Iterates of the Schur class operator-valued function and their conservative realizations
aa r X i v : . [ m a t h . F A ] A ug ITERATES OF THE SCHUR CLASS OPERATOR-VALUED FUNCTIONAND THEIR CONSERVATIVE REALIZATIONS
YURY ARLINSKI˘I
Abstract.
Let M and N be separable Hilbert spaces and let Θ( λ ) be a function from theSchur class S ( M , N ) of contractive functions holomorphic on the unit disk. The operatorgeneralization of the classical Schur algorithm associates with Θ the sequence of contractions(the Schur parameters of Θ) Γ = Θ(0) ∈ L ( M , N ) , Γ n ∈ L ( D Γ n − , D Γ ∗ n − ) and the sequenceof functions Θ = Θ, Θ n ∈ S ( D Γ n , D Γ ∗ n ) n = 1 , . . . (the Schur iterares of Θ) connected bythe relationsΓ n = Θ n (0) , Θ n ( λ ) = Γ n + λD Γ ∗ n Θ n +1 ( λ )( I + λ Γ ∗ n Θ n +1 ( λ )) − D Γ n , | λ | < . The function Θ( λ ) ∈ S ( M , N ) can be realized as the transfer functionΘ( λ ) = D + λC ( I − λA ) − B of a linear conservative and simple discrete-time system τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) with thestate space H and the input and output spaces M and N , respectively.In this paper we give a construction of conservative and simple realizations of the Schuriterates Θ n by means of the conservative and simple realization of Θ. Contents
1. Introduction 22. Completely non-unitary contractions 53. Contractions generated by a contraction 84. Passive discrete-time linear systems and their transfer functions 124.1. Basic definitions 124.2. Defect functions of the Schur class functions 144.3. Parametrization of contractive block-operator matrices 145. The M¨obius representations 196. Realizations of the Schur iterates 236.1. Realizations of the first Schur iterate 236.2. Schur iterates of the characteristic function 266.3. Conservative realizations of the Schur iterates 28References 30
Mathematics Subject Classification.
Key words and phrases.
Contraction, characteristic function, passive system, conservative system, transferfunction, realization, Schur class function. Introduction
The Schur class S of scalar analytic functions and bounded by one in the unit disc D = { λ ∈ C : | λ | < } plays a prominent role in complex analysis and operator theory as wellin their applications in linear system theory and mathematical engineering. Given a Schurfunction f ( λ ), which is not a finite Blaschke product, define inductively f ( λ ) = f ( λ ) , f n +1 ( λ ) = f n ( λ ) − f n (0) λ (1 − f n (0) f n ( λ )) , n ≥ . It is clear that { f n } 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 γ n := f n (0) are called the Schur parameters: S f = { γ , γ , . . . } . Note that f n ( λ ) = γ n + λf n +1 ( λ )1 + ¯ γ n λf n +1 = γ n + (1 − | γ n | ) λf n +1 ( λ )1 + ¯ γ n λf n +1 ( λ ) , n ≥ . The method of labeling f ∈ S by its Schur parameters is known as the Schur algorithm andis due to I. Schur [33]. In the case when f ( λ ) = e iϕ N Y k =1 λ − λ k − ¯ λ k λ is a finite Blaschke product of order N , the Schur algorithm terminates at the N -th step.The sequence of Schur parameters { γ n } Nn =0 is finite, | γ n | < n = 0 , , . . . , N −
