)ni and is a constant equal to Pk

p=0

hxp, xpi.

The converse is an immediate consequence of Lemma 3.6.

Proposition 3.9. Let E be a HilbertA-module and S:E →E an isometric A-module map. If S is ans-shift then there exists a closed A-submoduleL of E wandering forS such that

E= +^∞

n=0SⁿL.

Furthermore, if(E, s)is sequentially complete the converse also holds.

Proof. For the first part it is sufficient to prove that E = +^∞

n=0SⁿL, where L=U^∗F (we used the notations and results in the Remark 3.5). Indeed

E=U^∗ +^∞

n=0F

=n

U^∗(xn)n|(xn)n=s- lim

m→∞

Xm k=0

(0, . . . , xk,0, . . .)o

=n

x∈E |x=s- lim

m→∞

Xm k=0

U^∗(0, . . . , xk,0, . . .)o

=n

x∈E |x=s- lim

m→∞

Xm k=0

S^klk

o

= +^∞

n=0SⁿL,

the last equality, more precisely the inclusion from left to right (the other one being obvious), being obtained due to the relations

φ(hX

k∈F

xk−x,X

k∈F

xk−xi)

=φ(h Xn k=0

xk−x, Xn k=0

xk−xi)−φ(h X

k∈F\{0,···n}

xk, X

k∈F\{0,···n}

xki)

and

φ(h X

k∈F\{0,···n}

xk, X

k∈F\{0,···n}

xki)≤φ(h Xm k=n+1

xk, Xm k=n+1

xki), withx∈E,xk ∈S^kL,k= 0, n,n∈N,F ∈ F:{0, . . . , n} ⊂F,m= maxF.

For the converse let L be a closed A-submodule, wandering for S with the property E = +^∞

n=0SⁿL. We build U: E → +^∞

n=0L, U(x) = U(s- lim

n→∞

Pn

k=0S^klk) = (ln)n,x∈Eandln∈L,n∈N. We used here that ifx∈ +^∞

n=0SⁿL,xis defined by ans-summable family (Sⁿln)n∈N namelyx=s- lim

F∈F

P

k∈FS^klk. So we can deduce that x= s- lim

n→∞

Pn

k=0S^klk also. The facts that the definition is correct and U is isometric can be obtained from Lemma 3.6 and the following relations

hUx, Uxi=h(ln)n,(ln)ni= lim

n→∞

Xn k=0

hlk, lki= lim

n→∞

Xⁿ

k=0

S^klk, Xn k=0

S^klk

=hx, xi

(the symbol lim above indicates the ultraweak convergence, the last equality being a consequence of the first part in Remarks 3.8). Using the sequential completeness of (E, s) we can easily prove the surjectivity ofU. Therefore, with the results in Proposition 3.2,S=U^∗SFU, that is S is ans-shift.

4. The Wold-Type Decomposition

As we stated before A will denote a von Neumann algebra unless otherwise specified.

LetE be a HilbertA-module and [E, V] an isometry. As we detailedly proved in [7], for eachn∈N,

E=L⊕V L⊕ · · · ⊕VⁿL⊕Vⁿ⁺¹E, where L= kerV^∗. So eachx∈E can be written as

(2) x=

Xn k=0

V^klk+Vⁿ⁺¹zn+1, {lk}ⁿ_k=0⊂L,zn+1∈E being given by the formulas

lk= (IE−V V^∗)V^∗kx, zn+1=Vⁿ⁺¹V^∗n+1x, n∈N.

Before we enunciate the main theorem of this section let us give a definition.

Definition 4.1. [E, V] admits a Wold-type decomposition if there exist two submodulesE0,E1⊂E such that

(i) E=E0⊕E1;

(ii) E0 (and consequentlyE1) reducesV;

(iii) V|E0 is a unitary operator andV|E1 is ans-shift.

Theorem 4.2. Suppose that(E, s)is sequentially complete. Then the isometry [E, V]admits a Wold-type decomposition. This decomposition is unique.

Proof. Forx∈ E and xn =Pn

k=0V^k(IE−V V^∗)V^∗kxwe obtain, due to (2), the equality

(3) x=xn+Vⁿ⁺¹V^∗(n+1)x, n∈N.

