on the occasion of his 70th birthday
SOME CONNECTIONS BETWEEN
THE MAXIMAL FUNCTION AND LINEAR SYSTEMS
ILIE VALUS¸ESCU
The aim of this paper is to establish some connections between the maximal function of a contraction and the linear systems theory. Some relations between the maximal function and the characteristic function of the same contraction are recalled, and some new one are found. For the conservative system given by the rotation operator, an investigation with the maximal function is given, and some characterizations in terms of the maximal function are found. Implications for systems having the main operator a contraction are obtained, and a study using the maximal function is suggested. Some implicit utilizations of the maximal function are analyzed, and explicit forms are given.
AMS 2000 Subject Classification: 47A20, 47N70.
Key words: contraction, unitary dilation, maximal function, systems.
1. MAXIMAL FUNCTIONS
Let T be a contraction on a complex separable Hilbert space H. As usually, byDT = (I−T∗T)1/2,DT∗ = (I−T T∗)1/2 will be denoted thedefect operators of T, and by DT = DTH, DT∗ =DT∗H will be denoted the defect spaces of the contraction T.
In [14], the maximal outer function {H,DT∗,Θ1(λ)} attached to the semispectral measure of a contraction T was obtained in the form
(1.1) Θ1(λ) =DT∗(I−λT∗)−1, λ∈D, and was called the maximal function of the contractionT.
Analogously, the maximal function ofT∗ will be of the form (1.2) Θ2(λ) =DT(I−λT)−1, λ∈D.
Between the maximal function Θ1(λ) and the characteristic function of the contraction T, ΘT(λ) = [−T +λDT∗(I−λT∗)−1DT]|DT,besides the ob- vious relation ΘT(λ) = [−T+λΘ1(λ)DT]|DT, in [14] was found the following
MATH. REPORTS12(62),2 (2010), 189–199
connection between these two analytic functions, namely (1.3) ΘT(λ)DT = Θ1(λ)(λI−T), λ∈D, which can be also written into the matrix form
(1.4)
Θ1(λ) ΘT(λ)
T−λI DT
= 0.
In some investigations (see e.g. [7]), an operator T on His called stable ifTn→0, and∗-stable ifT∗n→0, strongly as n→0. Actually, in the linear system theory, a system with the main operatorT is called stable, if it is stable and ∗-stable.
To the maximal function Θ1(λ) the operator Θ1 : H → H2(DT∗) is attached by
(1.5) (Θ1h)(λ) = Θ1(λ)h.
This operator will be useful in applications of the maximal function in systems theory. Analogously, for the maximal function of T∗ the operator Θ2 :H → H2(DT) is defined by
(1.6) (Θ2h)(λ) = Θ2(λ)h.
Proposition 1.1. If T is a ∗-stable contraction, then its maximal func- tion{H,DT∗,Θ1(λ)}is bounded, and the attached operatorΘ1 defined by (1.5) is an isometry. Moreover, the Sz.-Nagy–Foias functional model[12] reduces to a functional representation given by the maximal function Θ1(λ). Namely, the imbedding of H into the functional model is given by
(1.7) H={u∈H2(DT∗)|u(λ) = Θ1(λ)h, h∈ H}, and the contraction T is represented by
(1.8) T∗u(λ) = 1
λ[Θ1(λ)h−Θ1(0)h].
Proof. The fact that for a ∗-stable contraction T the imbedding of H into the Sz.-Nagy–Foias functional model H is given by (1.7) was proved in [14], and is based on the fact that the functional model (see [12], Ch.VI) is obtained by a unitary imbedding of the dilation space into a functional space.
For any contractionT and h∈ H we have
n
X
k=0
kDT∗T∗khk2 =
n
X
k=0
D2T∗T∗kh, T∗kh
=
n
X
k=0
(kT∗khk2− kT∗k+1hk2) =
=
n
X
k=0
kT∗khk2−
n+1
X
k=1
kT∗khk2=khk2− kT∗k+1hk2.