1, and | γ N | = 1.The Schur algorithm for matrix valued Schur class functions has been considered in thepaper of Delsarte, Genin, and Kamp [27] and in the book of Dubovoj, Fritzsche, and Kirstein[28]. An operator extension of the Schur algorithm was developed by T. Constantinescu in[25] and with numerous applications is presented in the book of Bakonyi and Constantinescu[17].In what follows the class of all continuous linear operators defined on a complex Hilbertspace H and taking values in a complex Hilbert space H is denoted by L ( H , H ) and L ( H ) := L ( H , H ). The domain, the range, and the null-space of a linear operator T aredenoted by dom T , ran T , and ker T , respectively. The set of all regular points of a closedoperator T is denoted by ρ ( T ). We denote by I H the identity operator in a Hilbert space H and by P L 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 positiveintegers will be denoted by N . An operator T ∈ L ( H , H ) is said to be(a) contractive if k T k ≤ isometric if k T f k = k f k for all f ∈ H ⇐⇒ T ∗ T = I H ;(c) co-isometric if T ∗ is isometric ⇐⇒ T T ∗ = I H ;(d) unitary if it is both isometric and co-isometric.Given a contraction T ∈ L ( H , H ). The operators D T := ( I − T ∗ T ) / , D T ∗ := ( I − T T ∗ ) / EALIZATIONS OF THE SCHUR ITERATES 3 are called the defect operators of T , and the subspaces D T = ran D T , D T ∗ = ran D T ∗ the defect subspaces of T . The dimensions dim D T , dim D T ∗ are known as the defect numbers of T . The defect operators satisfy the following intertwining relations(1.1) T D T = D T ∗ T, T ∗ D T ∗ = D T T ∗ . It follows from (1.1) that T D T ⊂ D T ∗ , T ∗ D T ∗ ⊂ D T , and T (ker D T ) = ker D T ∗ , T ∗ (ker D T ∗ ) =ker D T . Moreover, the operators T ↾ ker D T and T ∗ ↾ ker D T ∗ are isometries and T ↾ D T and T ∗ ↾ D T ∗ are pure contractions, i.e., || T f || < || f || for f ∈ H \ { } .The Schur class S ( H , H ) is the set of all function Θ( λ ) analytic on the unit disk D withvalues in L ( H , H ) and such that k Θ( λ ) k ≤ λ ∈ D . The following theorem takesplace. Theorem 1.1. [25] , [17] . 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 ( D Θ(0) , D Θ ∗ (0) ) such that (1.2) Θ( λ ) = Θ(0) + D Θ ∗ (0) Z ( λ )( I + Θ ∗ (0) Z ( λ )) − D Θ(0) , λ ∈ D . In what follows we will call the representation (1.2) of a function Θ( λ ) from the Schur class the M¨obius representation of Θ( λ ) and the function Z ( λ ) we will call the M¨obius parameter of Θ( λ ). Clearly, Z (0) = 0 and by Schwartz’s lemma we obtain that || Z ( λ ) || ≤ | λ | , λ ∈ D . The operator Schur’s algorithm [17]. Fix Θ( λ ) ∈ S ( M , N ), put Θ ( λ ) = Θ( λ ) and let Z ( λ ) be the M¨obius parameter of Θ. DefineΓ = Θ(0) , Θ ( λ ) = Z ( λ ) λ ∈ S ( D Γ , D Γ ∗ ) , Γ = Θ (0) = Z ′ (0) . If Θ ( λ ) , . . . , Θ n ( λ ) and Γ , . . . , Γ n have been chosen, then let Z n +1 ( λ ) ∈ S ( D Γ n , D Γ ∗ n ) be theM¨obius parameter of Θ n . PutΘ n +1 ( λ ) = Z n +1 ( λ ) λ , Γ n +1 = Θ n +1 (0) . The contractions Γ ∈ L ( M , N ) , Γ n ∈ L ( D Γ n − , D Γ ∗ n − ), n = 1 , , . . . are called the Schurparameters of Θ( λ ) and the function Θ n ( λ ) ∈ S ( D Γ n − , D Γ ∗ n − ) we will call the n − th Schuriterate of Θ( λ ).Formally we haveΘ n +1 ( λ ) ↾ ran D Γ n = 1 λ D Γ ∗ n ( I D Γ ∗ n − Θ n ( λ )Γ ∗ n ) − (Θ n ( λ ) − Γ n ) D − n ↾ ran D Γ n . Clearly, the sequence of Schur parameters { Γ n } is infinite if and only if the operators Γ n are non-unitary. The sequence of Schur parameters consists of finite number operators Γ , Γ , . . . , Γ N if and only if Γ N ∈ L ( D Γ N − , D Γ ∗ N − ) is unitary. If Γ N is isometric (co-isometric)then Γ n = 0 for all n > N .The following theorem is the operator generalization of Schur’s result. Theorem 1.2. [25] , [17] . There is a one-to-one correspondence between the Schur classfunctions S ( M , N ) and the set of all sequences of contractions { Γ n } n ≥ such that (1.3) Γ ∈ L ( M , N ) , Γ n ∈ L ( D Γ n − , D Γ ∗ n − ) , n ≥ . YURY ARLINSKI˘I
