CONSERVATIVE DISCRETE TIME-INVARIANT SYSTEMS AND

CONSERVATIVE DISCRETE TIME-INVARIANT SYSTEMS AND BLOCK OPERATOR CMV MATRICES

arXiv:0808.1700v1 [math.FA] 12 Aug 2008

YURY ARLINSKI˘I Abstract. It is well known that an operator-valued function Θ from the Schur class S(M, N), where M and N are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function Θ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called ”Schur parameters”. In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero–Moral–Velázquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

Contents 1. 2. 3. 4. 5.

Introduction The Schur class functions and their iterates Conservative discrete-time linear systems and their transfer functions Conservative realizations of the Schur iterates Block operator CMV matrices and conservative realizations of the Schur class function (the case when the operator Γn is neither an isometry nor a co-isometry for each n) 5.1. Block operator CMV matrices 5.2. Truncated block operator CMV matrices 5.3. Simple conservative realizations of the Schur class function by means of its Schur parameters 6. Block operator CMV matrices (the rest cases) 7. Unitary operators with cyclic subspaces, dilations, and block operator CMV matrices 7.1. Carathéodory class functions associated with conservative systems 7.2. Unitary operators with cyclic subspaces 7.3. Unitary dilations of a contraction 7.4. The Naimark dilation 8. The block operator CMV matrix models for completely non-unitary contractions

2 9 11 13 16 17 20 22 28 35 35 38 39 41 42

1991 Mathematics Subject Classification. 47A48, 47A56, 47B36, 93B28 . Key words and phrases. Contraction, conservative system, transfer function, realization, Schur class function, Schur parameters, block Operator CMV matrices, unitary dilation. 1

YURY ARLINSKI˘I

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References

44

1. Introduction In what follows the class of all continuous linear operators defined on a complex Hilbert space H1 and taking values in a complex Hilbert space H2 is denoted by L(H1 , H2 ) and L(H) := L(H, H). We denote by IH the identity operator in a Hilbert space H and by PL the orthogonal projection onto the subspace (the closed linear manifold) L. The notation T ↾ L means the restriction of a linear operator T on the set L. The range and the null-space of a linear operator T are denoted by ran T and ker T , respectively. Recall that an operator T ∈ L(H1 , H2 ) is said to be • • • •

contractive if kT k ≤ 1; isometric if kT f k = kf k for all f ∈ H1 ⇐⇒ T ∗ T = IH1 ; co-isometric if T ∗ is isometric ⇐⇒ T T ∗ = IH2 ; unitary if it is both isometric and co-isometric.

Given a contraction T ∈ L(H1 , H2 ). The operators DT := (I − T ∗ T )1/2 ,

DT ∗ := (I − T T ∗ )1/2

are called the defect operators of T , and the subspaces DT = ran DT , DT ∗ = ran DT ∗ the defect subspaces of T . The dimensions dim DT , dim DT ∗ are known as the defect numbers of T . The defect operators satisfy the following intertwining relations (1.1)

T DT = DT ∗ T,

T ∗ D T ∗ = DT T ∗ .

It follows from (1.1) that T DT ⊂ DT ∗ , T ∗ DT ∗ ⊂ DT , and T (ker DT ) = ker DT ∗ , T ∗ (ker DT ∗ ) = ker DT . Moreover, the operators T ↾ ker DT and T ∗ ↾ ker DT ∗ are isometries and T ↾ DT and T ∗ ↾ DT ∗ are pure contractions, i.e., ||T f || < ||f || for f ∈ H \ {0}. The Schur class S(H1 , H2) is the set of all holomorphic and contractive L(H1 , H2 )-valued functions on the unit disk D = {λ ∈ C : |λ| < 1}. This class is a natural generalization of the Schur class S of scalar analytic functions mapping the unit disk D into the closed unit disk D [56] and is intimately connected with spectral theory and models for Hilbert space contraction operators [64], [25], [26], [27], [28], [29], the Lax-Phillips scattering theory [48], [1], [23], the theory of scalar and matrix orthogonal polynomials on the unit circle T = {ξ ∈ C : |ξ| = 1} [37], [60], [39], [40], the theory of passive (contractive) discrete time-invariant linear systems [45], [46], [13], [14], [15], [22], [21]. One of the characterization of the operator-valued Schur class is that any Θ ∈ S(M, N) can be realized as the transfer (characteristic) function of the form Θ(λ) = D + λC(IH − λA)−1 B, λ ∈ D of a discrete time-invariant system (colligation) D C τ= ; M, N, H B A

BLOCK OPERATOR CMV MATRICES

3

with the input space M, the output space N, and some state space H. Moreover, if the operator Uτ is given by the block operator matrix M N D C Uτ = : ⊕ → ⊕ , B A H H then the system τ can be chosen (a) passive (Uτ is contractive) and minimal, (b) co-isometric (Uτ is co-isometry) and observable, (c) isometric (Uτ is isometry) and controllable, (d) conservative (Uτ is unitary) and simple (see Section 3). The corresponding models of the systems τ and the state space operators A are well-known. We mention the de Branges– Rovnyak functional model of a co-isometric system [26], [7], [49], the Sz.-Nagy–Foias [64], the Pavlov [52], [53], [54], and the Nikol’ski˘i–Vasyunin [50], [51] functional models of completely non-unitary contractions, the Brodski˘i [29] functional model of a simple unitary colligation, the Arov–Kaashoek–Pik [15] functional model of a passive minimal and optimal system. All these models involve the Schur class function and/or the Hardy spaces, the de Branges– Rovnyak reproducing kernel Hilbert space. The main goal of the present paper is constructions of models for simple conservative systems and completely non-unitary contractions by means of the operator analogs of the scalar CMV matrices, which recently appeared in the theory of orthogonal polynomials on the unit circle [31], [60], [61], [37]. In the paper of M.J. Cantero, L. Moral, and L. Velázquez [31] it is established that the semi-infinite matrices of the form   α ¯0 α ¯ 1 ρ0 ρ1 ρ0 0 0 ...  ρ0 −¯ α1 α0 −ρ1 α0 0 0 . . .   0 α ¯ 2 ρ1 −¯ α2 α1 α ¯ 3 ρ2 ρ3 ρ2 . . .  (1.2) C = C({αn }) =  ρ2 ρ1 −ρ2 α1 −¯ α3 α2 −ρ3 α2 . . . 0   0 0 α ¯ 4 ρ3 −¯ α4 α3 . . . 0 .. .. .. .. .. .. . . . . . .

and



(1.3)

α ¯0 ρ0 0 0 0 α ¯ ρ −¯ α1 α0 α ¯ 2 ρ1 ρ2 ρ1 0  1 0  ρ1 ρ0 −ρ1 α0 −¯ α2 α1 −ρ2 α1 0 e n }) =  Ce = C({α 0 α ¯ 3 ρ2 −¯ α3 α2 α ¯ 4 ρ3  0  0 0 ρ ρ −ρ α −¯ α4 α3  3 2 3 2 .. .. .. .. .. . . . . .

 ... . . .  . . .  . . .  . . . .. .

give representations of the unitary operator (Uf )(ζ) = ζf (ζ) in L2 (T, dµ), where the dµ is a nontrivial probability measure on the unite circle, with respect to the orthonormal systems obtained by orthonormalization of the sequences {1, ζ, ζ −1, ζ 2, ζ −2, . . .} and {1, ζ −1, ζ, ζ −2, ζ 2, . . .}, respectively. The Verblunsky coefficients {αn }, |αn | < 1, arise in the Szeg˝o recurrence formula ¯ ζΦn (ζ) = Φn+1 (ζ) + α ¯ n ζ n Φn (1/ζ),

n = 0, 1, . . .

YURY ARLINSKI˘I

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p for monic orthogonal with respect to dµ polynomials {Φn }, and ρn := 1 − |αn |2 . The e n }) and are called the CMV matrices. Note that the matrix C e is matrices C({αn }) C({α transpose to C. Given a probability measure µ on T, define the Carathéodory function by Z Z ∞ X ζ +λ n F (λ) = F (λ, µ) := dµ(ζ) = 1 + 2 βn λ , βn = ζ −n dµ ζ − λ T T n=1

the moments of µ. F is an analytic function in D which obeys Re F > 0, F (0) = 1. The Schur class function f (λ) is then defined by f (λ) = f (λ, µ) :=

1 F (λ) − 1 , λ F (λ) + 1

Given a Schur function f (λ), which is not a finite Blaschke product, define inductively f0 (λ) = f (λ), fn+1 (λ) =

fn (λ) − fn (0)

λ(1 − fn (0)fn (λ))

, n ≥ 0.

It is clear that {fn } is an infinite sequence of Schur functions called the n − th Schur iterates and neither of its terms is a finite Blaschke product. The numbers γn := fn (0) are called the Schur parameters: Sf = {γ0 , γ1 , . . .}. Note that

fn (λ) =

γn + λfn+1 (λ) λfn+1 (λ) = γn + (1 − |γn |2 ) , n ≥ 0. 1 + γ¯n λfn+1 1 + γ¯n λfn+1 (λ)

The method of labeling f ∈ S by its Schur parameters is known as the Schur algorithm and is due to I. Schur [56]. In the case when N Y λ − λk f (λ) = e ¯kλ 1−λ k=1 iϕ

is a finite Blaschke product of order N, the Schur algorithm terminates at the N-th step. The sequence of Schur parameters {γn }N n=0 is finite, |γn | < 1 for n = 0, 1, . . . , N − 1, and |γN | = 1. Due to Geronimus theorem for the function f (λ, µ) the relations γn = αn hold true for all n = 0, 1, . . .. There is a nice multiplicative structure of the CMV matrices. In the semi-infinite case C and Ce are the products of two matrices: C = LM, Ce = ML, where L = Ψ(α0 ) ⊕ Ψ(α2 ) ⊕ . . . Ψ(α2m ) ⊕ . . . ,

M = 11×1 ⊕ Ψ(α1 ) ⊕ Ψ(α3 ) ⊕ . . . ⊕ Ψ(α2m+1 ) ⊕ . . . , α ¯ ρ and Ψ(α) = . The finite (N + 1) × (N + 1) CMV matrices C and Ce obey ρ −α α0 , α1 , . . . , αN −1 ∈ D and |αN | = 1, and also C = LM, Ce = ML, where in this case Ψ(αN ) = (α ¯ N ).


5

In the paper [12] it is established that the truncated CMV matrices   −¯ α1 α0 −ρ1 α0 0 0 ...  α ¯ ρ −¯ α2 α1 α ¯ 3 ρ2 ρ3 ρ2 . . .    2 1 ρ2 ρ1 −ρ2 α1 −¯ α3 α2 −ρ3 α2 . . .  T0 = T0 ({αn }) =    0 α ¯ 4 ρ3 −¯ α4 α3 . . .   0 .. .. .. .. .. .. . . . . . .

and

  −¯ α1 α0 α ¯ 2 ρ1 ρ2 ρ1 0 ...  −ρ1 α0 −¯ α2 α1 −ρ2 α1 0 . . .   0 α ¯ 3 ρ2 −¯ α3 α2 α ¯ 4 ρ3 . . . Te0 = Te0 ({αn }) =    ρ3 ρ2 −ρ3 α2 −¯ α4 α3 . . .  0 .. .. .. .. .. . . . . . e n }) by deleting the first obtained from the “full” CMV matrices C = C({αn }) and Ce = C({α row and the first column, provide the models of completely non-unitary contractions with rank one defect operators. As pointed out by Simon in [61], the history of CMV matrices is started with the papers of Bunse-Gerstner and Elsner [30] (1991) and Watkins [65] (1993), where unitary semi-infinite five-diagonal matrices were introduces and studied. In [31] Cantero, Moral, and Velazquez (CMV) re-discovered them. In a context different from orthogonal polynomials on the unit circle, Bourget, Howland, and Joye [24] introduced a set of doubly infinite family of matrices with three sets of parameters which for special choices of the parameters reduces to two-sided CMV matrices on ℓ2 (Z). The Schur algorithm for matrix valued Schur class functions and its connection with the matrix orthogonal polynomials on the unit circle have been considered in the paper of Delsarte, Genin, and Kamp [40] and in the book of Dubovoj, Fritzsche, and Kirstein [42]. The CMV matrices, connected with matrix orthogonal polynomials on the unit circle with respect to nontrivial matrix-valued measures are considered in [61], [37]. If the k × k matrixvalued non-trivial measure µ on T, µ(T) = Ik×k is given, then there are the left and the right orthonormal matrix polynomials. The Szeg˝o recursions take slightly different form than in the scalar case and the Verblunsky k × k matrix coefficients (the Schur parameters of the corresponding matrix-valued Schur function) {αn } satisfy the inequality ||αn || < 1 for all n. The latter condition is in fact equivalent to the non-triviality of the measure. The entries of the corresponding CMV matrix have the size k × k and the numbers ρn are replaced by the ∗ 1/2 , where α∗ k × k defect matrices ρLn = Dαn = (I − αn∗ αn )1/2 and ρR n = Dα∗n = (I − αn αn ) is the adjoint matrix. In these notations the CMV matrix is of the form [37]   ∗ α0 ρL0 α1∗ ρL0 ρL1 0 0 ... ρR −α0 α1∗ −α0 ρL1 0 0 . . .   0 ∗ R ∗ L ∗ L L  0 α2 ρ1 −α2 α1 ρ2 α3 ρ2 ρ3 . . .  R ∗ L (1.4) C = C({αn }) =  −ρR . . . .  0 ρR 2 ρ1 2 α1 −α2 α3 −α2 ρ3   0 0 α4∗ ρR −α4∗ α3 . . . 0 3 .. .. .. .. .. .. . . . . . .

The operator extension of the Schur algorithm was developed by T. Constantinescu in [34] and with numerous applications is presented in the monographs [20], [36]. The next theorem

YURY ARLINSKI˘I

6

goes back to Shmul’yan [57], [58] and T. Constantinescu [34] (see also [20], [8], [9]) and plays a key role in the operator Schur algorithm. Theorem 1.1. Let M and N be separable Hilbert spaces and let the function Θ(λ) be from the Schur class S(M, N). Then there exists a function Z(λ) from the Schur class S(DΘ(0) , DΘ∗ (0) ) such that (1.5)

Θ(λ) = Θ(0) + DΘ∗ (0) Z(λ)(I + Θ∗ (0)Z(λ))−1 DΘ(0) , λ ∈ D.