Let us observe that (VⁿV^∗nx)n is an s-Cauchy sequence. Indeed, because (hV^∗nx, V^∗nxi)n is a decreasing sequence of positive elements there exists an ele- menta∈Asuch that

φ(hV^∗nx, V^∗nxi)ⁿ−→^→∞φ(a), for allφ∈A⁺∗

(every functionalφ∈A⁺∗ being normal). Consequently (4) φ(hV^∗nx, V^∗nxi − hV^∗mx, V^∗mxi)−→^m,n0. Furthermore, form, n∈N,m < n,

hVⁿV^∗nx−V^mV^∗mx, VⁿV^∗nx−V^mV^∗mxi=hV^∗mx, V^∗mxi − hV^∗nx, V^∗nxi which, due to (4), permits us to conclude that (VⁿV^∗nx)n is s-Cauchy. Since (E, s) is sequentially complete there exists xu = s- lim

n→∞VⁿV^∗nx. By passing to limit in (3) it obtains the decomposition

x=xu+xs, xu=s- lim

n→∞VⁿV^∗nx, xs=s- lim

n→∞

Xn k=0

V^klk ∈ +^∞

n=0VⁿL.

Since, forn∈Nfixed, VⁿE is s-closed and, for m≥n,V^mV^∗mx∈VⁿE we obtain thatxu ∈ T

n≥0VⁿE. Furthermore, ifx∈ +^∞

n=0VⁿLandy∈ T

n≥0VⁿE then hx, yi=

s- lim

n→∞

Xn k=0

V^klk, y

= 0

because for eachn∈ N, hVⁿln, yi= 0. Using the notations E0 = T

n≥0VⁿE and E1= +^∞

n=0VⁿLit obtains the decomposition we are looking for, that isE=E0⊕E1. It is immediate thatE0,E1reduceV,V|E0 is a unitary operator, andV|E1is an s-shift.

For the uniqueness let us suppose that these exist other two submodules E₀⁰, E₁⁰ ⊂E with the properties (i)-(iii) in Definition 4.1. Since V|E₀⁰ is unitary, each x ∈ E0⁰ has the form x = VⁿV^∗nx, n ∈ N and so x ∈ T

n≥0VⁿE = E0, that is E₀⁰ ⊂E0. Now ifx∈E then the decompositionE =E₀⁰⊕E₁⁰ allows to assert that x=x⁰_u+x⁰_s, with x⁰_u ∈ E₀⁰ ⊂E0 ⊂ V E, and x⁰_s =s- lim

n→∞

Pn

k=0V^kl_k⁰ ∈ L⁰+V E, where L⁰ = ker(V^∗|E₁⁰)⊂L. HenceE =L⁰⊕V E, that isL=L⁰, E1 =E₁⁰ and

E0=E0⁰.

Because, for a self-dual Hilbert moduleE, (E, s) is sequentially complete as we have already stated in the second section of this paper we can formulate

Corollary 4.3. LetE be a self-dual Hilbert A-module. Every isometry [E, V] admits a unique Wold-type decomposition.

Remark 4.4.

•As we saw in [7],E0={x∈E| hx, xi=hV^∗nx, V^∗nxi,n∈N}. Furthermore E1={x∈E |V^∗nx−→^s 0}. Indeed, letx∈E. Thenx=xn+Vⁿ⁺¹V^∗(n+1)x∈ E1 if and only ifVⁿ⁺¹V^∗(n+1)x−→^s 0.

•If, in addition, (hV^∗nx, V^∗nxi)n converges in norm inA for all x∈ E then the Wold-type decomposition obtained above coincides with the decomposition in Theorem 2.2. In this case

M∞ n=0

VⁿL= +^∞

n=0VⁿL, whereL= kerV^∗.

• An isometry [E, V] is an s-shift if and only if V^∗nx −→^s 0, for all x ∈ E.

Indeed, if [E, V] is ans-shift andx∈E then

hV^∗nx, V^∗nxi=

V^∗n s- lim

m→∞

Xm k=0

V^klk

, V^∗n s- lim

m→∞

Xm k=0

V^klk

= s- lim

m→∞

Xm k=0

V^kln+k, s- lim

m→∞

Xm k=0

V^kln+k

which, by the first part of the Remarks 3.8, is the ultraweak limit of the se- quence (Pm

k=0hln+k, ln+ki)_m. If a∈A is the least upper bound of the sequence (Pn

k=0hlk, lki)_n, then φ(Pn

k=0hlk, lki)ⁿ−→^→∞ φ(a) for every φ∈A⁺∗, and the con- clusion follows.