Notice that a sequence of contractions of the form (1.3) is called the choice sequence [24].It is known [23], [11] that every Θ( λ ) ∈ S ( M , N ) can be realized as the transfer functionΘ( λ ) = D + λC ( I H − λA ) − B of a linear conservative and simple discrete-time system (see Section 4) τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) with the state space H and input and output spaces M and N , respectively. In this paperwe study the problem of the conservative realizations of the Schur iterates of the functionΘ( λ ) ∈ S ( M , N ) by means of the the conservative realization of Θ.In this connection it should be pointed out that the similar problem for a scalar generalizedSchur class function has been studied in papers [1], [2], [3], [4].Here we describe our main results. Let A be a completely non-unitary contraction [38] ina separable Hilbert space H . Define the subspaces and operators H m, = ker D A m , H ,l = ker D A ∗ l , H m,l = ker D A m ∩ ker D A ∗ l , m, l ∈ N ,A m,l = P m,l A ↾ H m,l , where P m,l is the orthogonal projection in H onto H m,l .We prove that1) if A is a completely non-unitary contraction in a Hilbert space then for every n ∈ N the operators A n, , A n − , , . . . , A ,n are unitary equivalent completely non-unitary contractions and their Sz.-Nagy– Foias char-acteristic functions [38] coincide with the pure contractive part [38], [17] for the n -th Schuriterate Φ n ( λ ) of the characteristic function Φ( λ ) of A ;2) if Θ( λ ) ∈ S ( M , N ) is the transfer function of a simple conservative system τ = (cid:26)(cid:20) Γ CB A (cid:21) ; M , N , H (cid:27) then the Schur parameters of Θ take the formΓ = D − ∗ C (cid:0) D − B ∗ (cid:1) ∗ , Γ = D − ∗ D − ∗ CA (cid:0) D − D − ( B ∗ ↾ H , ) (cid:1) ∗ , . . . , Γ n = D − ∗ n − · · · D − ∗ CA n − (cid:16) D − n − · · · D − ( B ∗ ↾ H n − , ) (cid:17) ∗ , . . . , and the n -th Schur iterate Θ n ( λ ) of Θ is the transfer function of the simple conservative andunitarily equivalent systems τ ( k ) n = (" Γ n D − ∗ n − · · · D − ∗ ( CA n − k ) A k (cid:16) D − n − · · · D − ( B ∗ ↾ H n, ) (cid:17) ∗ A n − k,k ; D Γ n − , D Γ ∗ n − , H n − k,k ) for k = 0 , . . . , n . Here D − m and D − ∗ m are the Moore– Penrose pseudo-inverses. For acompletely non-unitary contraction A with rank one defect operators it was proved in[10] that the characteristic functions of the operators A , = P ker D A A ↾ ker D A and A , = P ker D A ∗ A ↾ ker D A ∗ coincide with the first Schur iterate of the characteristic function of A .This result has been established using the model of A given by a truncated CMV matrix. EALIZATIONS OF THE SCHUR ITERATES 5
Here we use another approach based on the parametrization of a contractive block-operatormatrix T = (cid:20) D CB A (cid:21) : M ⊕ H → N ⊕ K established in [16], [26], [36], and the construction of the passive realization for the M¨obiusparameter of Θ( λ ) obtained in [8] by means of a passive realization of Θ.2. Completely non-unitary contractions
Let S be an isometry in a separable Hilbert space H . A subspace Ω in H is calledwandering for V if S p Ω ⊥ S q Ω for all p, q ∈ Z + , p = q . Since S is an isometry, the latter isequivalent to S n Ω ⊥ Ω for all n ∈ N . If H = P ∞ n =0 ⊕ S n Ω then S is called a unilateral shift and Ω is called the generating subspace. The dimension of Ω is called the multiplicity of theunilateral shift S . It is well known [38, Theorem I.1.1] that S is a unilateral shift if and onlyif T ∞ n =0 S n H = { } . Clearly, if an isometry V is the unilateral shift in H then Ω = H ⊖ SH is the generating subspace for S . An operator is called co-shift if its adjoint is a unilateralshift.A contraction A acting in a Hilbert space H is called completely