The representation (1.5) of a function Θ(λ) from the Schur class is called the Möbius representation of Θ(λ) and the function Z(λ) is called the Möbius parameter of Θ(λ) (see [8], [9]). Clearly, Z(0) = 0 and by Schwartz’s lemma we obtain that ||Z(λ)|| ≤ |λ|, λ ∈ D. The operator Schur’s algorithm [20]. Fix Θ(λ) ∈ S(M, N), put Θ0 (λ) = Θ(λ) and let Z0 (λ) be the Möbius parameter of Θ. Define Γ0 = Θ(0), Θ1 (λ) =

Z0 (λ) ∈ S(DΓ0 , DΓ∗0 ), Γ1 = Θ1 (0) = Z0′ (0). λ

If Θ0 (λ), . . . , Θn (λ) and Γ0 , . . . , Γn have been chosen, then let Zn+1 (λ) ∈ S(DΓn , DΓ∗n ) be the Möbius parameter of Θn . Put Θn+1 (λ) =

Zn+1 (λ) , Γn+1 = Θn+1 (0). λ

The contractions Γ0 ∈ L(M, N), Γn ∈ L(DΓn−1 , DΓ∗n−1 ), n = 1, 2, . . . are called the Schur parameters of Θ(λ) and the function Θn (λ) ∈ S(DΓn−1 , DΓ∗n−1 ) we will call the n − th Schur iterate of Θ(λ). Formally we have Θn+1 (λ)↾ ran DΓn =

1 DΓ∗ (ID ∗ − Θn (λ)Γ∗n )−1 (Θn (λ) − Γn )DΓ−1n ↾ ran DΓn . λ n Γn

Clearly, the sequence of Schur parameters {Γn } is infinite if and only if all operators Γn are non-unitary. The sequence of Schur parameters consists of a finite number of operators Γ0 , Γ1 , . . . , ΓN if and only if ΓN ∈ L(DΓN−1 , DΓ∗N−1 ) is unitary. If ΓN is isometric (co-isometric) then Γn = 0 for all n > N. The following generalization of the classical Schur result is proved in [34] (see also [20]). Theorem 1.2. There is a one-to-one correspondence between the Schur class functions S(M, N) and the set of all sequences of contractions {Γn }n≥0 such that (1.6)

Γ0 ∈ L(M, N), Γn ∈ L(DΓn−1 , DΓ∗n−1 ), n ≥ 1.

A sequence of contractions of the form (1.6) is called the choice sequence [32]. Such objects are used for the indexing of contractive intertwining dilations, of positive Toeplitz forms, and of the Naimark dilations of semi-spectral measures on the unit circle (see [32], [33], [35], [20], [36]). Observe that the Naimark dilation and the model of a simple conservative system are given in [33], [34], and [20] by infinite in all sides block operator matrix whose entries are expressed by means of the choice sequence or the Schur parameters.


7

Let us describe the main results of our paper. Given a choice sequence (1.6). We construct e0 = H e 0 ({Γn }n≥0 ) , the unitary operators the Hilbert spaces H0 = H0 ({Γn }n≥0 ), H

M N e0 Γ G Γ0 G0 0 U0 = U0 ({Γn }n≥0 ) = : ⊕ → ⊕ , Ue0 = Ue0 ({Γn }n≥0 ) = e e : F0 T0 F0 T0 H0 H0

and the unitarily equivalent simple conservative systems e0 Γ0 G0 Γ G 0 e0 , ζ0 = ; M, N, H0 , ζe0 = ; M, N, H F0 T0 Fe0 Te0

N M ⊕ → ⊕ , e0 e0 H H

such that the Schur parameters of the transfer function Θ of the systems ζ0 and ζe0 are precisely {Γn }n≥0 . Moreover, the operators U0 and Ue0 in such constructions are given by the operator analogs of the CMV matrices. In the case when the operators Γn are neither e 0 are of the form isometric nor co-isometric for each n = 0, 1, . . ., the Hilbert spaces H0 and H H0 =

X M DΓ∗2n X M DΓ2n e0 = ⊕ ⊕ , , H ∗ DΓ2n+1 DΓ2n+1 n≥0 n≥0

and the operators U0 and Ue0 are given by the products of unitary diagonal operator matrices    IM JΓ0    JΓ1 JΓ2 ,  U0 =  JΓ3 JΓ4    .. .. . .  IN  JΓ1 Ue0 =  JΓ3 

where JΓ0

..

.

 JΓ0  JΓ2  JΓ4 



..

.

 , 

DΓk−1 M DΓ∗k−1 N Γk DΓ∗k Γ0 DΓ∗0 : ⊕ → ⊕ , k = 1, 2, . . . → ⊕ , JΓk = = : ⊕ DΓ0 −Γ∗0 DΓk −Γ∗k ∗ ∗ D DΓ0 DΓ0 DΓk Γk

are the unitary operators called ”elementary rotations” [20]. The operators U0 and Ue0 take the form of five-diagonal block operator matrices   0 0 0 0 0 ... Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗1 0 0 0 0 0 . . .   ∗  0 0 0 0 . . . −Γ2 Γ1 DΓ∗2 Γ3 DΓ∗2 DΓ∗3 Γ2 DΓ1  U0 =  0 0 0 . . .  0 DΓ2 DΓ1 −DΓ2 Γ∗1 −Γ∗2 Γ3 −Γ∗2 DΓ∗3   0 0 Γ4 DΓ3 −Γ4 Γ∗3 DΓ∗4 Γ5 DΓ∗4 DΓ∗5 0 . . .   0 .. .. .. .. .. .. .. .. .. . . . . . . . . .

YURY ARLINSKI˘I

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and



Γ0 DΓ∗0 0 0 0 0 ∗  Γ1 DΓ0 −Γ1 Γ0 DΓ∗1 Γ2 DΓ∗1 DΓ∗2 0 0  DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ∗1 DΓ∗2 0 0 Ue0 =   0 −Γ3 Γ∗2 DΓ∗3 Γ4 DΓ∗3 DΓ∗4 0 Γ3 DΓ2  0 DΓ3 DΓ2 −DΓ3 Γ∗2 −Γ∗3 Γ4 −Γ∗3 DΓ∗4  0 .. .. .. .. .. .. . . . . . . Note that the following relation

0 0 0 0 0 .. .

... ... ... ... ... .. .

       

Ue0 ({Γn }n≥0 ) = (U0 ({Γ∗n }n≥0 )∗ .

holds true. Hence the CMV matrix (1.4) corresponds to the case M = N = DΓ0 = DΓ∗0 = DΓ1 = DΓ∗1 = . . . = DΓn = DΓ∗n = . . . = Ck , αn = Γ∗n , n = 0, 1, . . . , Thus,

∗ e n }) = Ue0 ({Γ∗ }n≥0 ) = (U0 ({Γn }n≥0)∗ . C({αn }) = U0 ({Γ∗n }n≥0 ) = Ue0 ({Γn }n≥0 , C({α n

The block operator truncated CMV matrices

e e T0 = T0 ({Γn }n≥0 ) := PH0 U0 ↾ H0 and Te0 = Te0 ({Γn }n≥0 ) := PH e 0 U0 ↾ H0

are given by  ∗ −Γ0  JΓ2 T0 =  JΓ4 



..

.



  

JΓ1

and can be rewritten in the three  B1 C1 0 A1 B2 C2 T0 =   0 A2 B3 .. .. .. . . .

JΓ3

..

.



 JΓ1 JΓ3  , Te0 = 

  −Γ∗0  JΓ2  JΓ4 ..  . ..

diagonal block operator matrix   Be1 Ce1 0 0 0 · Ae Be Ce 0 0 · 1 2 2  , Te0 =   e e C3 0 ·  0 A2 B3 .. .. .. .. .. .. . . . . . .

 .

form with 2 × 2 entries  0 0 · 0 0 ·  . e C3 0 · .. .. .. . . .

 , 

The constructions above and the corresponding results are presented in Section 5. We essentially rely on the constructions of simple conservative realizations of the Schur iterates {Θn (λ)}n≥1 by means of a given simple conservative realization of the function Θ ∈ S(M, N) [9]. A brief survey of the results in [9] are given in Section 4. The cases when the Schur parameter Γm ∈ L(DΓm−1 , DΓ∗m−1 ) of the function Θ ∈ S(M, N), is isometric, co-isometric, unitary are considered in detail in Section 6. Observe that in fact we give another prove of Theorem 1.2 (the uniqueness of the function from S(M, N) with with given its Schur parameters is proved in Section 2). In Section 7 we obtain in the block operator CMV matrix form the minimal unitary dilations of a contraction and the minimal Naimark dilations of a semi-spectral measure on the unite circle. Another and more complicated constructions of the minimal Naimark dilation and a simple conservative realization for a function Θ ∈ S(M, N) by means of its Schur parameters are given in [33] and in [47], respectively (see also [20]).


9

Simple conservative realizations of scalar Schur functions with operators A, B, C, and D expressed via corresponding Schur parameters have been obtained by V. Duboboj [41]. We also prove in Section 7 that a unitary operator U in a separable Hilbert space K having a cyclic subspace M (span {U n M, n ∈ Z} = K) is unitarily equivalent to the block operator CMV matrices U0 and Ue0 constructed by means of the Schur parameters of the function 1 ∗ ¯ ∗ ¯ (λ) − IM)(FM (λ) + IM)−1 , where FM(λ) = PM(U + λIK)(U − λIK)−1 ↾ M, Θ(λ) = (FM λ λ ∈ D. In the last Section 8 we prove that the Sz.-Nagy–Foias [64] characteristic functions of truncated block operator CMV matrices T0 and Te0 , constructing by means of the Schur parameters {Γn }n≥0 of a purely contractive function Θ ∈ S(M, N), coincide with Θ in the sense of [64]. 2. The Schur class functions and their iterates In the sequel we need the well known fact [64], [20] that if T ∈ L(H1 , H2 ) is a contraction which is neither isometric nor co-isometric, then the operator (elementary rotation [20]) JT given by the operator matrix H1 H2 T DT ∗ → ⊕ : ⊕ JT = DT −T ∗ DT DT ∗ is unitary. Clearly, J∗T = JT ∗ . If T is isometric or co-isometric, then the corresponding unitary elementary rotation takes the row or the column form H2 H1 T (r) (c) JT = T IDT ∗ : ⊕ → H 2 , JT = : H1 → ⊕ , DT DT ∗ DT

and

∗ (c) (r) = JT ∗ . JT

In Section 5 we will need the following statement.

Proposition 2.1. [11]. Let T be a contraction. Then T h = DT ∗ g if and only if there exists a vector ϕ ∈ DT such that h = DT ϕ and g = T ϕ. Recall that if Θ(λ) ∈ S(H1 , H2 ) then there is a uniquely determined decomposition [64, Proposition V.2.1] DΘ(0) DΘ∗ (0) Θp (λ) 0 , → ⊕ Θ(λ) = : ⊕ 0 Θu ker DΘ∗ (0) ker DΘ(0) where Θp (λ) ∈ S(DΘ(0) , DΘ∗ (0) ), Θp (0) is a pure contraction and Θu is a unitary constant. The function Θp (λ) is called the pure part of Θ(λ) (see [20]). If Θ(0) is isometric (respect., coisometric) then the pure part is of the form Θp (λ) = 0 ∈ S({0}, DΘ∗ (0) ) (respect., Θp (λ) = 0 ∈ S(DΘ(0) , {0})). The function Θ is called purely contractive if ker DΘ(0) = {0}. Two operator-valued functions Θ ∈ S(M, N) and Ω ∈ S(K, L) coincide [64] if there are two unitary operators V : N → L and U : K → M such that (2.1)

V Θ(λ)U = Ω(λ),

λ ∈ D.

YURY ARLINSKI˘I

10

For the corresponding Schur parameters and the Schur iterates relation (2.1) yields the equalities (2.2)

Gn = V Γn U, DGn = U ∗ DΓn , DG∗n = V DΓ∗n , DGn = U ∗ DΓn U, DG∗n = V DΓ∗n V ∗ , V Θn (λ)U = Ωn (λ), λ ∈ D

for all n = 0, 1, . . . . In what follows we give a proof of Theorem 1.6 different from the original one in [34]. First of all we will prove the uniqueness. The existence will be proved in Section 5. Theorem 2.2. Any choice sequence uniquely determines a Schur class function. Proof. Let Γ0 ∈ L(M, N), Γn ∈ L(DΓn−1 , DΓ∗n−1 ), n ≥ 1 be a choice sequence. Suppose b 0 (λ) from the Schur class S(M, N) have {Γn }∞ as their Schur the functions Θ0 (λ) and Θ 0 parameters. Then for every n = 0, 1, . . . hold the relations Θn (λ) = Γn + λDΓ∗n (I + λΘn+1 (λ)Γ∗n )−1 Θn+1 (λ)DΓn , b n (λ) = Γn + λDΓ∗ (I + λΘ b n+1 (λ)Γ∗ )−1 Θ b n+1 (λ)DΓn , Θ n n

b n (λ)} are the Schur iterates of Θ and Θ, b respectively. Then one has where {Θn (λ)} and {Θ for every n the equalities (2.3) b n (λ) = Θn (λ) − Θ b n+1 (λ))(I + λΘ b n+1 (λ)Γ∗ )−1 DΓn , λ ∈ D. = λDΓ∗n (I + λΘn+1 (λ)Γ∗n )−1 (Θn+1 (λ) − Θ n b n+1 (λ)|| ≤ 2 for all λ ∈ D and Θn+1 (0) = Θ b n+1 (0) = Γn+1 , by Schwartz’s Since ||Θn+1(λ) − Θ lemma we get b n+1 (λ)|| ≤ 2 |λ|, λ ∈ D. ||Θn+1(λ) − Θ Further ||(I + λΘn+1 (λ)Γ∗n )f || ≥ (1 − |λ|)||f ||, b n+1 (λ)Γ∗ )f || ≥ (1 − |λ|)||f || ||(I + λΘ n for all λ ∈ D and for all f ∈ DΓ∗n−1 . These relations imply ||(I + λΘn+1 (λ)Γ∗n )−1 || ≤

1 1 b n+1 (λ)Γ∗ )−1 || ≤ , ||(I + λΘ n 1 − |λ| 1 − |λ|

for all λ ∈ D and for all n = 0, 1, . . . . Hence and from (2.3) we have b n (λ)|| ≤ 2|λ| ||Θn (λ) − Θ

|λ| , λ ∈ D. (1 − |λ|)2

b n−1 in the left hand side, we see that Then applying (2.3) for Θn−1 and Θ 2 |λ| b , λ ∈ D, ||Θn−1 (λ) − Θn−1 (λ)|| ≤ 2|λ| (1 − |λ|)2 and finally n+1 |λ| b 0 (λ)|| ≤ 2|λ| , λ∈D (2.4) ||Θ0(λ) − Θ (1 − |λ|)2 for all n = 0, 1, . . ..