Conversely, ifV^∗nx−→^s 0, for everyx∈Erelation (3) shows thatE= +^∞

n=0VⁿL.

The sequential completeness of (E, s) and Proposition 3.9 prove thatV iss-shift.

Definition 4.5. LetE be a Hilbert A-module. An isometry [E, V] is called completely non-unitary(c.n.u.) if the restriction to every submoduleF reduc- ing forV is not a unitary operator (excepting the caseF ={0}).

Corollary 4.6. LetE be a HilbertA-module and[E, V] an isometry. IfV is an s-shift thenV is c.n.u. Conversely, if (E, s) is sequentially complete andV is c.n.u. then V is ans-shift.

Proof. Without any difficulty we obtain that V is c.n.u. if and only if T

n≥0VⁿE = {0}. If V is an s-shift then V^∗nx −→^s 0, for all x ∈ E. If there exists anA-submoduleE⁰ ⊂Ereducing forV such thatV|E⁰ is a unitary opera- tor then for eachx∈E⁰,hx, xi=hV^∗nx,V^∗nxi, n∈Nthat is x= 0.

Conversely, we write the Wold-type decomposition corresponding to V. The unitary part being null it obtains thatV is ans-shift.

5. Application to Stationary Processes

LetHbe a Hilbert space,L(H) the space of all linear and bounded operators onH, E a rightL(H)-module andh·,·i:E×E→ L(H) an application.

Definition 5.1. {H, E,h·,·i}is called the correlated action of L(H) on E if (E,h·,·i) is a pre-HilbertL(H)-module.

Example 5.2. LetK be a Hilbert space. If we define

h·,·i:L(H,K)× L(H,K)→ L(H), hS, Ti:=S^∗T, S, T ∈ L(H,K) we obtain a correlated action{H,L(H,K),h·,·i}called theoperator model.

Every correlated action can be embedded in a correlated action of the type presented in the previous example as it results from the following proposition. Its complete proof can be found for example in [13].

Proposition 5.3. Let{H, E,h·,·i}be a correlated action. Then there exist a Hilbert spaceK and an algebraic embeding

E3x7−→^ϕ ϕ(x)∈ L(H,K)with the properties:

(i) hx, yi=ϕ(x)^∗ϕ(y), x, y ∈E;

(ii) K={ϕ(x)h|x∈E, h∈ H}.

This decomposition is unique (up to a unitary equivalence).

If the mapϕfrom Proposition 5.3 is surjective{H, E,h·,·i}is called acomplete correlated action.

In the following we shall use only complete correlated actions.

Adiscrete stationary processis a sequence{fn}_n of elements ofEwith the property that hfn, fmi depends only on the differencem−n and not onm orn separately.

The stationary process{gn}_n∈Zis calledwhite noiseifhgn, gmi= 0 form6=n.

The stationary process{fn}_n contains the white noise{gn}_n if:

(i) hfn, gmidepends only on the differencem−nand is equal to 0 form > n;

(ii) ϕ(g0)H ⊂ W⁰

k=−∞ϕ(fk)H; (iii) Rehfn−gn, gni ≥0.

The stationary process {fn}_n is calleddeterministic if it contains no non-null white noise. {fn}_n is called amoving average for the white noise{gn}_n if

(i) {fn}_n contains{gn}_n; (ii) ^∞W

n=−∞ϕ(fn)H= W^∞

n=−∞ϕ(gn)H.

Remark 5.4. As we detailedly presented in [8], for every complete correlated action {H, E,h·,·i} E is a self-dual Hilbert L(H)-module. Because an s-closed submodule in a self-dual Hilbert C^∗-module is also self-dual, it will be comple- mentable (the complete proof can be found in [1]). Consequently, if E1 is an L(H)-submodule in E we can consider the orthogonal projection associated to E1s

and denoted by P_E1^s.

Proposition 5.5. Let {H, E,h·,·i} be a complete correlated action and E1 a L(H)-submodule ofE. Then

ϕ(P_E1^sx) =PK₁ϕ(x), x∈E, where K₁ =W

x∈E1ϕ(x)H, and P^K1 is the orthogonal projection inK associated to its closed subspace K₁.