non-unitary if there isno nontrivial reducing subspace of A , on which A generates a unitary operator. Given acontraction A in H then there is a canonical orthogonal decomposition [38, Theorem I.3.2] H = H ⊕ H , A = A ⊕ A , A j = A ↾ H j , j = 0 , , where H and H reduce A , the operator A is a completely non-unitary contraction, and A is a unitary operator. Moreover, H = \ n ≥ ker D A n ! \ \ n ≥ ker D A ∗ n ! . Since n − \ k =0 ker( D A A k ) = ker D A n , n − \ k =0 ker( D A ∗ A ∗ k ) = ker D A ∗ n , we get \ n ≥ ker D A n = H ⊖ span { A ∗ n D A H , n = 0 , , . . . } , \ n ≥ ker D A ∗ n = H ⊖ span { A n D A ∗ H , n = 0 , , . . . } . (2.1)It follows that(2.2) A is completely non-unitary ⇐⇒ (cid:18) T n ≥ ker D A n (cid:19) T (cid:18) T n ≥ ker D A ∗ n (cid:19) = { } ⇐⇒⇐⇒ span { A ∗ n D A , A m D A ∗ , n, m ≥ } = H . Note that ker D A ⊃ ker D A ⊃ · · · ⊃ ker D A n ⊃ · · · ,A ker D A n ⊂ ker D A n − , n = 2 , , . . . . YURY ARLINSKI˘I
From (2.1) we get that the subspaces T n ≥ ker D A n and T n ≥ ker D A ∗ n are invariant with respectto A and A ∗ , respectively, and A ↾ T n ≥ ker D A n and A ∗ ↾ T n ≥ ker D A ∗ n are unilateral shifts,moreover, these operators are the maximal unilateral shifts contained in A and A ∗ , respec-tively [29, Theorem 1.1, Corollary 1]. Thus, for a completely non-unitary contraction A wehave(2.3) T n ≥ ker D A n = { } ⇐⇒ A does not contain a unilateral shift , T n ≥ ker D A ∗ n = { } ⇐⇒ A ∗ does not contain a unilateral shift . By definition [29] the operator A contains a co-shift V if the operator A ∗ contains theunilateral shift V ∗ .The function (see [38, Chapter VI])(2.4) Φ A ( λ ) = (cid:0) − A + λD A ∗ ( I − λA ∗ ) − D A (cid:1) ↾ D A is known as the Sz.-Nagy – Foias characteristic function of a contraction A [38]. This functionbelongs to the Schur class S ( D A , D A ∗ ) and Θ A (0) is a pure contraction. The characteristicfunctions of A and A ∗ are connected by the relationΦ A ∗ ( λ ) = Φ ∗ A (¯ λ ) , λ ∈ D . Two operator-valued functions Θ ∈ S ( M , N ) and Θ ∈ S ( M , N ) coincide [38] if thereare two unitary operators V : N → N and W : M → M such that V Θ ( λ ) W = Θ ( λ ) , λ ∈ D . The result of Sz.-Nagy–Foias [38, Theorem VI.3.4] states that two completely non-unitarycontractions A and A are unitary equivalent if and only if their characteristic functionsΦ A and Φ A coincide.It is well known that a function Θ( λ ) from the Schur class S ( M , N ) has almost everywherenon-tangential strong limit values Θ( ξ ), ξ ∈ T , where T = { ξ ∈ C : | ξ | = 1 } stands for theunit circle; cf. [38]. A function Θ ∈ S ( M , N ) is called inner if Θ ∗ ( ξ )Θ( ξ ) = I M and co-inner if Θ( ξ )Θ ∗ ( ξ ) = I N almost everywhere on ξ ∈ T . A function Θ ∈ S ( M , N ) is called bi-inner ,if it is both inner and co-inner. A contraction T on a Hilbert space H belongs to the class C · ( C · ), if s − lim n →∞ A n = 0 ( s − lim n →∞ A ∗ n = 0) , respectively. By definition C := C · ∩ C · . A completely non-unitary contraction A belongsto the class C · , C · , or C if and only if its characteristic function Φ A ( λ ) is inner, co-inner,or bi-inner, respectively (cf. [38, Section VI.2]). Note that for a completely non-unitarycontraction A the equality ker D A = ker D A ∗ = { } is impossible because otherwise thesubspace ker D A reduces A and A ↾ ker D A is a unitary operator.We complete this section by a description of completely non-unitary contractions withconstant characteristic functions. Note that Φ A ( λ ) = 0 ∈ S ( { } , D A ∗ ) ⇐⇒ A is a unilateralshift, and Φ A ( λ ) = 0 ∈ S ( D A , { } ) ⇐⇒ A is a co-shift. Theorem 2.1.