SinceT is∗-stable, the previous relation becomes
∞
P
n=0
kDT∗T∗nhk2=khk2,and taking into account that Θ1(λ) =
∞
P
n=0
DT∗T∗nλn,it follows that the attached operator Θ1 :H →H2(DT∗) is an isometry.
A dual representation can be found in the stable case forT∗.
Other relations between the maximal function and the characteristic function will be given in the next section, in the context of linear systems.
As a remark, even for the characteristic function we have the dual formula ΘT∗(λ) = ΘT(λ)∗, between the maximal functions there exists no such a duality relation except the case when T is a normal operator. Moreover, the characteristic function is a contractive analytic function, while the maximal function is generally not a bounded one, but anL2-bounded analytic function, i.e., there exists M >0 such that
(1.9) sup
0≤r<1
1 2π
Z 2π 0
kΘ(reit)hk2dt≤M2khk2, h∈ H, or, equivalently,
(1.10)
∞
X
n=0
kΘnhk2dt≤M2khk2, h∈ H, where Θnare the coefficients of the analytic function Θ(λ).
Some conditions for the boundedness of the maximal function of a con- traction can be found in [14].
The maximal function proofs to be a usefull tool in prediction theory, especially in investigation and obtaining the Wiener filter for prediction, and in estimation of the prediction-error operator (see [10], [11]). In this paper we are concerned in the connection between the maximal function and some discrete linear systems.
2. CONNECTIONS WITH LINEAR SYSTEMS
Let H,U,Y be separable Hilbert spaces and A ∈ L(H), B ∈ L(U,H), C ∈ L(H,Y), D∈ L(U,Y). A linear systemσ = (A, B, C, D;H,U,Y) of the form
(2.1)
hn+1 =Ahn+Bun
yn=Chn+Dun , n≥0,
where {hn} ⊂ H, {un} ⊂ U, {yn} ⊂ Y, is called a discrete-time system.
Usually, the spaces H, U, Y are called, respectively, the state space, the input space, and the output space, and the operators A, B, C and D are
called, respectively, the main operator, the control operator, theobservation operator, and the feedthrough operator of the systemσ.
Let us consider the bloc operator matrix (colligation)S :H⊕U → H⊕Y,
(2.2) S =
A B C D
: H
U
→ H
Y
.
Then (2.1) can be written into the matrix form hn+1
yn
=S hn
un
.
The system σ will be called: passive, isometric, co-isometric, conserva- tive, ifS is, respectively, a contraction, an isometry, a co-isometry, or unitary.
The operator valued function Θσ(λ) : U → Y, λ ∈ D, attached to a system σ by
(2.3) Θσ(λ) =D+λC(IH−λA)−1B, λ∈D,
is called the transfer function (or frequency response function) of the system.
The transfer function is the basic connection between state-space and frequency-domain in the linear systems theory.
The references for the linear systems are very large, we mention here only a few of them [5, 6, 8, 2, 9, 1]. The aim of this paper is not an exhaustive study on linear systems, but only to analyze some connections between the maximal function and discrete linear systems.
Ifσis a passive system, then Θσ(λ) is a contractive holomorphic function on D, i.e., Θσ(λ) belongs to the Schur class S(U,Y).
For a systemσ, the following subspaces of Hare considered
(2.4) Cσ = _
n≥0
AnBU (the controllable space), and
(2.5) Oσ = _
n≥0
A∗nC∗Y (the observable space).
Generally, we haveH= CσW Oσ
⊕ Cσ⊥∩ Oσ⊥
. The systemσ is called controllable if Cσ = H, observable if Oσ = H, and minimal if σ is both observable and controllable. The system σ is simple ifCσ∨ Oσ =H.
From (2.4) it follows that Cσ⊥
=
∞
T
n=0
ker(B∗A∗n), and from (2.5) we have Oσ⊥
=
∞
T
n=0
ker(CAn).Hence the following characterizations occur: the system σ is, respectively, controllable iff
∞
T
n=0
ker(B∗A∗n) ={0}, observable iff
∞
T
n=0
ker(CAn) ={0}, and simple iff ∞ T
n=0
ker(B∗A∗n)
∩ ∞ T
n=0
ker(CAn)
={0}.