Let |λ| < (3 −

√

11

5)/2. Then

Letting n → ∞ in (2.4) we get

|λ| < 1. (1 − |λ|)2

√ 3 − 5 b 0 (λ), |λ| < Θ0 (λ) = Θ . 2

b 0 are holomorphic in D, they are equal on D. Since Θ0 and Θ

3. Conservative discrete-time linear systems and their transfer functions D C Let M, N, and H be separable Hilbert spaces. A linear system τ = ; M, N, H B A with bounded linear operators A, B, C, D of the form σk = Chk + Dξk , (3.1) k ≥ 0, hk+1 = Ahk + Bξk where {ξk } ⊂ M, {σk } ⊂ N, {hk } ⊂ H is called a discrete time-invariant system. The Hilbert spaces M and N are called the input and the output spaces, respectively, and the Hilbert space H is called the state space. The operators A, B, C, and D are called the state space operator, the control operator, the observation operator, and the feedthrough operator of τ , respectively. Put M N D C Uτ = : ⊕ → ⊕ B A H H If Uτ is contractive, then the corresponding discrete-time system is said to be passive [13]. If the operator Uτ is isometric (respect., co-isometric, unitary), then the system is said to be isometric (respect., co-isometric, conservative). Isometric, co-isometric, conservative, and passive discrete time-invariant systems have been studied in [25], [26], [7], [64], [45], [46], [27], [29], [22], [6], [13], [14], [15], [16], [17], [18], [62], [63], [10], [8], [9], [43]. It is relevant to remark that a brief history of System Theory is presented in the recent preprint of B. Fritzsche, V. Katsnelson, and B. Kirstein [43]. The subspaces (3.2)

Hc := span {An BM : n = 0, 1, . . .} and Ho := span {A∗n C ∗ N : n = 0, 1, . . .}

are said to be the controllable and observable subspaces of the system τ , respectively. The system τ is said to be controllable (respect., observable) if Hc = H (respect., Ho = H), and it is called minimal if τ is both controllable and observable. The system τ is said to be simple if H = clos {Hc + Ho } = span {Ak BM, A∗l C ∗ N, k, l = 0, 1, . . .}

It follows from (3.2) that

(Hc )⊥ =

∞ \

n=0

ker(B ∗ A∗n ),

(Ho )⊥ =

∞ \

ker(CAn ),

n=0

and therefore there are the following alternative characterizations:

YURY ARLINSKI˘I

12

(a) τ is controllable ⇐⇒

∞ T

n=0 ∞ T

ker(B ∗ A∗n ) = {0};

(b) τ is observable ⇐⇒ ker(CAn ) = {0}; n=0 ∞ ∞ T T ∗ ∗n n (c) τ is simple ⇐⇒ ker(B A ) ∩ ker(CA ) = {0}. n=0

n=0

A contraction A acting in a Hilbert space H is called completely non-unitary [64] if there is no nontrivial reducing subspace of A, on which A generates a unitary operator. Given a contraction A in H then there is a canonical orthogonal decomposition [64, Theorem I.3.2] H = H0 ⊕ H1 ,

A = A0 ⊕ A1 ,

Aj = A↾ Hj ,

j = 0, 1,

where H0 and H1 reduce A, the operator A0 is a completely non-unitary contraction, and A1 is a unitary operator. Moreover, ! ! \ \ \ H1 = ker DAn ker DA∗n . n≥1

Since

n−1 \

n≥1

n−1 \

ker(DA Ak ) = ker DAn ,

k=0

k=0

we get

\

n≥1

(3.3)

\

n≥1

It follows that (3.4)

ker(DA∗ A∗k ) = ker DA∗n ,

ker DAn = H ⊖ span {A∗n DA H, n = 0, 1, . . .} , ker DA∗n = H ⊖ span {An DA∗ H, n = 0, 1, . . .} .

A is completely non-unitary ⇐⇒

T

n≥1

ker DAn

T

T

ker DA∗n

n≥1

= {0} ⇐⇒

⇐⇒ span {A∗n DA , Am DA∗ , n, m ≥ 0} = H. D C If τ = ; M, N, H is a conservative system then τ is simple if and only if the state B A space operator A is a completely non-unitary contraction [29], [22]. The transfer function Θτ (λ) := D + λC(IH − λA)−1 B,

λ ∈ D,

of the passive system τ belongs to the Schur class S(M, N) [13]. Conservative systems are also called the unitary colligations and their transfer functions are called the characteristic functions [29]. The examples of conservative systems are given by −A DA∗ −A∗ DA Σ= ; DA , DA∗ , H , Σ∗ = ; DA∗ , DA , H . DA ∗ A DA A∗ The transfer functions of these systems

ΦΣ (λ) = −A + λDA∗ (IH − λA∗ )−1 DA ↾ DA ,

λ∈D


and

ΦΣ∗ (λ) = −A∗ + λDA (IH − λA)−1 DA∗ ↾ DA∗ ,

13

λ∈D

are precisely the Sz.Nagy–Foias characteristic functions [64] of A and A∗ , correspondingly. It is well known that every operator-valued function Θ(λ) from the Schur class S(M, N) can be realized as the transfer function of some passive system, which can be chosen as controllable isometric (respect., observable co-isometric, simple conservative, minimal passive); cf. [26], [64], [29], [7] [13], [15], [6]. Moreover, two controllable isometric (respect., observable co-isometric, simple conservative) systems with the same transfer function are unitarily equivalent: two discrete-time systems D C1 D C2 τ1 = ; M, N, H1 and τ2 = ; M, N, H2 B1 A1 B2 A2 are said to be unitarily equivalent if there exists a unitary operator V from H1 onto H2 such that (3.5)

A1 = V −1 A2 V, B1 = V−1 B2 , C1 = C2 V ⇐⇒ IN 0 D C1 D C2 IM 0 ⇐⇒ = 0 V B1 A1 B2 A2 0 V

cf. [25], [26], [7], [29], [6]. 4. Conservative realizations of the Schur iterates Let A be a completely non-unitary contraction in a separable Hilbert space H. Suppose ker DA 6= {0}. Define the subspaces and operators (see [9])   H0,0 := H Hn,0 = ker DAn , H0,m := ker DA∗m , (4.1)  H n,m := ker DAn ∩ ker DA∗m , m, n ∈ N, An,m := Pn,m A↾ Hn,m ∈ L(Hn,m ),

(4.2)

where Pn,m are the orthogonal projections in H onto Hn,m . The next results have been established in [9]. Theorem 4.1. [9]. (1) Hold the relations ker DAkn,m = Hn+k,m, ker DA∗k = Hn,m+k , k = 1, 2, . . . , n,m DAn,m = ran (Pn,m DAn+1 ), , DA∗n,m = ran (Pn,m DA∗m+1 ) AHn,m = Hn−1,m+1 , n ≥ 1, , A∗ Hn,m = Hn+1,m−1 , m ≥ 1 (4.3)

(An,m )k,l = An+k,m+l .

(2) The operators {An,m } are completely non-unitary contractions.

YURY ARLINSKI˘I

14

(3) The operators An,0 , An−1,1 , . . . , An−k,k , . . . , A0,n are unitarily equivalent and An−1,m+1 Af = AAn,m f, f ∈ Hn,m , n ≥ 1. The relation (4.3) yields the following picture for the creation of the operators An,m :

A1,0 A2,0 A3,0

zz zz z z z| z

y yy yy y y |y y

z zz zz z |z z

DD DD DD DD "

DD DD DD DD "

zz zz z z |z z

A2,1

A EE

A1,1

EE EE EE E"

A0,1 z zz zz z z| z

DD DD DD DD "

DD DD DD DD "

zz zz z z z| z

A1,2

A0,2

DD DD DD DD "

··· ··· ··· ··· ··· ··· The process terminates on the N-th step if and only if

A0,3 ···

ker DAN = {0} ⇐⇒ ker DAN−1 ∩ ker DA∗ = {0} ⇐⇒ . . . ker DAN−k ∩ ker DA∗k = {0} ⇐⇒ . . . ker DA∗N = {0}. Theorem 4.2. [9]. Let A be a completely non-unitary contraction in a separable Hilbert space H. Assume ker DA 6= {0} and let the contractions An,m be defined by (4.1) and (4.2). Then the characteristic functions of the operators An,0 , An−1,1, . . . , An−m,m , . . . A1,n−1 , A0,n coincide with the pure part of the n-th Schur iterate of the characteristic function Φ(λ) of A. Moreover, each operator from the set {An−k,k }nk=0 is (1) a unilateral shift (respect., co-shift) if and only if the n-th Schur parameter Γn of Φ is isometric (respect., co-isometric), (2) the orthogonal sum of a unilateral shift and co-shift if and only if (4.4)

DΓn−1 6= {0}, DΓ∗n−1 6= {0} and Γm = 0

for all m ≥ n.

Each subspace from the set {Hn−k,k }nk=0 is trivial if and only if Γn is unitary. Theorem 4.3. [9]. Let Θ(λ) ∈ S(M, N) and let Γ0 C τ0 = ; M, N, H B A

be a simple conservative realization of Θ. Then the Schur parameters {Γn }n≥1 of Θ can be calculated as follows ∗ ∗ Γ1 = DΓ−1∗0 C DΓ−10 B ∗ , Γ2 = DΓ−1∗1 DΓ−1∗0 CA DΓ−11 DΓ−10 (B ∗ ↾ H1,0 ) , . . . , ∗ (4.5) Γn = DΓ−1∗n−1 · · · DΓ−1∗0 CAn−1 DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn−1,0 ) , . . . .


15

Here the operators DΓ−1k and DΓ−1∗ , k = 0, 1, . . . are the Moore-Penrose pseudo inverses, the k operator ∗ DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn−1,0 ) ∈ L(DΓn−1 , Hn−1,0 ) is the adjoint to the operator and

DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn−1,0) ∈ L(Hn−1,0 , DΓn−1 ), ran DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn,0 ) ⊂ ran DΓn , −1 −1 ran DΓ∗n−1 · · · DΓ∗0 (C↾ H0,n ) ⊂ ran DΓ∗n

for every n ≥ 1. Moreover, for each n ≥ 1 the unitarily equivalent simple conservative systems (4.6) (" # ) Γn DΓ−1∗n−1 · · · DΓ−1∗0 (CAn−k ) (k) ∗ τn = ; DΓn−1 , DΓ∗n−1 , Hn−k,k , Ak DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn,0) An−k,k k = 0, 1, . . . , n are realizations of the n-th Schur iterate Θn of Θ. Here the operator ∗ Bn = DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn,0) ∈ L(DΓn−1 , Hn,0 ) is the adjoint to the operator

DΓ−1n−1 · · · DΓ−10 (B ∗ ↾ Hn,0) ∈ L(Hn,0 , DΓn−1 ). Note that if 1) Γm is isometric then DΓn = 0, Γ∗n = 0 ∈ L(DΓ∗m , {0}), DΓ∗n = DΓ∗m , and H0,n = H0,m for n ≥ m. The unitarily equivalent observable conservative systems Γm DΓ−1∗m−1 · · · DΓ−1∗0 (CAm−k ) (k) ; DΓm−1 , DΓ∗m−1 , Hm−k,k , k = 0, 1, . . . , m τm = 0 Am−k,k

have transfer functions Θm (λ) = Γm and the operators Am−k,k are unitarily equivalent coshifts of multiplicity dim DΓ∗m , the Schur iterates Θn are null operators from L({0}, DΓ∗m ) for n ≥ m + 1 and are transfer functions of the conservative observable system 0 DΓ−1∗m−1 · · · DΓ−1∗0 C ; {0}, DΓ∗m , H0,m . τm+1 = 0 A0,m 2) Γm is co-isometric then DΓ∗n = 0, DΓn = DΓm , and Γn = 0 ∈ L(DΓm , {0}), Hn,0 = Hm,0 for n ≥ m. The unitarily equivalent controllable conservative systems (" ) # Γ 0 m ∗ (k) τm = ; DΓm−1 , DΓ∗m−1 , Hm−k,k Ak DΓ−1m−1 · · · DΓ−10 (B ∗ ↾ Hm,0 ) Am−k,k

have transfer functions Θm (λ) = Γm and the operators Am−k,k are unitarily equivalent unilateral shifts of multiplicity dim DΓm , the Schur iterates Θn are null operators from L(DΓm , {0}) for n ≥ m + 1 and are transfer functions of the conservative controllable system 0 0 ∗ τm+1 = ; DΓm , {0}, Hm,0 . DΓ−1m · · · DΓ−10 (B ∗ ↾ Hm+1,0 ) Am,0

YURY ARLINSKI˘I

16

We also mention that if Θ(λ) ∈ S(M, N), Γ0 = Θ(0), Θ1 (λ) is the first Schur iterate of Θ, and if Γ0 C τ= ; M, N, H B A is a simple conservative system with transfer function Θ, then the simple conservative systems −1 DΓ∗0 C(DΓ−10 B ∗ )∗ DΓ−1∗0 C↾ ker DA∗ ; DΓ0 , DΓ∗0 , ker DA∗ , ζ0,1 = −1 ∗ AP D P A↾ ker D ∗B ker D ker D A ∗ A A A −1 (4.7) DΓ∗0 C(DΓ−10 B ∗ )∗ DΓ−1∗0 CA↾ ker DA ζ1,0 = ; DΓ0 , DΓ∗0 , ker DA −1 Pker DA DA Pker DA A↾ ker DA ∗B −1 have transfer functions Θ1 (λ) (see [9]). Here the operators DΓ−10 , DΓ−1∗0 , and DA ∗ are the Moore–Penrose pseudo-inverses. In the sequel the transformations of the conservative system

τ → ζ0,1 , τ → ζ1,0

we will denote by Ω0,1 (τ ) and Ω1,0 (τ ), respectively.

Remark 4.4. The problem of isometric, co-isometric, and conservative realizations of the Schur iterates for a scalar function from the generalized Schur class has been studied in [2], [3], [4], [5]. For a scalar finite Blaschke product the realizations of the Schur iterates are constructed in [43]. 5. Block operator CMV matrices and conservative realizations of the Schur class function (the case when the operator Γn is neither an isometry nor a co-isometry for each n) Let Γ0 ∈ L(M, N), Γn ∈ L(DΓn−1 , DΓ∗n−1 ), n ≥ 1 be a choice sequence. In this and next Section 6 we are going to construct by means of {Γn }n≥0 two unitary equivalent simple conservative systems with such transfer function Θ ∈ S(M, N) that {Γn }n≥0 are its Schur parameters. In particular, this leads to the existence part of Theorem 1.2 and to the well known result that any Θ ∈ S(M, N) admits a realization as the transfer function of a simple conservative system. We begin with constructions of block operator CMV matrices for given choice sequence {Γn }n≥0 and will suppose that all operators Γn are neither isometries nor co-isometries. We will use the well known constructions of finite and infinite orthogonal sums of Hilbert spaces. Namely, if {Hk }∞ k=1 is a given sequence of Hilbert spaces, then N M X H= Hk k=1

is the Hilbert space with the inner product (f, g) =

N X

(fk , gk )Hk

k=0

for f = (f1 , . . . , fN )T and g = (g1 , . . . , gN )T , fk , gk ∈ Hk , k = 1, . . . , N and the norm ||f ||2 =

N X k=0

||fk ||2Hk .


The Hilbert space H=

∞ M X

Hk

k=0

consists of all vectors of the form f = (f1 , f2 , . . .)T , fk ∈ Hk , k = 1, 2, . . . , such that 2

||f || = The inner product is given by

∞ X k=1

||fk ||2Hk < ∞.