Proof. Letx∈E.Thenx⊥E1if and only ifhx, yi= 0, that isϕ(x)^∗ϕ(y) = 0, for everyy∈E1. Because{ϕ(z)h|z∈E1,h∈ H}is dense inK₁we obtain that ϕ(x)^∗|K₁= 0,which is the same withϕ(x)^∗PK₁= [PK₁ϕ(x)]^∗= 0. The conclusion follows using thatϕ⁻¹P^K1ϕis also an orthogonal projection as it isP_E1^s. The next theorem is called the Wold decomposition theorem for discrete sta- tionary processes.

Theorem 5.6. Let {fn}_n∈Z be a discrete stationary process in the complete correlated action {H, E,h·,·i}. There exists a unique decomposition of the form

fn =un+vn, n∈Z where

(a) {un}_n is a moving average of the maximal white noise contained in{fn}; (b) {vn}_n is a deterministic process;

(c) hun, vmi= 0, for allm, n∈Z.

Proof. ConsiderE∞^f and respectively E_n^f (n ∈ Z), the s-closed L(H)-submo- dules generated by{fm}_m∈Zand respectively{fm}_m≤n. We build

Uf:E∞^f →E∞^f , Uf(fn) =fn+1, n∈Z.

The relation Uf

X

n

0fnTn , Uf

X

m

0fmSm

=X

n,m

0T_n^∗hfn+1, fm+1iSm

=X

n,m

0T_n^∗hfn, fmiSm=X

n

0fnTn,X

m

0fmSm

, Tn, Sm∈ L(H)

(the notationP⁰represents finite sums) shows thatUf is well-defined and isomet- ric. The surjectivity proves thatUf is a unitary operator.

Because E₀^f is invariant toU_f^∗ we can define the isometryV =U_f^∗|E₀^f which, by Proposition 2.1, is adjointable. Using Corollary 4.3 V admits a Wold-type decomposition

(5) E₀^f = +^∞

n=0VⁿL⊕ \

n≥0

VⁿE₀^f, whereL⊕V E₀^f =E₀^f. Write the decomposition (5) in an operator form

I_E^f

0 =P+Q, P, Qprojections.

Define, for everyn∈Z,

un=U_fⁿP f0 and vn =U_fⁿQf0.

We shall prove in the following that{un}_n and{vn}_n verify the theorem conclu- sion.

It is obvious thatfn=un+vn, n∈Zand because

hun, umi=hU_fⁿP f0, U_f^mP f0i=hU_f^n−mP f0, P f0i

{un}_n is a discrete stationary process. Analogously it proves that{vn}_n is also a stationary process.

Furthermore, form, n∈Z,

hun, vmi=hU_fⁿP f0, U_f^mQf0i=hP f0, U_f^m−nQf0i= 0 because (U_f^∗|E₀^f)|T

n≥0VⁿE₀^f is unitary and soU_f^∗ T

n≥0VⁿE₀^f

=T

n≥0VⁿE₀^f, that isU_f^m−nQf0∈T

n≥0VⁿE₀^f. This proves (c).

Let us denote by gn = U_fⁿPLf0, n ∈ Z. We shall prove that {gn}_n is the maximal white noise contained in{fn}_n.

Because for example form > n,

hgn, gmi=hPLf0, U_f^m−nPLf0i=hV^m−nPLf0, PLf0i= 0,

{gn}_n is a white noise. Furthermore {gn}_n is contained in {fn}_n. Indeed (i) it obtains by

hfn, gmi=hU_f^mf0, U_fⁿPLf0i=hV^n−mf0, PLf0i= 0, forn > m. For (ii) it is sufficient to observe that, forh∈ H,

ϕ(g0)h=ϕ(PLf0)h=ϕ(f0)h−ϕ(P_{V E}^f

0f0)h

=ϕ(f0)h−ϕ(P_Ef

−1f0)h=ϕ(f0)h−P_Kf

−1ϕ(f0)h∈ K^f

0.

We used here Proposition 5.5 and the notationsK^f_n=W

k≤nϕ(fk)H,n∈Z. Since Rehfn−gn, gni=RehU_fⁿf0−U_fⁿPLf0, U_fⁿPLf0i

=Rehf0−PLf0, PLf0i= 0, (iii) is also proved.

The maximality, the facts that{un}_n is a moving average for{gn}_n,{vn}_n is a deterministic process and the uniqueness of the Wold decomposition can be proved

in a way similar to the classic case.

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D. Popovici, Department of Mathematics, West University of Timi¸soara, Bd. V. Pˆarvan no. 4, 1900 Timi¸soara, Romania,e-mail: [email protected], [email protected]