In this paper, we are mainly concerned on the system J given by the unitary operator (the rotation operator of T, or Julia operator)
(2.6) J(T) =RT =
T DT∗
DT −T∗
.
In this particular case, the controllable and the observable subspaces of H will be, respectively,
(2.7) C=
∞
_
n=0
TnDT∗DT∗, O=
∞
_
n=0
T∗nDTDT,
and the corresponding orthogonals in the state space Hwill be (2.8) C⊥=
∞
\
n=0
ker(DT∗T∗n) =\
n
kerDT∗n ={h∈ H; kT∗nhk=khk},
(2.9) O⊥=
∞
\
n=0
ker(DTTn) =\
n
kerDTn ={h∈ H; kTnhk=khk}.
Thus T|O⊥ and T∗|C⊥ are isometric operators and
(2.10) C⊥∩ O⊥={h∈ H; kTnhk=khk=kT∗nhk}=H0,
whereH0 is the subspace of the unitary part from the canonical decomposition [12] of the contraction T =T0⊕T1 =
T0 0 0 T1
into its unitary part and the completely non-unitary (c.n.u.) part on H=H0⊕ H1.
Also, let us remark that the transfer function ofJ is just the character- istic function of T∗, namely ΘJ(λ) =−T∗+λDT(I−λT)−1DT∗ = ΘT∗(λ).
If we consider the system J∗ given by the unitary bloc matrix (the rotation of T∗),J(T∗) =RT∗ =
T∗ DT DT∗ −T
, then the transfer function ofJ∗ will be given by {DT,DT∗,ΘT(λ)}, the characteristic function of T.
Obviously, RT∗ =R∗T, and the corresponding linear systems J and J∗ are dual, namely, if J is observable, thenJ∗ is controllable, and conversely.
Proposition2.1. The systemJ given by J(T) is conservative and sim- ple, if and only if the main operatorT is a completely non-unitary contraction.
Moreover, if TO⊥=O⊥, thenJ is observable, and ifT∗C⊥=C⊥, thenJ is a controllable system.
Proof. The systemJ is conservative, being governed by the unitary bloc matrix operatorJ(T). IfJ is simple, thenH=C ∨O, andC⊥∩O⊥={h∈ H;
kTnhk=khk=kT∗nhk}=H0 ={0}, i.e., the subspace of the unitary part of T is the null space, andT is a completely non-unitary contraction.
Conversely, ifT is c.n.u., thenH0 ={0}, andH=C ∨ O, i.e., the system J is simple.
It is obvious that O⊥ is invariant to T, and T restricted to O⊥ is an isometry. If TO⊥ = O⊥ then O⊥ reduces T to a unitary operator. Since J is simple, i.e., T is c.n.u., it follows that O⊥ = {0}, or O = H, and J is observable.
Analogously, if T∗C⊥ = C⊥, then C⊥ reduces T∗ to a unitary operator, which implies that C⊥={0}, i.e., the systemJ is controllable.
From the prediction point of view, we are interested in the cases when the maximal function attached to the distribution of a process is not a null function, to obtain the best linear predictor and the prediction-error in terms of the coefficients of the maximal function. In the case of discrete linear systems with the main operator a contraction, the fact that the maximal function Θ1(λ) of the main operator is the null function, gives some information about the corresponding structure.
Proposition2.2. LetJ be the conservative system given by the rotation operator of a completely non-unitary contraction T.
1) If the maximal function {H,DT∗,Θ1(λ)} of the main operator T is the null function, then the system J is observable.
2) If the maximal function {H,DT,Θ2(λ)} of T∗ is the null function, then the system J is controllable.
Proof. If the maximal function of the contractionT Θ1(λ) =DT∗(I−T∗)−1=
∞
X
k=0
Ckλk=
∞
X
k=0
DT∗T∗kλk≡ {0},
then the coefficientsCk= 0, i.e.,DT∗T∗nh= 0 for any h∈ H, and by (2.8) it follows that C⊥ =H, i.e., C ={0}. By the previous proposition, the system J is simple, thus it follows that O=H, and the systemJ is observable.