∞ X (f, g) = (fk , gk )Hk . k=1

5.1. Block operator CMV matrices. Define the Hilbert spaces (5.1)

X M DΓ∗2n X M DΓ2n e e ⊕ ⊕ , H0 = H0 ({Γn }n≥0) := H0 = H0 ({Γn }n≥0 ) := . ∗ D DΓ2n+1 n≥0 n≥0 Γ2n+1

From these definitions it follows, that

e 0 ({Γ∗ }n≥0 ) = H0 ({Γn }n≥0 ), H0 ({Γ∗ }n≥0 ) = H e 0 ({Γn }n≥0 ). H n n L Le The spaces N H0 and M H 0 we represent in the form

Let JΓ0 JΓk

N

M

M M X M DΓ∗2n−1 , H0 = ⊕ ⊕ DΓ0 n≥1 DΓ2n

M

M

M M X M DΓ2n−1 e ⊕ . H0 = ⊕ DΓ∗2n DΓ∗0 n≥1

M N Γ0 DΓ∗0 ⊕ ⊕ , → = : DΓ0 −Γ∗0 DΓ0 DΓ∗0 DΓk−1 DΓ∗k−1 Γk DΓ∗k : ⊕ → ⊕ , k = 1, 2, . . . = DΓk −Γ∗k DΓ∗k DΓk

be the elementary rotations. Define the following unitary operators L Le LPL M0 = M0 ({Γn }n≥0 ) := IM JΓ2n−1 : M H0 → M H 0, n≥1 L Le f0 = M f0 ({Γn }n≥0 ) := IN L P L JΓ H0 → N H M 0, 2n−1 : N (5.2) n≥1 Le L LPL H0 . JΓ2n : M H L0 = L0 ({Γn }n≥0 ) := JΓ0 0 → N n≥1

Observe that

(L0 ({Γn }n≥0 ))∗ = L0 ({Γ∗n }n≥0 ).

17

YURY ARLINSKI˘I

18

Let V0 = V0 ({Γn }n≥0 ) :=

(5.3)

Clearly, the operator V0 is unitary and M0 = IM

(5.4)

M

XM n≥1

e 0. JΓ2n−1 : H0 → H

f0 = IN V0 , M

M

V0 .

It follows that ∗ f0 ({Γn }n≥0 ) = M0 ({Γ∗ }n≥0 ), (M0 ({Γn }n≥0 ))∗ = M f0 ({Γ∗ }n≥0 ) M n n Finally define the unitary operators

L L U0 = U0 ({Γn }n≥0 ) := L0 M0 : M H0 → N H0 , f0 L0 : M L H e0 → N L H e 0. Ue0 = Ue0 ({Γn }n≥0 ) := M

(5.5)

By calculations we get 

Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 0 0 0 0 ... DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗1 0 0 0 0 0 ...  ∗ ∗ ∗ ∗  0 Γ2 DΓ1 −Γ2 Γ1 DΓ2 Γ3 DΓ2 DΓ3 0 0 0 ...   0 DΓ2 DΓ1 −DΓ2 Γ∗1 −Γ∗2 Γ3 −Γ∗2 DΓ∗3 0 0 0 ... U0 =  0 0 Γ4 DΓ3 −Γ4 Γ∗3 DΓ∗4 Γ5 DΓ∗4 DΓ∗5 0 ...  0  0 0 DΓ4 DΓ3 −DΓ4 Γ∗3 −Γ∗4 Γ5 −Γ4 DΓ∗5 0 ...  0  ∗ 0 0 0 0 Γ6 DΓ5 −Γ6 Γ5 DΓ∗6 Γ7 DΓ∗6 DΓ∗7  0 .. .. .. .. .. .. .. .. .. . . . . . . . . .

 ... . . .  . . .  . . .  . . .  . . .  . . . .. .

and 

Γ0 DΓ∗0 0 0 0 0 0 0 ∗ ∗ ∗ ∗  Γ1 DΓ0 −Γ1 Γ0 DΓ1 Γ2 DΓ1 DΓ2 0 0 0 0  DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ∗1 DΓ∗2 0 0 0 0   0 0 0 0 Γ3 DΓ2 −Γ3 Γ∗2 DΓ∗3 Γ4 DΓ∗3 DΓ∗4  e ∗ U0 =  0 0 0 0 DΓ3 DΓ2 −DΓ3 Γ2 −Γ∗3 Γ4 −Γ∗3 DΓ∗4  0 0 0 Γ5 DΓ4 −Γ5 Γ∗4 DΓ∗5 Γ6 DΓ∗5 DΓ∗6  0  0 0 0 DΓ5 DΓ4 −DΓ5 Γ∗4 −Γ∗5 Γ6 −Γ∗5 DΓ∗6  0 .. .. .. .. .. .. .. .. . . . . . . . . Let

C0 = DΓ∗0 Γ1 DΓ∗0 DΓ∗1

DΓ0 DΓ0 DΓ0 → N, A0 = :M→ ⊕ , : ⊕ 0 DΓ∗1 DΓ∗1

... ... ... ... ... 0 0 .. .

 ... . . .  . . .  . . .  . . . .  . . .  . . . .. .


(5.6)

 ∗ DΓ2n−2 DΓ2n−2   −Γ2n−2 Γ2n−1 −Γ∗2n−2 DΓ∗2n−1   Bn = : ⊕ → ⊕ ,   Γ2n DΓ2n−1 −Γ2n Γ∗2n−1   ∗ ∗ D D  Γ2n−1 Γ2n−1    DΓ2n DΓ2n−2  0 0 Cn = : ⊕ → ⊕ , ∗ ∗ D Γ D D  Γ2n 2n+1 Γ2n Γ∗2n+1  ∗ ∗ D D  Γ Γ 2n+1 2n−1    DΓ2n−2 D  Γ2n ∗  DΓ2n DΓ2n−1 −DΓ2n Γ2n−1   ⊕ ⊕ : A = → ,  n  0 0  ∗ ∗ DΓ2n−1 DΓ2n+1

e0 = DΓ∗ C 0

(5.7)

19

DΓ∗0 DΓ∗0 Γ D 1 Γ 0 0 : ⊕ :M→ ⊕ , → N, Ae0 = DΓ1 DΓ0 DΓ1 DΓ1

   −Γ2n−1 Γ∗2n−2 DΓ∗2n−1 Γ2n   e Bn = :   −DΓ2n−1 Γ∗2n−2 −Γ∗2n−1 Γ2n       DΓ∗  2n DΓ∗2n−1 DΓ∗2n 0 e ⊕ → Cn = : −Γ∗2n−1 DΓ∗2n 0   D  Γ2n+1    DΓ∗  2n−2   en = 0 Γ2n+1 DΓ2n : ⊕  A →   0 DΓ2n+1 DΓ2n  DΓ2n−1

DΓ∗2n−2 DΓ∗2n−2 → ⊕ , ⊕ DΓ2n−1 DΓ2n−1 DΓ∗2n−2 , ⊕ DΓ2n−1 DΓ∗2n ⊕ DΓ2n+1

It is easy to see that the operators U0 and Ue0 take the following three-diagonal block operator matrix form   Γ0 C0 0 0 0 · · A0 B1 C1 0 0 · ·  U0 = U0 ({Γn }n≥0 ) =   0 A1 B2 C2 0 · · , .. .. .. .. .. .. .. . . . . . . .   Γ0 Ce0 0 0 0 · · Ae Be Ce 0 0 · · 1 1  0  e e U0 = U0 ({Γn }n≥0 ) =  . e e e 0 A B C 0 · ·   1 2 2 .. .. .. .. .. .. .. . . . . . . . The block operator matrices U0 and Ue0 we will call block operator CMV matrices. Observe that f0 U0 = Ue0 M0 , (5.8) M and the following equalities hold true (5.9)

(U0 ({Γn }n≥0 )) = Ue0 ({Γ∗n }n≥0 ), ∗

Ue0 ({Γn }n≥0 )

∗

= U0 ({Γ∗n }n≥0 )

Therefore the matrix Ue0 can be obtained from U0 by passing to the adjoint U0∗ and then by replacing Γn (respect., Γ∗n ) by Γ∗n (respect., Γn ) for all n. In the case when the choice sequence consists of complex numbers from the unit disk the matrix Ue0 is the transpose to U0 , i.e., Ue0 = U0t .

YURY ARLINSKI˘I

20

5.2. Truncated block operator CMV matrices. Define two contractions (5.10)

T0 = T0 ({Γn }n≥0 ) := PH0 U0 ↾ H0 : H0 → H0 ,

(5.11)

e e e e Te0 = Te0 ({Γn }n≥0 ) := PH e 0 U0 ↾ H0 : H0 → H0 .

The operators T0 and Te0  B1 A1 T0 =  0 .. .

take on the three-diagonal block operator matrix forms    Be1 Ce1 0 0 0 · C1 0 0 0 · Ae Be Ce 0 0 · B2 C2 0 0 · 1 2 2   , Te0 =   , e e e A2 B3 C3 0 ·  0 A2 B3 C3 0 · .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . .

where An , Bn , Cn , Aen , Ben , and Cen are given by (5.6) and (5.7). Since the matrices T0 and Te0 are obtained from U0 and Ue0 by deleting the first rows and the first columns, we will call them truncated block operator CMV matrices. Observe that from the definitions of L0 , M0 , f0 , T0 , and Te0 it follows that T0 and Te0 are products of two block-diagonal matrices M    ∗ JΓ1 −Γ0    JΓ3 JΓ2    ..    JΓ4 .    .. (5.12) T0 = T0 ({Γn }n≥0 ) =  ,   . JΓ2n+1       .. JΓ2n    . .. . 

    e e (5.13) T0 = T0 ({Γn }n≥0 ) =    

JΓ1 JΓ3

..

. JΓ2n+1 .. .

In particular, it follows that (5.14)

 ∗ −Γ0  JΓ2   JΓ4  ..  .   JΓ2n 



..

.

    .   

(T0 ({Γn }n≥0 ))∗ = Te0 ({Γ∗n }n≥0 ).

From (5.12) and (5.13) we have

V0 T0 = Te0 V0 ,

where the unitary operator V0 is defined by (5.3). Therefore, the operators T0 and Te0 are unitarily equivalent.

Proposition 5.1. Let Θ ∈ S(M, N) and let {Γn }n≥0 be the Schur parameters of Θ. Suppose Γn is neither isometric nor co-isometric for each n. Let the function Ω ∈ S(K, L) coincides with Θ and Let {Gn }n≥0 be the Schur parameters of Ω. Then truncated block operator CMV matrices T0 ({Γn }n≥0 ) and T0 ({Gn }n≥0 ) (respect., Te0 ({Γn }n≥0 ) and Te0 ({Gn }n≥0 )) are unitarily equivalent.


21

Proof. Since Ω(λ) = V Θ(λ)U, where U ∈ L(K, M) and V ∈ L(N, L) are unitary operators, we get relations (2.2). It follows that DGn 6= {0} and DG∗n 6= {0} for all n. Hence, we have ∗ U 0 V 0 = J , n = 0, 1, . . . . (5.15) J Gn 0 V 0 U ∗ Γn Define the Hilbert space

HΩ 0 = H0 ({Gn }n≥0 ) :=

X M DG2n ⊕ DG∗2n+1 n≥0

and truncated block operator CMV matrix  −G∗0  J G2   J G4  .. T0 ({Gn }n≥0 ) :=  .   JG2n 

..

Define the unitary operator

 ∗ U  V  W=  

U



    ,   



∗

V ..

From (5.12) and (5.15) we obtain

.

 J G1  J G3  ..  .   JG2n+1   ..  .

.

   : H0 → HΩ 0  

WT0 ({Γn }n≥0 ) = T ({Gn }n≥0 )W.

Thus T0 ({Γn }n≥0) and T ({Gn }n≥0 ) are unitarily equivalent.

Now we are going to find the defect operators and defect subspaces for T0 and Te0 . Let f = (f~0 , f~1 , . . .)T ∈ H0 , where DΓ2n h ~ fn = n ∈ ⊕ , n = 0, 1, . . . . gn ∗ DΓ2n+1 Then

2

h0

= ||DΓ∗ (Γ1 h0 + DΓ∗ g0 )||2, ||f|| − ||T0 f|| = ||PNU0 f|| = C 0

1 0 g0

2

∗ h0 2

||f||2 − ||T0∗ f||2 = ||PMU0∗ f||2 =

A0 g0 = ||DΓ0 h0 || . 2

(5.16)

2

2

e 0 , where Let x = (x0 , x1 , . . .)T ∈ H DΓ∗2n h , n = 0, 1, . . . . xn = n ∈ ⊕ gn DΓ∗2n+1

YURY ARLINSKI˘I

22

Then

2

2 2 e e e0 h0 = ||DΓ∗ h0 ||2 , ||x|| − ||T0 x|| = ||PNU0 x|| = C

0 g

0 2

∗ h0 ∗ 2 e

||x||2 − ||Te0∗ x||2 = ||PMUe0∗ x||2 =

A0 g0 = ||DΓ0 (Γ1 h0 + DΓ1 g0 )|| . 2

Now from Proposition 2.1 it follows that    L P L DΓ2n DΓ1 ϕ   ⊕ , ker D = , ϕ ∈ D  T0 Γ1  −Γ1 ϕ  n≥1  ∗ DΓ2n+1     D   L P L Γ2n   ∗ = DΓ∗ ⊕ , ker D  T  1 0  n≥1 DΓ∗2n+1 (5.17) ∗   L P L DΓ2n Γ ψ  1  ~ ~ ∗ ⊕ D T0 = , ψ ∈ DΓ0 0, 0 ∈ ,   DΓ∗1 ψ  n≥1  ∗ D Γ  2n+1    D  Γ 2n  L LPL   ⊕ DT0∗ = DΓ0 ~0, ~0 ∈ DΓ∗1 .    n≥1 ∗ DΓ2n+1

(5.18)

 ∗  L P L DΓ2n    ⊕ , ker DTe0 = DΓ1    n≥1  D Γ2n+1    ∗   L P L DΓ2n DΓ∗1 ϕ   ⊕  , , ϕ ∈ DΓ∗0 ker DTe ∗ =  0 −Γ∗1 ϕ  n≥1 DΓ2n+1 ∗ D   L P L Γ2n L  ~0, ~0 ∈ DΓ1  ∗ ⊕ , D = D  e Γ T0 0   n≥1  D  Γ2n+1    ∗  L P L DΓ2n  Γ ψ 1  ~0, ~0 ∈  ⊕ . , ψ ∈ DΓ0   DTe0∗ = DΓ1 ψ n≥1 DΓ2n+1

5.3. Simple conservative realizations of the Schur class function by means of its Schur parameters. Let G0 = G0 ({Γn }n≥ 0) = DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 0 . . . : H0 → N, e 0 → N, Ge0 = Ge0 ({Γn }n≥0 ) = DΓ∗0 0 0 . . . : H     Γ1 DΓ0 DΓ0 DΓ1 DΓ0   0    e 0.  : M → H0 , Fe0 = Fe0 ({Γn }n≥0 ) =  0  : M → H F0 = F0 ({Γn }n≥0 ) =   0    0   .. .. . .


23

The operators U0 and Ue0 can be represented by 2 × 2 block operator matrices

M Γ0 G0 U0 = : ⊕ → F0 T0 H0 M Γ Ge Ue0 = e0 e0 : ⊕ → F0 T0 e0 H

N ⊕ , H0 N ⊕ . e0 H

Define the following conservative systems Γ0 G0 ζ0 = ; M, N, H0 = {U0 ({Γn }n≥0 ); M, N, H0({Γn }n≥0 )} , F0 T0 n (5.19) o Γ0 Ge0 e e 0({Γn }n≥0 ) . e e0 ({Γn }n≥0 ); M, N, H ; M, N, H = U ζ0 = 0 Fe0 Te0

The equalities (5.4) and (5.8) yield that systems ζ0 and ζe0 are unitarily equivalent. Hence, ζ0 and ζe0 have equal transfer functions. Observe that   IM 0  DΓ0 , G0 = DΓ∗ Γ1 DΓ∗ 0 0 . . . , F0 =  0 1 0   .. .  Γ1 DΓ1    e 0  DΓ0 , Ge0 = DΓ∗ , IN 0 0 0 . . . F0 =    0  0  .. . 

and

Γ1 DΓ∗1

    Γ1 IM D   0   Γ1   0  = Γ1 .  0 0 ...    0  = IN 0 0 0 . . .   0  .. .. . .

Theorem 5.2. The unitarily equivalent conservative systems ζ0 and ζe0 given by (5.19) are simple and the Schur parameters of the transfer function of ζ0 and ζe0 are {Γn }n≥0 .

Proof. The main step is a proof that the systems Ω0,1 (ζ0 ) and Ω1,0 (ζe0 ) given by (4.7) take the form n o e 0 ({Γn }n≥1 ) , e0 ({Γn }n≥1 ) , DΓ0 , DΓ∗ , H Ω0,1 (ζ0 ) = U 0 (5.20) e Ω1,0 (ζ0 ) = U0 ({Γn }n≥1 ) , DΓ , DΓ∗ , H0 ({Γn }n≥1 ) . 0

0

First of all we will prove that the systems ζ0 and ζe0 are simple.

YURY ARLINSKI˘I

24

Define the subspaces H2k−1

X M DΓ∗2n−1 X M DΓ2n ⊕ = , k = 1, 2, . . . , H2k = ⊕ ∗ DΓ2n+1 n≥k n≥k DΓ2n

Clearly, H0 ⊃ H1 ⊃ H2 ⊃ · · · ⊃ Hm ⊃ · · · . From (5.1) it follows the equality \ Hm = {0}. m≥0

Let Γ−1 = 0 : M → N. Then DΓ−1 = M, DΓ∗−1 = N. We can consider U0 as acting from e 0 . Fix m ∈ N e 0 onto DΓ∗ ⊕ H DΓ−1 ⊕ H0 onto DΓ∗−1 ⊕ H0 and Ue0 as acting from DΓ−1 ⊕ H −1 and define Γn(m) = Γn+m , n = −1, 0, 1, . . . (m)

Then {Γn }n≥0 = {Γk }k≥m, and

e 0 ({Γ(2k−1) e 0 ({Γn }n≥2k−1 ) , H2k−1 = H }n≥0 ) = H n (2k) H2k = H0 ({Γn }n≥0 ) = H0 ({Γn }n≥2k ) .

Let

DΓ∗2k−2 DΓ2k−2 , → ⊕ W2k−1 = = Ue0 ({Γn }n≥2k−1 ) : ⊕ H2k−1 H2k−1 DΓ∗2k−1 DΓ2k−1 (2k) , k ≥ 1. → ⊕ W2k = U0 ({Γn }n≥0 ) = U0 ({Γn }n≥2k ) : ⊕ H2k H2k Define the operators (2k−1) Ue0 ({Γn }n≥0 )

Tm = PHm Wm ↾ Hm , m = 1, 2, . . . .

(5.21) Then

(2k−1) }n≥0 ) = Te0 ({Γn }n≥2k−1), T2k−1 = Te0 ({Γn (2k) T2k = T0 ({Γn }n≥0 ) = T0 ({Γn }n≥2k ).

(5.22)

From (5.17), (5.18), (5.21), and (5.22) we get

ker DT0∗ = H1 , ker DT1 = H2 , . . . , ker DT2k∗ = H2k+1 , ker DT2k−1 = H2k , . . . . From (5.12), (5.13), and (5.22) it follows that Pker DT ∗ T0 ↾ ker DT0∗ = T1 , Pker DT1 T1 ↾ ker DT1 = T2 , . . . , 0 Pker DT2k−1 T2k−1 ↾ ker DT2k−1 = T2k , Pker DT ∗ T2k ↾ ker DT2k∗ = T2k+1 , . . . . 2k

Thus, H2k−1 = ker DT0∗k ∩ ker DT k−1 , 0 H2k = ker DT0∗k ∩ ker DT0k . In notations of Section 4 the operators T2k−1 and T2k coincide with the operators (T0 )k−1,k and (T0 )k,k , respectively. From the definition of H0 we get ! ! \ \ \ \ \ H2k = {0}. ker DT0∗k ∩ ker DT0k = ker DT0∗k ker DT0k = k≥1

k≥1

k≥1

k≥1


25

So, the operators T0 , Te0 , and {Tk }k≥1 are completely non-unitary. It follows that the conservative systems e0 Γ0 G0 Γ G 0 e0 ; M, N, H ζ0 = ; M, N, H and ζe0 = F0 T0 Fe0 Te0 are simple. The operators Wm takes the following 2 × 2 block operator matrix form

DΓm−1 DΓ∗m−1 Γm Gm → ⊕ , Wm = : ⊕ Fm Tm Hm Hm

where

G2k−1 = Ge0 ({Γn }n≥2k−1) = DΓ∗2k−1 0 0 . . . : H2k−1 → DΓ∗2k−2 , G2k = G0 ({Γn }n≥2k ) = DΓ∗2k Γ2k+1 DΓ∗2k DΓ∗2k+1 0 0 . . . : H2k → DΓ∗2k−1 ,   Γ2k DΓ2k−1 DΓ2 k DΓ2k−1    e  : DΓ 0 F2k−1 = F0 ({Γn }n≥2k−1 ) =  → H2k−1 , 2k−2   0   .. .   DΓ2k  0   F2k = F0 ({Γn }n≥2k ) =   0  : DΓ2k−1 → H2k . .. .

Suppose that the system

ζ0 = has transfer function Ψ(λ), i.e.,

Γ0 G0 ; M, N, H F0 T0

Ψ(λ) = Γ0 + λG0 (IH0 − λT0 )−1 F0 . Then Ψ(0) = Γ0 . Let Ψ1 (λ) be the first Schur iterate of Ψ. By (4.7) the transfer function of the simple conservative system −1 DΓ∗0 C(DΓ−10 B ∗ )∗ DΓ−1∗0 C↾ ker DA∗ ; DΓ0 , DΓ∗0 , ker DA∗ Ω0,1 (ν) = −1 APker DA DA Pker DA∗ A↾ ker DA∗ ∗B is the first Schur iterate of the transfer function of the simple conservative system Γ0 C ν= ; M, N, H . B A We will construct the system ζ1 = Ω0,1 (ζ0) from the system ζ0 . In our case ) # (" DΓ−1∗0 G0 ↾ ker DT0∗ DΓ−1∗0 G0 (DΓ−10 F0∗ )∗ ; DΓ0 , DΓ∗0 , ker DT0∗ ζ1 = Ω0,1 (ζ0 ) = Pker DT ∗ T0 ↾ ker DT0∗ T0 Pker DT0 DT−1 ∗ F0 0 0

YURY ARLINSKI˘I

26

Clearly,

Therefore,

DΓ−1∗0 G0 = Γ1 DΓ∗1 0 0 . . . : H0 → DΓ∗0 ,   IM  0  (DΓ−10 F0∗ )∗ =   0  : DΓ0 → H0 . .. .   IM  0   0 0 ...   0  = Γ1 . .. .

Γ1 DΓ∗1

Thus, the first Schur parameter of Ψ is equal to Γ1 . From (5.17) it follows that ker DT0∗ = H1 and DT0∗ = DΓ0 PDΓ0 . Hence

As has been proved above

  IDΓ0  0    D−1 T0∗ F0 =  0  : DΓ0 → H0 . .. . Pker DT ∗ T0 ↾ ker DT0∗ = T1 . 0

Let h ∈ DΓ0 . Let us find the projection Pker DT0 h. According to (5.17) we have to find the vectors ϕ ∈ DΓ1 and ψ ∈ DΓ∗0 such that ∗ Γ1 ψ h DΓ1 ϕ = + . 0 −Γ1 ϕ DΓ∗1 ψ We have

h = DΓ1 ϕ + Γ∗1 ψ Γ1 ϕ = DΓ∗1 ψ.

From the second equation and Proposition 2.1 it follows ϕ = DΓ1 g, ψ = Γ1 g, where g ∈ DΓ0 . Therefore h = DΓ2 1 g + Γ∗1 Γ1 g, i.e., g = h. Hence  DΓ2 1 h −Γ1 DΓ1 h   −1 . 0 Pker DT0 DT0∗ F0 h = Pker DT0 h =    0   .. . 


27

Now we get T0 Pker DT0 D−1 T0∗ F0 h =  0 0 0 0 0 ... DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗1  0 Γ2 DΓ1 −Γ2 Γ∗1 DΓ∗2 Γ3 DΓ∗2 DΓ∗3 0 0 0 ...  ∗ ∗ ∗  0 DΓ2 DΓ1 −DΓ2 Γ1 −Γ2 Γ3 −Γ2 DΓ∗3 0 0 0 ...  ∗ ∗ ∗ ∗ 0 0 0 Γ4 DΓ3 −Γ4 Γ3 DΓ4 Γ5 DΓ4 DΓ5 0 ... =  0 ... 0 0 DΓ4 DΓ3 −DΓ4 Γ∗3 −Γ∗4 Γ5 −Γ4 DΓ∗5  0  ∗ ∗ 0 0 0 0 Γ6 DΓ5 −Γ6 Γ5 DΓ6 Γ7 DΓ∗6 DΓ∗7  0 .. .. .. .. .. .. .. .. .. . . . . . . . . .     0 DΓ2 1 h  Γ2 DΓ1 h   −Γ1 DΓ1 h   DΓ2 DΓ1 h   ∈ H1 . = 0 × 0     0     0   .. .. . .

 ... . . .  . . .  . . . ×  . . .  . . . .. .

Thus we get that ζ1 is of the form n o Γ1 G1 e 0 ({Γn }n≥1 ) . e0 ({Γn }n≥1 ) , DΓ0 , DΓ∗ , H ζ1 = Ω0,1 (ζ0 ) = ; DΓ0 , DΓ∗0 , H1 = U 0 F1 T1 Similarly

Ω1,0 (ζe0 ) = U0 ({Γn }n≥1 ) , DΓ0 , DΓ∗0 , H0 ({Γn }n≥1 ) .

The transfer functions of these systems are equal to Ψ1 (λ) (see Section 4), and Γ1 is exactly is the first Schur parameter of Ψ(λ). Let Ψ2 (λ) is the second Schur iterate of Ψ. Constructing the simple conservative system ζ2 = Ω1,0 (ζ1 ) of the form (4.7) with the transfer function Ψ2 we will get the system Γ2 G2 ζ2 = ; DΓ1 , DΓ∗1 , H2 = U0 ({Γn }n≥2 ); DΓ1 , DΓ∗1 , H0 ({Γn }n≥2 ) . F2 T2

Let Ψm (λ) be the m−th Schur iterate of Ψ. Arguing by induction we get that Ψm (λ) is transfer function of the system Γm Gm ζm = ; DΓm−1 , DΓ∗m−1 , Hm = m Tm Fn o e  Ue0 ({Γn }n≥2k−1); DΓ ∗ , DΓ2k−2 , H0 ({Γn }n≥2k−1 ) , m = 2k − 1 2k−2 o n =  U0 ({Γn }n≥2k ); DΓ ∗ , H ({Γ } ) , m = 2k , D 0 n n≥2k Γ 2k−1 2k−1 for all m. Observe that

ζ2k−1 = Ω0,1 (ζ2k−2), ζ2k = Ω1,0 (ζ2k−1 ), k ≥ 1.

Thus, {Γn }n≥0 are the Schur parameters of Ψ.

From Theorem 5.2 and Theorem 2.2 we immediately arrive at the following result. Theorem 5.3. Let Θ(λ) ∈ S(M, N) and let {Γn }n≥0 be the Schur parameters of Θ. Then the systems (5.19) are simple conservative realizations of Θ. Moreover, for each natural number

28

YURY ARLINSKI˘I

k the k-th Schur iterate Θk of Θ is theotransfer n function of the simple conservative systems o n e 0 (Γn }n≥k ) . U0 (Γn }n≥k ); DΓk−1 , DΓ∗k−1 , H0 (Γn }n≥k ) and Ue0 (Γn }n≥k ); DΓk−1 , DΓ∗k−1 , H Observe that in fact we have proved Theorem 1.2 and our proof is different from given in [34] and [20].

Remark 5.4. More complicated construction of the state Hilbert space and simple conservative realization for a Schur function Θ ∈ S(M, N) by means of a block operator matrix are given in [47] (see [20]). These constructions also involve Schur parameters of Θ and some additional Hilbert spaces and operators. One more model based on the Schur parameters of a scalar Schur class function Θ is obtained in [41]. In terms of this model in [41] are established the necessary and sufficient conditions in order to Θ has a meromorphic pseudocontinuation of bounded type to the exterior of the unit disk. In recent preprint [43] a construction of a minimal conservative realization of a scalar finite Blaschke product in terms of the Hessenberg matrix is given. 6. Block operator CMV matrices (the rest cases) Let {Γn } be the Schur parameters of the function Θ ∈ S(M, N). Suppose Γm is an isometry (respect., co-isometry, unitary) for some m ≥ 0. Then Θm (λ) = Γm for all λ ∈ D and Θm−1 (λ) = Γm−1 + λDΓ∗m−1 Γm (IDΓm−1 + λΓ∗m−1 Γm )−1 DΓm−1 , Θm−2 (λ) = Γm−2 + λDΓ∗m−2 Θm−1 (λ)(IDΓm−2 + λΓ∗m−2 Θm−1 (λ))−1 DΓm−2 , ... ... ... ... ... ... ... ... ... ... ... ..., Θ(λ) = Γ0 + λDΓ∗0 Θ1 (λ)(IDΓ0 + λΓ∗0 Θ1 (λ))−1 DΓ0 , λ ∈ D. In this case the function Θ also is the transfer function of the simple conservative systems constructed similarly to the situation in Section 5 by means of its Schur parameters and corresponding block operator CMV matrices U0 and Ue0 . Observe that if Γm is isometric (respect., co-isometric) then Γn = 0, DΓ∗n = DΓ∗m , DΓ∗n = IDΓ∗m (respect., DΓn = DΓm , DΓn = IDΓm ) for n > m. The constructions of the state spaces H0 = H0 ({Γn }n≥0 ) and e0 = H e 0 ({Γn }n≥0 ) are similar to (5.1) but one have to replace DΓn by {0} (respect., DΓ∗ by H n {0}) for n ≥ m, and DΓ∗n by DΓ∗m (respect., DΓn by DΓm ) for n > m. The relation e 0 ({Γ∗ }n≥0 ) = H0 ({Γn }n≥0 ) H n

remains true. If, in addition, the operator Γm is isometry ( ⇐⇒ the operator DΓ∗m is the orthogonal projection in DΓm−1 onto ker Γ∗m ) or co-isometry ( ⇐⇒ the operator DΓm is the orthogonal projection in DΓ∗m−1 onto ker Γm ), then the corresponding unitary elementary rotation takes the the row or the column form DΓm−1 (r) → DΓ∗m−1 , JΓm = Γm IDΓ∗m : ⊕ DΓ∗m DΓ∗m−1 Γm (c) : DΓm−1 → ⊕ JΓm = . DΓm DΓm Therefore, in definitions (5.2) of the block diagonal operator matrices f0 = M f0 ({Γn }n≥0 ) L0 = L0 ({Γn }n≥0 ), M0 = M0 ({Γn }n≥0 ), and M


29

one should replace (r)

• JΓm by JΓm and JΓn by IDΓ∗m for n > m, when Γm is isometry, (c)

• JΓm by JΓm , and JΓn by IDΓm for n > m, when Γm is co-isometry, • JΓm by Γm , when Γm is unitary. As in Section 5 in all these cases the block operators CMV matrices U0 = U0 ({Γn }n≥0 ) and Ue0 = Ue0 ({Γn }n≥0 ), are given by the products f0 L0 . U0 = L0 M0 , Ue0 = M

These matrices are five block-diagonal. In the case when the operator Γm , is unitary the block operator CMV matrices U0 and Ue0 are finite and otherwise they are semi-infinite. As in Section 5 the truncated block operator CMV matrices T0 = T0 (({Γn }n≥0 ) and Te0 = Te0 ({Γn }n≥0 ) are defined by (5.10) and (5.11) e e T0 = PH0 U0 ↾ H0 , Te0 = PH e 0 U0 ↾ H0 .