Analogous, if{H,DT,Θ2(λ)≡ {0}}, then by (2.9) we haveO⊥=H, and it follows that C=H, i.e., the system J is controllable.
Corollary2.3. If the main operatorT of the systemJ is an isometric (co-isometric) operator, then J is controllable (observable). If T is unitary, then the system J is minimal.
The fact that the maximal function of the main operator is the null function is not a necessary condition. It is sufficient to consider the subspaces Ω =O⊥ TO⊥ and Ω∗ =C⊥ T∗C⊥ and we have
Proposition 2.4. The conservative simple systemJ is 1)observable if and only if Θ1(λ)|Ω = 0;
2)controllable if and only if Θ2(λ)|Ω∗ = 0,
where Θ1(λ), andΘ2(λ), are the maximal functions ofT, andT∗, respectively.
Proof. If J is observable, then O = H, i.e., O⊥ = {0}, and by the definition of Ω it follows that Ω ={0}, and Θ1(λ)|Ω = 0.
Conversely, if Θ1(λ)|Ω = 0, then for any g ∈ DT∗, and ω ∈ Ω we have 0 =
Θ1(λ)ω, g
=
ω,Θ1(λ)∗g
=
ω, PΩΘ1(λ)∗g
,that is, 0 =PΩΘ1(λ)∗g= PΩ[DT∗(I−λT∗)−1]∗g=PΩ(I−λT)−1DT∗g=PΩP∞
n=0λnTnDT∗g.It follows that for any n ≥ 0, and g ∈ DT∗, we have PΩTnDT∗g = 0, and by (2.7) it results that PΩW∞
n=0TnDT∗DT∗ = PΩC = {0}, i.e., Ω ⊂ C⊥. But Ω ⊂ O⊥, and it results that Ω ⊂ C⊥ ∩ O⊥ = H0 = {0}, where H0 is the space of unitary part of the c.n.u. contraction T. Therefore, Ω = {0}, and we have O⊥ TO⊥={0}, or equivalently,TO⊥=O⊥. By (2.9)T|O⊥is an isometry, therefore O⊥ reducesT to a unitary operator. T being c.n.u.,O⊥={0}, and it follows that O=H, i.e., J is observable.
IfJ is controllable, thenC=H, i.e., C⊥={0}, which implies Ω∗ ={0}, and Θ2(λ)|Ω∗= 0.
Conversely, if Θ2(λ)|Ω∗ = 0 then for anyv∈Ω∗, and anyg∈ DT we have Θ2(λ)v, g
=
v,Θ2(λ)∗g
=
v, PΩ∗Θ2(λ)∗g
= 0, that is PΩ∗Θ2(λ)∗g = PΩ∗[DT(I −λ)−1]∗g = PΩ∗(I −λT∗)−1DTg = PΩ∗P∞
n=0λnT∗nDTg = 0. It follows that for any n ≥ 0 and g ∈ DT we have PΩ∗T∗nDTg = 0, and by (2.7) it results that PΩ∗W∞
n=0T∗nDTDT =PΩ∗O={0},i.e., Ω∗ ⊂ O⊥. Since Ω∗ ⊂ C⊥, we have Ω∗ ⊂ O⊥∩ C⊥ =H0 ={0}, i.e., Ω∗ = {0}, which implies T∗C⊥=C⊥, and the fact that C⊥ reduces T∗ to a unitary operator. But T is completely non-unitary, so we have C⊥ ={0}, or equivalent, C =H, and the system J is controllable.
A stronger characterization for the controllability (observability) of the system J can be done with the maximal function of the main operator T as follows.
Proposition 2.5. The discrete linear system J is controllable if and only if the operator Θ1 defined by the maximal function of T is one to one, and J is observable if and only if the operator Θ2 defined by the maximal function of T∗ is one to one.
Proof. If the system J is controllable, then CJ =H, where CJ is given by (2.7), or equivalently, CJ⊥ =T
ker(DT∗T∗n) ={0}. That is,DT∗T∗nh= 0 for any n ≥ 0 if and only if h = 0. Taking into account that Θ1(λ)h = DT∗h+DT∗T∗λh+DT∗T∗2λ2h+· · ·,it follows that ker Θ1 = 0.
Conversely, if ker Θ1 = 0 then Θ1h = 0 if and only if h = 0, i.e., DT∗T∗nh= 0 for anyn≥0 if and only ifh= 0, which implies thatCJ⊥={0}, or equivalentlyCJ =H, and the system is controllable.
Analogously can be proved the same facts for the systemJ∗, and taking into account the duality of the systemsJ andJ∗, the proof is completed.
Therefore the operators Θ1 and Θ2 corresponding to the maximal func- tions Θ1(λ) and Θ2(λ) of T and T∗, respectively, contain the information about the structure of the corresponding systems.
Usefull in the study of the linear systems can be the fact that the defect functions of the transfer function of J and J∗ generate positive definite ker- nels, expressible in terms of the maximal functions Θ1(λ) and Θ2(λ) as follows.
Proposition 2.6. Let {DT,DT∗,ΘT(λ)} be the characteristic function of the contraction T. The relations
(2.11) KT(λ, µ) = I−ΘT(λ)ΘT(µ)∗
1−λµ = Θ1(λ)Θ1(µ)∗ and
(2.12) KT∗(λ, µ) = I−ΘT(µ)∗ΘT(λ)
1−λµ = Θ2(µ)Θ2(λ)∗,
holds where Θ1(λ) and Θ2(λ) are the maximal functions ofT and T∗, respec- tively.
Proof. It is known ([12], Ch. VI, (1.4)) that the defect function of the characteristic function {DT,DT∗,ΘT(λ)} is obtained by
f, f
−
ΘT(λ)f,ΘT(µ)f
= (1−λµ)
(I−λT∗)−1DTf,(I −µT∗)−1DTf , hence ∆2Θ
T(λ, µ) =I−ΘT(µ)∗ΘT(λ) = (1−λµ)DT(I−µT)−1(I−λT∗)−1DT, and taking into account by (1.2) it follows that
I−ΘT(µ)∗ΘT(λ)
1−λµ = Θ2(µ)Θ2(λ)∗. An analogous calculus leads to
∆2ΘT∗(λ, µ) =I−ΘT(λ)ΘT(µ)∗ = (1−λµ)DT∗(I−λT∗)−1(I−µT)−1DT∗, and by (1.1) it follows that
I −ΘT(λ)ΘT(µ)∗
1−λµ = Θ1(λ)Θ1(µ)∗. Taking into account Proposition 2.5 we have the following
Corollary2.7. The discrete linear systemJ is controllable if and only the positive kernel KT(λ, λ)is strictly positive, andJ is observable if and only if the kernel KT∗(λ, λ) is strictly positive.
For a general study of linear systems, we are interested in bloc matri- ces operators of the form S =
A B C D
. By definition, if S is, respectively, contraction, isometry, co-isometry, unitary, then the corresponding system is, respectively, passive, isometric, co-isometric, conservative. In the following, some explicit relations show the utility of the maximal function in the in- vestigation of other type of systems. For the study of passive systems, the following theorem ([3], Theorem 1.3), which gives a characterization of such a bloc matrix S to be a contraction, is very helpful.
Theorem 2.8. The formula
(2.13) X =−Γ2A∗Γ1+DΓ∗
2ΓDΓ1
establishes a one-to-one correspondence between all operators X ∈ L(H2,K2) such that S =
A DA∗Γ1
Γ2DA X
is a contraction and all Γ ∈ L(DΓ1,DΓ∗
2).
Moreover, DS can be identified withDΓ2⊕ DΓ, andDS∗ can be identified with DΓ∗
1 ⊕ DΓ∗.