As before the operators T0 and Te0 are unitarily equivalent completely non-unitary contractions and, moreover, the equalities (5.9), (5.14), and Proposition 5.1 hold true. Unlike Section 5 the operators given by truncated block operator CMV matrices Tm and Tem obtaining from U0 and Ue0 by deleting first m + 1 rows and m + 1 columns are • co-shifts of the form   DΓ∗ DΓ∗m m 0 IDΓ∗m 0 0 ... ⊕ ⊕ 0 0 IDΓ∗m 0 . . .   Tm = Tem = 0 : DΓ∗m → DΓ∗m , 0 0 IDΓ∗m . . .   ⊕ ⊕ .. .. .. .. .. . .. . . . . . .. . when Γm is isometry, • the unilateral shifts of the form   0 0 0 0 ... IDΓm 0 0 0 . . .  Tm = Tem =  IDΓm 0 0 . . . :  0 .. .. .. .. .. . . . . .

DΓm DΓm ⊕ ⊕ DΓm → DΓm , ⊕ ⊕ .. .. . .

when Γm is co-isometry. One can see that Proposition 5.1 remains true. Similarly to (5.19) let us consider the conservative systems e 0}. ζ0 = {U0 ; M, N, H0}, ζe0 = {Ue0 ; M, N, H

One can check that the systems ζ0 and ζe0 are simple and unitarily equivalent. Moreover, relations (5.20) and, therefore, Theorem 5.2 and Theorem 5.3 remain valid for a situations considered here. In order to obtain precise forms of U0 and Ue0 one can consider the following cases: (1) Γ2N is isometric (co-isometric) for some N, (2) Γ2N +1 is isometric (co-isometric) for some N,

YURY ARLINSKI˘I

30

(3) the operator Γ2N is unitary for some N, (4) the operator Γ2N +1 is unitary for some N. We shall give several examples. Example 6.1. The operator Γ4 is isometric. Define the state spaces DΓ0 L DΓ2 L L L L L ⊕ DΓ∗4 DΓ∗4 . . . DΓ∗4 ..., H0 := ⊕ DΓ∗3 DΓ∗1 DΓ∗0 L DΓ∗2 L L L L L e 0 := ⊕ ⊕ DΓ∗4 DΓ∗4 . . . DΓ∗4 .... H DΓ3 DΓ1 Le L Then the spaces M H H0 can be represented as follows 0 and N

M L DΓ1 L DΓ3 L L Le L ⊕ ⊕ DΓ∗4 DΓ∗4 M H0 = ⊕ ..., DΓ∗0 DΓ∗2 DΓ∗4 N L DΓ∗1 L L L L L L ⊕ DΓ∗3 DΓ∗4 . . . DΓ∗4 .... N H0 = ⊕ DΓ0 DΓ2

Define the unitary operators M M M M M M M e 0, IDΓ∗ JΓ3 M0 = IM JΓ1 IDΓ∗ H ... : M H0 → M 4 4 M M M M M M M e 0, f0 = IN IDΓ∗ JΓ3 M JΓ1 IDΓ∗ H ... : N H0 → N 4 4 M (r) M M M M M M e0 → N JΓ4 JΓ2 IDΓ∗ L0 = JΓ0 IDΓ∗ ... : M H H0 . 4

Then



4

Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 0 0 0 0 0 DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗1 0 0 0 0 0 0  ∗ ∗ ∗ ∗  0 Γ2 DΓ1 −Γ2 Γ1 DΓ2 Γ3 DΓ2 DΓ3 0 0 0 0  0 0 0 0  0 DΓ2 DΓ1 −DΓ2 Γ∗1 −Γ∗2 Γ3 −Γ∗2 DΓ∗3  U0 = L0 M0 =  0 0 0 Γ4 DΓ3 −Γ4 Γ∗3 IDΓ∗ 0 0 0 4   0 0 0 0 0 0 0 0 IDΓ∗  4  0 0 0 0 0 0 0 IDΓ∗ 0  4 .. .. .. .. .. .. .. .. .. . . . . . . . . .  Γ0 DΓ∗0 0 0 0 0 0 0 ∗  Γ1 DΓ0 ∗ ∗ ∗ −Γ Γ D Γ D D 0 0 0 0 1 0 Γ1 2 Γ1 Γ2  DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ1 DΓ∗ 0 0 0 0 2  ∗  0 ∗ ∗ 0 0 Γ D −Γ Γ D 0 Γ D Γ3 3 2 Γ3 4 3 Γ2 f0 L0 =  Ue0 = M  0 0 0 0 DΓ3 DΓ2 −DΓ3 Γ∗2 −Γ∗3 Γ4 −Γ∗3   0 0 0 0 0 0 IDΓ∗ 0  4  0 0 0 0 0 0 0 IDΓ∗  4 .. .. .. .. .. .. .. .. . . . . . . . .

 ... . . .  . . .  . . .  . . . ,  . . .  . . .  .. .  0 ... 0 . . .  0 . . .  0 . . .  0 . . . ,  0 . . .  0 . . .  .. .. . .


31

Example 6.2. The operator Γ0 is co-isometric. Then

e0 = H0 = H

∞ M X

DΓ0 ,

n=0

L L L M0 = IM IDΓ0 IDΓ0 ..., L L L f0 = IN ID M ID ..., Γ0 L Γ0 L (c) L L0 = JΓ0 IDΓ0 IDΓ0 ..., 

Γ0 0 0 0 0 DΓ0  0 IDΓ0 0 U0 = Ue0 =   0 IDΓ0  0 .. .. .. . . .

0 0 0 0 .. .

 ... . . .  . . . .  . . . .. .

Example 6.3. The operator Γ2 is co-isometric. In this case DΓ0 L L L L L ..., . . . DΓ2 DΓ2 DΓ2 H0 = ⊕ DΓ∗1 DΓ∗0 L L L L L e ..., . . . DΓ2 DΓ2 DΓ2 H0 = ⊕ DΓ1 

   U0 = L0 M0 =    

JΓ0

(c)

JΓ2

IDΓ2

IDΓ2

..

0 0 Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 ∗ ∗ DΓ0 −Γ0 Γ1 −Γ0 DΓ∗1 0 0  −Γ2 Γ∗1 0 0 Γ2 DΓ1  0  0 0 0 DΓ2 DΓ1 −DΓ2 Γ∗1 =  0 0 0 0 I  DΓ2  0 0 0 IDΓ2  0 .. .. .. .. .. . . . . .

 IM  JΓ1   IDΓ2   IDΓ2 

. 0 0 0 0 0 0 .. .

 ... . . .  . . .  . . . ,  . . .  . . . .. .

 ..

.

  =  

YURY ARLINSKI˘I

32



IM

  f0 L0 =  Ue0 = M  

JΓ1 IDΓ2

IDΓ2

..

. 0 0 0 0 0



 J Γ0 (c)  JΓ2   IDΓ2   IDΓ2 

Γ0 DΓ∗0 0 0  Γ1 DΓ0 −Γ1 Γ∗0 DΓ∗1 Γ2 0  0 DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2  0 0 D 0 Γ2 =  0 0 IDΓ2  0  0 0 0 IDΓ2  0 .. .. .. .. .. . . . . .

 ... . . .  . . .  . . . .  . . .  . . . .. .

0 0 0 0 0 0 .. .

 ..

.

   =  

Example 6.4. The operator Γ1 is isometric. In this case DΓ0 L L L L L DΓ∗1 DΓ∗1 . . . DΓ∗1 ..., H0 = ⊕ DΓ∗1 e 0 = DΓ∗ L DΓ∗ L DΓ∗ L . . . L DΓ∗ L . . . . H 0 1 1 1 

   U0 = L0 M0 =    



JΓ0

IDΓ∗ 1

IDΓ∗ 1

IDΓ∗ 1

..

.

     



IM

Γ0 DΓ∗0 Γ1 DΓ∗0 0 0 0 0 0 ∗ ∗ DΓ0 −Γ0 Γ1 −Γ0 0 0 0 0 0   0 0 0 IDΓ∗ 0 0 0 0 1  = 0 0 0 0 0 0 0 IDΓ∗  1  0 0 0 0 0 IDΓ∗ 0 0  1 .. .. .. .. .. .. .. .. . . . . . . . .   JΓ0 IN (r)    JΓ1     I e f D ∗ Γ U0 = ML0 =   1   I D ∗   Γ1 .. .  Γ0 DΓ∗0 0 0 0 0 0 0 Γ1 DΓ0 −Γ1 Γ∗0 IDΓ∗ 0 0 0 0 0 1   0 0 IDΓ∗ 0 0 0 0 = 0 1  0 0 0 0 IDΓ∗ 0 0 0  1 .. .. .. .. .. .. .. .. . . . . . . . .

(r)

JΓ1

IDΓ∗ 1

IDΓ∗ 1

 ... . . .  . . .  , . . .  . . .  .. .

..

.



IDΓ∗ 1

IDΓ∗ 1

IDΓ∗ 1

 ... . . .  . . . . . . .  .. .

   =  

..

.

   =  


Example 6.5. The operator Γ3 is isometric. DΓ0 L DΓ2 L L L L L ⊕ DΓ∗3 H0 = ⊕ DΓ∗3 . . . DΓ∗3 ..., DΓ∗3 DΓ∗1 DΓ∗0 L L L L L L e0 = ⊕ .... . . . DΓ∗1 DΓ∗3 DΓ∗3 DΓ∗2 H DΓ∗1    JΓ0 IM    JΓ2 JΓ1    (r) IDΓ∗    J Γ 3 3    U0 = L0 M0 =  =  IDΓ∗ IDΓ∗     3 3    I I DΓ∗ DΓ∗    3 3 .. .. . .   Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 0 0 0 0 ... DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗ 0 0 0 0 0 . . . 1   ∗   0 ∗ Γ3 ∗ Γ D −Γ Γ D D 0 0 0 . . . 2 Γ 2 Γ Γ 1 1 2 2   ∗ ∗ ∗  0 0 0 . . . =  0 DΓ2 DΓ1 −DΓ2 Γ1 −Γ2 Γ3 −Γ2 ,  0 0 0 0 0 IDΓ∗ 0 0 . . .   3   0 0 0 0 0 0 I 0 . . . D ∗ Γ   3 .. .. .. .. .. .. .. .. .. . . . . . . . . .    JΓ0 IN    JΓ2 JΓ1    (r)   IDΓ∗  J Γ 3 3   f0 L0 =  Ue0 = M    = IDΓ∗ IDΓ∗     3 3    I I DΓ∗ DΓ∗    3 3 .. .. . .   Γ0 DΓ∗0 0 0 0 0 0 0 ... ∗ ∗ ∗ ∗  Γ1 DΓ0 −Γ1 Γ0 DΓ1 Γ2 DΓ1 DΓ2 0 0 0 0 . . .   DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ∗1 DΓ∗2 0 0 0 0 . . .   0 Γ3 DΓ2 −Γ3 Γ∗2 IDΓ∗ 0 0 0 . . .  0 = 3 .   0 0 0 0 0 I 0 0 . . . D ∗ Γ   3   0 0 . . . 0 0 0 0 0 I D ∗   Γ 3 .. .. .. .. .. .. .. .. .. . . . . . . . . .

Example 6.6. The operator Γ5 is co-isometric.

DΓ0 L DΓ2 L L L L L ⊕ ..., . . . DΓ5 DΓ5 DΓ4 H0 = ⊕ ∗ ∗ DΓ3 DΓ1 DΓ∗0 L DΓ∗2 L DΓ∗4 L L L L L e0 = ⊕ ⊕ ⊕ ..., . . . DΓ5 DΓ5 DΓ5 H DΓ5 DΓ3 DΓ1

33

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34

L L L L L IDΓ5 IDΓ5 JΓ4 ..., JΓ2 L0 = JΓ0 L (c) L L L L L JΓ5 JΓ3 IDΓ5 M0 = IM JΓ1 IDΓ5 ..., L L L L L L (c) f0 = IN JΓ1 JΓ5 IDΓ5 IDΓ5 .... JΓ3 M



Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 0 0 0 0 0 ∗ ∗ DΓ0 −Γ0 Γ1 −Γ0 DΓ∗1 0 0 0 0 0 0   0 Γ2 DΓ1 −Γ2 Γ∗1 DΓ∗2 Γ3 DΓ∗2 DΓ∗3 0 0 0 0   0 DΓ2 DΓ1 −DΓ2 Γ∗1 −Γ∗2 Γ3 −Γ∗2 DΓ∗3 0 0 0 0  0 0 Γ4 DΓ3 −Γ4 Γ∗3 DΓ∗4 Γ5 0 0 0  0 U0 = L0 M0 =  0 0 DΓ4 DΓ3 −DΓ4 Γ∗3 −Γ∗4 Γ5 0 0 0  0  0 0 0 0 0 D 0 0 0  Γ5  0 0 0 0 0 IDΓ5 0 0  0  0 0 0 0 0 0 IDΓ5 0  0 .. .. .. .. .. .. .. .. .. . . . . . . . . .  Γ0 DΓ∗0 0 0 0 0 0 0 ∗ ∗ ∗ ∗  Γ1 DΓ0 −Γ1 Γ0 DΓ1 Γ2 DΓ1 DΓ2 0 0 0 0  DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ∗1 DΓ∗2 0 0 0 0   0 0 Γ3 DΓ2 −Γ3 Γ∗2 −DΓ∗3 Γ4 DΓ∗3 DΓ∗4 0 0  DΓ∗4 Γ5 0 0 0 0 Γ4 DΓ3 −Γ4 Γ∗3  0 f0 L0 =  Ue0 = M 0 0 0 Γ5 DΓ4 −Γ5 Γ∗4 0 0  0  0 0 0 DΓ5 DΓ4 −DΓ5 Γ∗4 0 0  0  0 0 0 0 0 IDΓ5 0  0  0 0 0 0 0 0 0 I  DΓ5 .. .. .. .. .. .. .. .. . . . . . . . .

Example 6.7. The operator Γ2N is unitary. In this case

DΓ2n ⊕ , n=0 DΓ∗2n+1 , NP −1 L DΓ∗2n e0 = ⊕ H n=0 DΓ2n+1   IM   JΓ1   .. ,  JΓ3 .   ..    . JΓ2(N−1) JΓ2N−1 Γ2N   JΓ0 JΓ2     . .  JΓ3 ..    ..    . JΓ H0 =



  U0 =   

JΓ0 JΓ2

 IN  JΓ1  Ue0 =   

NP −1 L

2(N−1)

JΓ2N−1

Γ2N

 ... . . .  . . .  . . .  . . .  . . . ,  . . .  . . .  . . . .. . 0 0 0 0 0 0 0 0 0 .. .

 ... . . .  . . .  . . .  . . .  . . . .  . . .  . . .  . . . .. .


35

If N = 1 (Γ2 is unitary) then we have     Γ0 DΓ∗0 Γ1 DΓ∗0 DΓ∗1 Γ0 DΓ∗0 0 −Γ1 Γ∗0 DΓ∗1 Γ2  U0 = DΓ0 −Γ∗0 Γ1 −Γ∗0 DΓ∗1  , Ue0 =  Γ1 DΓ0 0 Γ2 DΓ1 −Γ2 Γ∗1 DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2

Example 6.8. The operator Γ2N +1 is unitary. e 0 = DΓ∗ if N = 0, H0 = DΓ0 , H 0 D DΓ∗2n L Γ 2n N −1 NP −1 L L , e0 = P L ⊕ ⊕ DΓ∗2N if N ≥ 1 DΓ2N , H H0 = n=0 n=0 DΓ∗2n+1 DΓ2n+1 Γ0 DΓ∗0 IM 0 Γ0 DΓ∗0 Γ1 U0 = = , ∗ 0 Γ1 DΓ0 −Γ∗0 Γ1 DΓ0 −Γ 0 I 0 Γ0 DΓ∗0 Γ0 DΓ∗0 Ue0 = N , if N = 0, ∗ = 0 Γ1 DΓ0 −Γ0 Γ1 DΓ0 −Γ1 Γ∗0   IM     JΓ1 JΓ0      JΓ3 JΓ2 ,  U0 =  . .    .. ..     JΓ2N−1 JΓ2N Γ2N +1   IN     JΓ0 JΓ1      JΓ3 JΓ2   , if N ≥ 1. Ue0 =  . .    . . . .     J J Γ2N−1

Γ2N

Γ2N +1

If N = 1 (Γ3  Γ0 DΓ0 U0 =   0 0

is unitary) then

  DΓ∗0 Γ1 DΓ∗0 DΓ∗1 0 Γ0 DΓ∗0 0 0 ∗ ∗ ∗   ∗ ∗ ∗ −Γ0 Γ1 −Γ0 DΓ1 0  e −Γ1 Γ0 DΓ1 Γ2 DΓ1 DΓ∗2  Γ1 DΓ0 , U = 0 ∗ DΓ1 DΓ0 −DΓ1 Γ∗0 −Γ∗1 Γ2 −Γ∗1 DΓ∗ Γ2 DΓ1 −Γ2 Γ1 DΓ∗2 Γ3  2 DΓ2 DΓ1 −DΓ2 Γ∗1 −Γ∗2 Γ3 0 0 Γ3 DΓ2 −Γ3 Γ∗2



 . 

7. Unitary operators with cyclic subspaces, dilations, and block operator CMV matrices

7.1. Carath´ eodory class functions associated with conservative systems. Definition 7.1. Let M be a separable Hilbert space. The class C(M) of L(M)-valued functions holomorphic on the unit disk D and having positive real part for all λ ∈ D is called the Carathéodory class. D C Consider a conservative systems τ = ; M, M, H whose input and output spaces B A coincide. Put H= M⊕H

YURY ARLINSKI˘I

36

and let the function Fτ (z) be defined as follows (7.1) where

Fτ (λ) = PM(Uτ + λIH )(Uτ − λIH )−1 ↾ M, λ ∈ D,

M M D C Uτ = : ⊕ → ⊕ B A H H is unitary operator in H associated with the system τ . The function Fτ (z) is holomorphic in D and ¯ H )−1 (Uτ − λIH )−1 ↾ M. Fτ (λ) + Fτ∗ (λ) = 2(1 − |λ|2 )PM(Uτ∗ − λI It follows that Fτ (λ) + Fτ∗ (λ) ≥ 0 for all λ ∈ D. The function Fτ (λ) defined by (7.1) belongs to the Carathéodory class C(M) and, in addition, Fτ (0) = IM. We also shall consider the function ¯ = PM(IH + λUτ )(IH − λUτ )−1 . Feτ (λ) := F ∗ (λ) τ

e The functions Fτ andFτ we will call the Carathéodory functions associated with conservative D C system τ = ; M, M, H . B A Proposition 7.2. Let

D C τ= ; M, M, H B A be a conservative system. Then the transfer function Θτ (λ) and the Carathéodory function Fτ (λ) are connected by the following relations (7.2)

1 (Fτ (λ) − IM)(Fτ (λ) + IM)−1 , λ ∗ ¯ −1 ¯ Fτ (λ) = (IM + λΘ∗τ (λ))(I M − λΘτ (λ)) , λ ∈ D.

¯ = Θ∗τ (λ)

Proof. We use the well known Schur–Frobenius formula for the inverse of block operators. Let Φ be a bounded linear operator given by the block operator matrix M M X Y Φ= : ⊕ → ⊕ . Z W H H Suppose that W −1 ∈ L(H) and (X − Y W −1 Z)−1 ∈ L(M). Then Φ−1 ∈ L(M ⊕ H, M ⊕ H) and K −1 −K −1 Y W −1 −1 Φ = , −W −1 ZK −1 W −1 + W −1 ZK −1 Y W −1 where K = X − Y W −1 Z. Applying this formula for IM − λD −λC Φ = IH − λUτ = , λ ∈ D, −λB IH − λA

we get K = IM − λD − λ2 C(IH − λA)−1 B = IM − λΘτ (λ). Therefore Hence

PM(IH − λUτ )−1 ↾ M = (IM − λΘτ (λ))−1 ,

λ ∈ D.

¯ −1 , PM(IH − λUτ∗ )−1 ↾ M = (IM − λΘ∗τ (λ))

λ ∈ D.


37

Since Uτ is unitary, from (7.1) we get Fτ (λ) = PM(IH + λUτ∗ )(IH − λUτ∗ )−1 ↾ M = ¯ −1 = = −IM + 2PM(IH − λUτ∗ )−1 ↾ M = −IM + 2(IM − λΘ∗τ (λ)) ∗ ¯ ∗ ¯ −1 = (IM + λΘτ (λ))(IM − λΘτ (λ)) , λ ∈ D. The following theorem is well known (see [29]). Theorem 7.3. Let M be a separable Hilbert space and let F (λ) ∈ C(M). Then (1) F (λ) admits the integral representation

1 F (λ) = (F (0) − F ∗ (0)) + 2

Z2π 0

eit + λ dΣ(t), eit − λ

where Σ(t) is a non-decreasing and nonnegative L(M)-valued function on [0, 2π]; (2) under the condition F (0) = IM there exists a Hilbert space H containing M as a subspace, and a unitary operator U in H such that F (λ) = PM(U + λIH )(U − λIH )−1 ↾ M; moreover, the pair {H, U} can be chosen minimal in the sense span {U n M, n ∈ Z} = H. Proposition 7.4. [41]. The conservative system D C τ= ; M, M, H B A is simple if and only if span {Uτn M, n ∈ Z} = H. Proof. Let τ be a simple conservative system. Suppose h ∈ H and h is orthogonal to Uτn M for all n ∈ Z. Then the vectors Uτ∗n h are orthogonal to M in H for all n ∈ Z. It follows that h ∈ H and Ch = CAh = CA2 h = . . . = CAn h = . . . = 0, B ∗ h = B ∗ A∗ h = B ∗ A∗2 h = . . . B ∗ A∗n h = . . . = 0. T T n ∗ ∗n Hence h ∈ n≥0 ker(CA ) ∩ n≥0 ker(B A ) . Since τ is simple we get h = 0, i.e., (7.3)

span {Uτn M, n ∈ Z} = H.

Conversely, let span {Uτn M, n ∈ Z} = H. Suppose thar relations (7.3) hold for some h ∈ H. Then h ⊥ Uτn M for all n ∈ Z. Hence h = 0 and τ is simple.

YURY ARLINSKI˘I

38

7.2. Unitary operators with cyclic subspaces. Let U be a unitary operator in a separable Hilbert space K and let M be a subspace of K. Put H = K ⊖ M. Then U takes the block operator matrix form M M D C U= : ⊕ → ⊕ . B A H H Since U is unitary, the system

D C η= ; M, M, H B A

is conservative. By Proposition 7.4 the system η is simple if and only if span {U n M, n ∈ Z} = K.

(7.4)

A subspace M of K is called cyclic for U if the condition (7.4) is satisfied. Define the Carathéodory function and a Schur function

FM(λ) = PM(U + λIH )(U − λIH )−1 ↾ M, λ ∈ D

1 (FM(λ) − IM)(FM(λ) + IM)−1 , λ ∈ D. λ According to Proposition 7.2 the transfer function Θ(λ) of the system η and the function EM(λ) are connected by the relation ¯ λ ∈ D. Θ(λ) = E ∗ (λ), EM(λ) =

M

Theorem 7.5. Let U be a unitary operator in a separable Hilbert space and let M be a cyclic subspace for U. Then U is unitarily equivalent to the block operator CMV matrices e = M⊕ U0 ({Γn }n≥0 ) and Ue0 ({Γn }n≥0) in the Hilbert spaces H = M ⊕ H0 ({Γn }n≥0 ) and H e 0 ({Γn }n≥0 ), respectively, where {Γn }n≥0 are the Schur parameters of the function H

1 ∗ ¯ ∗ ¯ (F (λ) − IM)(FM (λ) + IM)−1 . λ M Proof. Because M is a cyclic subspace for U, the conservative system η is simple. By Theorem 5.3 the system η is unitarily equivalent to the systems ζ0 and ζe0 given by (5.19). From (3.5) it follows that U is unitarily equivalent to U0 ({Γn }n≥0 ) and Ue0 ({Γn }n≥0 ). Θ(λ) =

Suppose that the cyclic subspace M for unitary operator U in K is one-dimensional. Let ϕ ∈ M, ||ϕ|| = 1, and let µ(ζ) = (E(ζ)ϕ, ϕ)K, where E(ζ), ζ ∈ T, is the resolution of the identity for U. Then the scalar Carathéodory function F (λ) is of the form Z ζ +λ −1 F (λ) = (U + λIH )(U − λIH ) ϕ, ϕ K = dµ(ζ) , λ ∈ D. T ζ −λ Thus, the function F (λ) is associated with the probability measure µ on T. The Schur function associated with µ [60] is the function E(λ) =

1 F (λ) − 1 , λ ∈ D. λ F (λ) + 1


39

By Geronimus theorem [44] the Schur parameters of the function E(λ) coincide with Verblun¯ λ ∈ D and let sky coefficient {αn }n≥0 of the measure µ (see [60]). Let Θ(λ) := E(λ), {γn }n≥0 be the Schur parameters of Θ. Then α ¯ n = γn for all n and the CMV matrices e e U0 = U0 ({γn }n≥0 ) and U0 = U0 ({γn }n≥0 ) coincide with the CMV matrices C and Ce given by (1.2) and (1.3), correspondingly. Observe that dim K = m ⇐⇒ the function E(λ) is the Blaschke product of the form m Y λ − λk iϕ E(λ) = e ¯k λ . 1−λ k=1 7.3. Unitary dilations of a contraction. Let T be a contraction acting in a Hilbert space H. The unitary operator U in a Hilbert space H containing H as a subspace is called the unitary dilation of T if T n = PH U n for all n ∈ N [64]. Two unitary dilations U in H and U ′ in H′ of T are called isomorphic if there exists a unitary operator W ∈ L(H, H′ ) such that W ↾ H = IH

and W U = U ′ W.

It is established in [64] that for every contraction T in the Hilbert space H there exists a unitary dilation U in a space H such that U is is minimal [64], i.e., span {U n H, n ∈ Z} = H.

Moreover, two minimal unitary dilations of T are isomorphic [64]. The minimal unitary dilation by means of the infinite matrix form is constructed in [64] on the base of Schäffer paper [55]. Below we show that the minimal unitary dilations can be given by the operator CMV matrices. Theorem 7.6. Let T be a contraction in a Hilbert space H. Define the Hilbert spaces

(7.5)

DT L DT L ⊕ ··· , H0 = ⊕ ∗ ∗ DT DT DT ∗ L DT ∗ L e0 = ⊕ ⊕ H ··· , DT DT

e 0 . Let e0 = H ⊕ H and the Hilbert spaces H0 = H ⊕ H0 , and H DT DT ∗ 0 IDT ∗ J0 = : ⊕ → ⊕ IDT 0 DT ∗ DT Define operators (7.6) and (7.7)

L L L e0 , M0 = IH J J · · · : H0 → H L 0L 0L e0 → H0 , L 0 = JT J0 J0 · · · : H

e0 → H e0 . U0 = L0 M0 : H0 → H0 , Ue0 = M0 L0 : H

e0 , Ue0 } are unitarily equivalent minimal unitary dilations of the operator Then {H0 , U0 } and {H T.

YURY ARLINSKI˘I

40

Proof. Define the L(H)-valued function Fe(λ) = (IH + λT )(IH − λT )−1 , λ ∈ D.

Then the function Fe belongs to the Carathéodory class C(H) and Θ(λ) = T =

1 e (F (λ) − IH )(Fe(λ) + IH )−1 , λ ∈ D, λ

belongs to the Schur class S(H, H). The Schur parameters of Θ is the sequence Γ0 = T, Γn = 0 ∈ S(DT , DT ∗ ), n ∈ N. e 0 be defined by (7.5), H0 = H ⊕ H0 , H e 0 . Then the operators U0 and e0 = H ⊕ H Let H0 and H Ue0 defined by (7.7) are the block operator CMV matrices constructed by means of the Schur e 0} be the corresponding parameters of Θ. Let ζ0 = {H0 , M, M, H0} and ζe0 = {Ue0 , M, M, H conservative systems. By Theorem 5.3 the systems ζ0 and ζe0 are simple, unitary equivalent, and their transfer functions are equal Θ. By Proposition 7.2 we have

Hence

(IH + λT )(IH − λT )−1 = Fe(λ) = (IH + λΘ(λ))(IH − λΘ(λ))−1 = = PH (IH0 + λU0 )(IH0 − λU0 )−1 ↾ H. T n = PH U0n ↾ H, n = 0, 1, . . . .