Therefore, in our case, the block matrixS =
A B C D
, whereA∈ L(H), B ∈ L(U,H), C ∈ L(H,Y), and D ∈ L(U,Y), is a contraction if and only if there exist the unique determined contractions Γ1 ∈ L(U,DA∗), Γ2∈ L(DA,Y) and Γ∈ L(DΓ1,DΓ∗
2), such that
(2.14) S =
A DA∗Γ1
Γ2DA −Γ2A∗Γ1+DΓ∗
2ΓDΓ1
.
If for the passive system governed by the contractive bloc matrix S, having the main operator the contraction A, we construct the conservative system governed by the unitary bloc matrix RA =
A DA∗ DA −A∗
, then S can be obviously written in the form
(2.15) S =
I 0 0 Γ2
A DA∗ DA −A∗
| {z }
RA
I 0 0 Γ1
+
0 0 0 DΓ∗
2ΓDΓ1
.
Using (2.15), the following relation between the controllable and ob- servable subspaces of a passive system, and the corresponding conservative attached system can be proved.
Proposition2.9. Letσ = (A, B, C, D;H,U,Y)be a passive system cor- responding to the bloc matrix S, and J = (A, DA∗, DA,−A∗;H,DA,DA∗) the conservative system corresponding to RA, where A is the main operator ofσ.
The observable and the controllable subspaces of the system σ are contained, respectively, into the observable and the controllable subspaces of J.
Proof. The system σ is passive, hence the bloc matrix S is contraction, and if we use the structure given by (2.14), then the proof is a straightforward verification, starting from the definition of the corresponding subspaces. Let Cσ and CJ be the controllable subspaces of σ and J, respectively, and Oσ, OJ the corresponding observable subspaces. Then, taking into account that Γ1∈ L(U,DA∗) and Γ∗2∈ L(Y,DA), we have
Cσ =
∞
_
n=0
AnDA∗Γ1Y ⊆
∞
_
n=0
AnDA∗DA∗ =CJ and
Oσ =
∞
_
n=0
A∗n(Γ2DA)∗Y=
∞
_
n=0
A∗nDAΓ∗2Y ⊆
∞
_
n=0
A∗nDADA=OJ. Therefore Cσ ⊆ CJ and Oσ ⊆ OJ.
From these inclusions we have the following
Corollary 2.10. If the passive system σ is controllable, observable, minimal, or simple, then the attached conservative system J becomes, respec- tively, controllable, observable, minimal, or simple.
These facts suggest again that the maximal function can be an investi- gation tool for general systems having as the main operator a contraction.
Finally, let us recall some more results and connections with the maxi- mal function. One of them is that between the maximal function and the characteristic function there exist the following relation, useful in solving in- terpolation problems (see [7], Ch. IX, Theorem 6.4): if T is∗-stable, then the operatorWT from the space of the minimal isometric dilationK =H⊕H2(DT) into H2(DT∗) defined by WT(h⊕f) = Θ1h+ ΘTf, or into matricial form
WT =
Θ1 ΘT
:H ⊕H2(DT)→H2(DT∗), is unitary. Generally, WT is not unitary, as we can see by (1.4).
As a remark, the stable and∗-stable systems, having a contraction as the main operator, can be analyzed also in a functional form, taking account by Proposition 1.1, into a functional model generated by the maximal function of the main operator.
Another explicit relation between the maximal function and the charac- teristic function of a contraction was obtained in [13], where using Redheffer products, the composing of the system generated by J RT∗ =
DT∗ −T T∗ DT
,
where J = 0 I
I 0
, with the extended feedback system {I, λI}is expressed as the extended feedback system of the maximal function Θ1(λ) and the charac- teristic function ΘT(λ). Also an extended form for the maximal function Θ1(λ) to Θ1(X), whereX∈ L(H), is used to obtain a generalized form of the charac- teristic function.
All previous facts and the strong relations between the maximal function and transfer functions show up that the maximal function is used implicitly in the study of the linear systems, and can become an explicit tool for inves- tigation.
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Received 15 September 2009 Romanian Academy
“Simion Stoilow” Institute of Mathematics P.O. Box 1-764
014700 Bucharest, Romania Ilie.Valusescu@imar.ro