Therefore U0 is a unitary dilation of T in H0 . By Proposition 7.4 this dilation is minimal. e0. Similarly the operator Ue0 is a minimal unitary dilation of T in H Taking into account (7.6), (5.6), and (5.7) we obtain the for minimal unitary dilations U0 and Ue0 :  0 0 0 0 T 0 DT ∗ DT 0 −T ∗ 0 0 0 0   0 0 0 0 IDT ∗ 0 0   0 IDT 0 0 0 0 0 U0 =  0 0 0 0 0 IDT ∗  0  0 0 IDT 0 0 0  0  0 0 0 0 0 0  0 .. .. .. .. .. .. .. . . . . . . . 

0 0 0 0 T DT ∗  0 0 0 IDT ∗ 0 0  DT −T ∗ 0 0 0 0   0 0 0 0 0 IDT ∗ Ue0 =  0 IDT 0 0 0  0  0 0 0 0 0  0  0 0 0 0 I 0  DT .. .. .. .. .. .. . . . . . .

following operator matrix forms 0 0 0 0 0 0 0 0 0 0 0 0 0 IDT ∗ .. .. . .

0 0 0 0 0 0 0 0 0 0 0 IDT ∗ 0 0 .. .. . .

0 0 0 0 0 0 0 .. .

 ... . . .  . . .  . . .  . . . ,  . . .  . . . .. .

 ... . . .  . . .  . . .  . . . .  . . .  . . . .. .


41

7.4. The Naimark dilation. Let M be a separable Hilbert space. Denote by B(T) the σalgebra of Borelian subsets of the unit circle T = {ξ ∈ C : |ξ| = 1}. Let µ be a L(M)-valued Borel measure on B(T), i.e., (a) for any δ ∈ B(T) the operator µ(δ) is nonnegative, (b) µ(∅) = 0, (c) µ is σ-additive with respect to the strong operator convergence. Denote by M(T, M) the set of all L(M)-valued Borel measures. Definition 7.7. [33], [20], [41]. Let µ ∈ M(T, M) be a probability measure ( µ(T) = IM) and let the operators {Sn }n∈Z be the sequence of Fourier coefficients of µ, i.e., Z Sn = ξ −n µ(dξ), n ∈ Z. T

A Naimark dilation of µ is a pair {H, U}, where H is a separable Hilbert space containing M as a subspace, U is unitary operator in H such that Sn = PMU n ↾ M, n ∈ Z.

A Naimark dilation is called minimal if

span {U n M, n ∈ Z} = H.

Proposition 7.8. [33], [20], [41]. Let {H1 , U1 } and {H2 , U2 } be two minimal Naimark dilations of a probability measure µ ∈ M(T, M). Then there exists a unitary operator W ∈ L(H1 , H2 ) such that WU1 = U2 W and W↾ M = IM. The minimal Naimark dilation is constructed by T. Constantinescu in [33] by means of the infinite in both sides block operator matrix whose entries depend on some choice sequence. Here we construct the minimal Naimark dilations in the form of block operator CMV matrices. Theorem 7.9. Let M be a separable Hilbert space and let µ ∈ M(T, M) be a probability measure. Define the functions Z ξ+λ F (λ) = µ(dξ), λ ∈ D, ξ−λ T

1 E(λ) = (F (λ) − IM)(F (λ) + IM)−1 . λ Then E(λ) belongs to the Schur class S(M, M). Let {Gn }n≥0 be the Schur parameters of E. Construct the Hilbert spaces e0 = H e 0 ({Gn }n≥0 ) H0 = H0 ({Gn }n≥0 ), H and the Hilbert spaces

Let

e0 = M ⊕ H e 0. H0 = M ⊕ H0 , H

U0 = U0 ({Gn }n≥0 ), Ue0 = Ue0 ({Gn }n≥0 ) be the block operator CMV matrices constructing by means of {Gn }. Then the pairs {H0 , U0 } e0 , Ue0 } are unitarily equivalent minimal Naimark dilations of the measure µ. and {H

YURY ARLINSKI˘I

42

Proof. The function F (λ) has the Taylor expansion Z ∞ ∞ X X n ξ −n µ(dξ) = IM + 2 λ n Sn . F (λ) = IM + 2 λ n=1

n=1

T

Then ¯ = IM + 2 F (λ) ∗

∞ X

λn S−n .

n=1

Because F (λ) + F ∗ (λ) ≥ 0 for λ ∈ D, the L(M)-valued function E(λ) belongs to the Schur class S(M, M). Construct the Hilbert spaces H0 = H0 ({Gn }n≥0 ), H0 = M ⊕ H0 and let U0 = U0 ({Gn }n≥0 ) = (U0 ({Gn }n≥0 ))∗ be the block operator CMV matrix. Then U0 is unitary operator in the Hilbert space H0 . The system ζ0 = {U0 ; M, M, H0} is a conservative and simple, and its transfer function is equal to E(λ) (see Subsection 5.3, (5.19), Theorem 5.3). ¯ By Hence the transfer of the adjoint system ζ0∗ = {U0∗ ; M, M, H0} is equal to Θ(λ) = E ∗ (λ). definition of F (λ) and E(λ), and by Proposition 7.2, and (7.2) we have ∗ ¯ −1 ¯ F (λ) = (IM + λE(λ))(IM − λE(λ))−1 = (IM + λΘ∗ (λ))(I = M − λΘ (λ)) ∗ ∗ −1 = PM(U0 + λIH0 )(U0 − λIH0 ) ↾ M. Hence

F (λ) = IM + 2

∞ P

λn PMU0n ↾ M,

n=1 ∞ P

¯ = IM + 2 F (λ) ∗

n=1

λn PMU0−n ↾ M.

Thus, the pair {H0 , U0 } is the minimal Naimark dilation of the measure µ. The same is true e 0 , Ue0 }. for the pair {H 8. The block operator CMV matrix models for completely non-unitary contractions

Theorem 8.1. Let T be a completely non-unitary contraction in a separable Hilbert space H. Let ΦT (λ) = (−T + λDT ∗ (IH − λT ∗ )−1 DT )↾ DT be the Sz.-Nagy–Foias characteristic function of T [64]. If {Γn }n≥0 are the Schur parameters of ΦT (λ), then the operator T is unitarily equivalent to the truncated block operator CMV matrices T0 ({Γ∗n }n≥0 ) and Te0 ({Γ∗n }n≥0 ).

Proof. Consider the simple conservative system ∗ −T DT ; DT ∗ , DT , H . η= DT ∗ T

The transfer function of η is given by Since

Θη (λ) = −T ∗ + λDT (IH − λT )−1 DT ∗ ↾ DT ∗ , λ ∈ D.

ΦT (λ) = −T + λDT ∗ (IH − λT ∗ )−1 DT ↾ DT , λ ∈ D, ¯ λ ∈ D. Hence, if {Γn }n≥0 are the Schur parameters of ΦT (λ), then we get ΦT (λ) = Θ∗η (λ), ∗ {Γn }n≥0 are the Schur parameters of Θη (λ). Construct the Hilbert spaces H0 = H0 ({Γ∗n }n≥0 ),


43

e0 = H e 0 ({Γ∗ }n≥0 ), the block operator CMV matrices U0 = U0 ({Γ∗ }n≥0 ), Ue0 = Ue0 ({Γ∗ }n≥0 ), H n n n ∗ ∗ e e truncated block CMV matrices T0 = T0 ({Γn }n≥0 ) and T0 = T0 ({Γn }n≥0 ). Consider the corresponding conservative systems n o e0 . ζ0 = {U0 ; DT ∗ , DT , H0 } , ζe0 = Ue0 ; DT ∗ , DT , H

By Theorem 5.3 the systems ζ0 and ζe0 are simple conservative realizations of the function Θ. It follows that the operator T is unitarily equivalent to the operators T0 ({Γ∗n }n≥0 ) and Te0 ({Γ∗n }n≥0 ). ∗ Observe that T0 ({Γ∗n }n≥0 ) = Te0 ({Γn }n≥0 ) and Te0 ({Γ∗n }n≥0 ) = (T0 ({Γn }n≥0 ))∗ . The results of Sz.-Nagy–Foias [64, Theorem VI.3.1] states that if the function Θ ∈ S(M, N) is purely contractive (||Θ(0)f || < ||f || for all f ∈ M \ {0}) then there exists a completely non-unitary contraction T whose characteristic function coincides with Θ. Here we give another proof of this result. Theorem 8.2. Let the function Θ(λ) ∈ S(M, N) be purely contractive. If {Γn }n≥0 are the Schur parameters of Θ(λ) then the characteristic functions of completely non-unitary contractions given by truncated block operator CMV matrices T0 ({Γ∗n }n≥0 ) and Te0 ({Γ∗n }n≥0 ) coincide with Θ.

¯ Then {Γ∗ }n≥0 are the Schur parameters of Θ. e Construct the e Proof. Let Θ(λ) := Θ∗ (λ). n ∗ ∗ e e Hilbert spaces H0 = H0 ({Γn }n≥0 ), H0 = H0 ({Γn }n≥0 ), the block operator CMV matrices U0 = U0 ({Γ∗n }n≥0 ), Ue0 = Ue0 ({Γ∗n }n≥0 ), truncated block CMV matrices T0 = T0 ({Γ∗n }n≥0 ), Te0 = Te0 ({Γ∗n }n≥0 ), and consider the corresponding conservative systems n o e0 . ζ0 = {U0 ; M, N, H0} , ζe0 = Ue0 ; M, N, H e Then the transfer functions of ζ0 and ζe0 are equal to Θ(λ). Since the operator ∗ M N Γ0 G0 ⊕ → ⊕ U0 = : F0 T0 H0 H0

is a contraction, there exist contractions (see [19], [38], [59]) K ∈ L(DT0 , N), M ∈ L(M, H0 ), X ∈ L(DM , DK∗ ) such that G0 = KDT0 , F0 = DT0∗ M, Γ∗0 = −KT0∗ M + DK∗ X DM .

Because U0 is unitary, the operators K, M∗ are isometries and X is unitary (see [8],[9]). The characteristic function of T0∗ and the transfer function of the system ζ0 are connected by the relation (see [10], [9]) e Θ(λ) = KΦT0∗ (λ)M + X DM . Because the operator DM is the orthogonal projection in M onto ker M, and we have for f ∈ ker M

e Θ(λ)↾ ker M = X ,

e ||Γ∗0 f || = ||Θ(0)f || = ||X f || = ||f ||.

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YURY ARLINSKI˘I

Since Γ∗0 is a pure contraction, we obtain ker M = {0}. Similarly ker K∗ = {0}, i.e., K and e M are unitary operators, and Θ(λ) = KΦT0∗ (λ)M, λ ∈ D. Thus the characteristic function ΦT0 of T0 coincides with Θ. Similarly, the characteristic function ΦTe0 of Te0 coincides with Θ.

Remark 8.3. For completely non-unitary contractions with one-dimensional defect operators and for a scalar Schur class functions Theorem 8.1 and Theorem 8.2 have been established in [12]. References [1] V.M. Adamjan and D.Z. Arov, On unitary coupling of semi-unitary operators, Math Issled. Kishinev I(2) (1966), 3–65 [Russian]. English translation in AMS Tranl. 95 (1970), 75–129. [2] D. Alpay, T. Azizov, A. Dijksma, and H. Langer, The Schur algorithm for generalized Schur functions. I. Coisometric realizations, Oper. Theory: Adv. Appl., 129 (2001), 1–36. [3] D. Alpay, T. Azizov, A. Dijksma, H. Langer, and G. Wanjala, The Schur algorithm for generalized Schur functions. II. Jordan chains and transformations of characteristic functions. Monatsh. Math. 138 (2003), no. 1, 1–29. [4] D. Alpay, T. Azizov, A. Dijksma, H. Langer, and G. Wanjala, The Schur algorithm for generalized Schur functions. IV. Unitary realizations, Oper. Theory Adv. Appl., 149 (2004), 23–45. [5] D. Alpay, A. Dijksma, and H. Langer, The transformation of Issai Schur and related topics in indefinite setting, Oper. Theory Adv. Appl., 176 (2007), 1–98. [6] D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo, Schur functions, operator colligations, and Pontryagin spaces, Oper. Theory Adv. Appl., 96, Birkhäuser Verlag, Basel-Boston, 1997. [7] T. Ando, De Branges spaces and analytic operator functions, Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan, 1990. [8] Yu. Arlinski˘ı, The Kalman–Yakubovich–Popov inequality for passive discrete time-invariant systems, Operators and Matrices 2(2008), No.1, 15–51. [9] Yu.M. Arlinski˘ı, Iterates of the Schur class operator-valued function and their conservative realizations, arXiv: 0801.4267 [math.FA] v 1, 28 Jan 2008. [10] Yu.M. Arlinski˘ı, S. Hassi, and H.S.V. de Snoo, Parametrization of contractive block-operator matrices and passive disrete-time systems, Complex Analysis and Operator Theory, 1 (2007), No.2, 211–233. [11] Yu.M. Arlinski˘ı, S. Hassi, and H.S.V. de Snoo. Q-functions of Hermitian contractions of Kre˘ın – Ovcharenko type, Integral Equations and Operator Theory. 53 (2005), No.2, 153–189. [12] Yu. Arlinski˘ı, L. Golinski˘ı, and E. Tsekanovski˘ı, Contractions with rank one defect operators and truncated CMV matrices, Journ. of Funct. Anal., 254 (2008) 154–195. [13] D.Z. Arov, Passive linear stationary dynamical systems, Sibirsk. Math. Journ.(1979) 20, No.2., 211228 [Russian]. English translation in Siberian Math. Journ., 20 (1979), 149–162. [14] D.Z. Arov, Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory, 1 (1979), 95–126 (Russian). English translation in Interpolation Theory, Systems, Theory and Related Topics. The Harry Dym Anniversary Volume, Oper. Theory: Adv. Appl. 134 (2002), 99–136, Birkhauser Verlag. [15] D.Z. Arov, M.A. Kaashoek, and D.P. Pik, Minimal and optimal linear discrete time-invariant dissipative scattering systems, Integral Equations Operator Theory, 29 (1997), 127–154. [16] D.Z. Arov and M.A. Nudel’man, A criterion for the unitary similarity of minimal passive systems of scattering with a given transfer function, Ukrain. Math. J., 52 no. 2 (2000), 161–172 [Russian]. English translation in Ukrainian Math. J. 52 (2000), no. 2, 161–172. [17] D.Z. Arov and M.A. Nudel’man, Tests for the similarity of all minimal passive realizations of a fixed transfer function (scattering and resistance matrix), Mat. Sb., 193 no. 6 (2002), 3–24 [Russian]. English translation in Sb. Math. 193 (2002), no. 5-6, 791–810. [18] D.Z. Arov and O. Staffans, State/signal linear time-invariant systems theory: passive discrete time systems. Internat. J. Robust Nonlinear Control 17 (2007), no. 5-6, 497–548. [19] Gr. Arsene and A. Gheondea, Completing matrix contractions, J. Operator Theory, 7 (1982), 179-189.


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