Let H be a separable Hilbert space. A completely non-unitary contraction A with nonzero defect operators has a constant characteristic function if and only if H is theorthogonal sum H = H ⊕ H EALIZATIONS OF THE SCHUR ITERATES 7 and A takes the operator matrix form (2.5) A = (cid:20) S Γ0 S ∗ (cid:21) : H ⊕H → H ⊕H , where S and S are unilateral shifts in H and H , respectively, and Γ is a contraction suchthat (2.6) ran Γ ⊂ D S ∗ , ran Γ ∗ ⊂ D S ∗ , || Γ f || < || f || , f ∈ D S ∗ \ { } , || Γ ∗ h || < || h || , h ∈ D S ∗ \ { } . In particular, the characteristic function of A is identically equal zero if and only if A is theorthogonal sum of a shift and co-shift.Proof. Suppose that the contraction A takes the form (2.5) with unilateral shifts S and S ,and the contraction Γ with the properties (2.6). Then(2.7) D A = (cid:20) D S ∗ − Γ ∗ Γ (cid:21) : H ⊕H → H ⊕H , and(2.8) D A ∗ = (cid:20) D S ∗ − ΓΓ ∗
00 0 (cid:21) : H ⊕H → H ⊕H . Since D S ∗ = ker S ∗ , D S ∗ = ker S ∗ , and D S ∗ and D S ∗ are the orthogonal projections in H onto D S ∗ and D S ∗ , respectively, we get from (2.6) the relations(2.9) D A = D S ∗ , D A ∗ = D S ∗ . Taking into account that H is an invariant subspace for A ∗ , we have D A ∗ ( I H − λA ∗ ) − D A = 0 . Hence Φ A ( λ ) = Γ ↾ D S ∗ = const. Because S and S are unilateral shifts, we get H = X n ≥ ⊕ S n D S ∗ , H = X n ≥ ⊕ S n D S ∗ . Since H = H ⊕ H , the operator A is completely non-unitary. If Γ = 0 then A is theorthogonal sum of a shift and co-shift.Now suppose that the characteristic function of A is a constant. From (2.4) we get D A ∗ A ∗ n D A = 0 , D A A n D A ∗ = 0 , n = 0 , , , . . . . It follows span { D A ∗ n D A , n = 0 , , . . . } ⊂ ker D A ∗ ⇐⇒ T n ≥ ker D A n ⊃ D A ∗ , span { D A n D A ∗ , n = 0 , , . . . } ⊂ ker D A ⇐⇒ T n ≥ ker D A ∗ n ⊃ D A . YURY ARLINSKI˘I
Let H = \ n ≥ ker D A n , H = \ n ≥ ker D A ∗ n . Since A H ⊂ H and A H ⊥ D A ∗ , we get H ⊖ A H ⊃ D A ∗ and similarly H ⊖ A ∗ H ⊃ D A . Let h ∈ H and h ⊥ D A ∗ . Itfollows h ∈ ker D A ∗ \ \ n ≥ ker D A n ! . Then h = Ag , g ∈ ker D A . Hence g ∈ T n ≥ ker D A n = H , i.e., H ⊖ A H = D A ∗ . Similarly H ⊖ A ∗ H = D A .Since A is completely non-unitary contraction, the operators A ↾ H and A ∗ ↾ H are uni-lateral shifts. Therefore(2.10) H = ∞ X n =0 ⊕ A n D A ∗ , H = ∞ X n =0 ⊕ A ∗ n D A . Note that for all ϕ, ψ ∈ H ( A m D A ∗ ϕ, A ∗ k D A ψ ) = ( D A A m + k D A ∗ ϕ, ψ ) = 0 , m, k = 0 , , . . . . Hence H ⊥ H . Taking into account (2.10) and the relation H ⊖ H = span { A ∗ n D A , n = 0 , , . . . } , we get H ⊖ H = H . Because H is invariant with respect to A , the matrix form of A is ofthe form (2.5) with unilateral shifts S := A ↾ H , S := A ∗ ↾ H , and some operator Γ ∈ (cid:0) H , H . Since A is a contraction, we have || Γ f || ≤ || D S ∗ f || , f ∈ H , || Γ ∗ h || ≤ || D S ∗ h || , h ∈ H . From (2.7) and (2.8) we getran ( D S ∗ − Γ ∗ Γ) = D A , ran ( D S ∗ − ΓΓ ∗ ) = D A ∗ . It follows that (2.6) holds true and Φ A ( λ ) = Γ.If A is the orthogonal sum of a shift and co-shift then clearly the characteristic functionof A is identically zero. (cid:3) Contractions generated by a contraction
In this section we define and study the subspaces and the corresponding operators obtainedfrom a completely non-unitary contraction A in a separable Hilbert space H .Suppose ker D A = { } . Define the subspaces(3.1) H , := HH n, = ker D A n , H ,m := ker D A ∗ m , H n,m := ker D A n ∩ ker D A ∗ m , m, n ∈ N EALIZATIONS OF THE SCHUR ITERATES 9
Let P n,m be the orthogonal projection in H onto H n,m . Define the contractions(3.2) A n,m := P n,m A ↾ H n,m ∈ L ( H n,m )and(3.3) A n,m := A n,m P n +1 ,m ↾ H n,m ∈ L ( H n,m ) . In the next theorem we establish the main properties of A n,m and A n,m . Theorem 3.1. (1)
Hold the relations (3.4) (cid:26) ker D A kn,m = H n + k,m ker D A ∗ kn,m = H n,m + k , k = 1 , , . . . , (3.5) (cid:26) D A n,m = ran ( P n,m D A n +1 ) , D A ∗ n,m = ran ( P n,m D A ∗ m +1 ) , (3.6) (cid:26) A H n,m = H n − ,m +1 , n ≥ ,A ∗ H n,m = H n +1 ,m − , m ≥ , (3.7) (cid:26) ker D A kn,m = H n + k,m ker D A ∗ kn,m = H n,m + k k = 1 , , . . . , (3.8) (cid:26) D A n,m = D A n +1 ,m D A ∗ n,m = D A ∗ n +1 ,m , (3.9) ( A n,m ) k,l = A n + k,m + l . (2) The operators { A n,m } and {A n,m } are completely non-unitary contractions. (3) The operators A n, , A n − , , . . . , A n − k,k , . . . , A ,n are unitarily equivalent and (3.10) A n − ,m +1 Af = AA n,m f, f ∈ H n,m , n ≥ . (4) The operators A n, , A n − , , . . . , A n − k,k , . . . , A ,n are unitarily equivalent and (3.11) A n − ,m +1 Af = A A n,m f, f ∈ H n,m , n ≥ . (5) The following statements are equivalent (a) A n, ∈ C · ( A n, ∈ C · ) for some n , (b) A n +1 − k,k ∈ C · ( A n +1 − k,k ∈ C · ) for all k = 0 , , . . . , n + 1 . Proof.
It is sufficient to prove the first equality from (3.4). From (3.1) and (3.2) we have f ∈ H n,m , f ∈ ker D A kn,m ⇐⇒ (cid:26) || f || = || A n f || = || A ∗ m f |||| f || = || A kn,m f ||⇐⇒ Af, . . . , A k f ∈ H n,m ⇐⇒ f ∈ H n + k,m . This proves (3.4). Hence D A n,m = H n,m ⊖ H n +1 ,m = H n,m ⊖ (ker D A n +1 ∩ ker D A ∗ m ) == H n,m ∩ D A n +1 + D A ∗ m = ran ( P n,m D A n +1 ) , D A ∗ n,m = H n,m ⊖ H n,m +1 = H n,m ⊖ (ker D A n ∩ ker D A ∗ m +1 ) == H n,m ∩ D A n + D A ∗ m +1 = ran ( P n,m D A ∗ m +1 ) , i.e., relations (3.5) are valid. Furthermore if n ≥ f ∈ H n,m ⇐⇒ Af ∈ ker D A n − ,A ∗ Af = f,f ∈ ker D A ∗ m (for m ≥ ⇐⇒ Af ∈ ker D A n − ∩ ker D A ∗ m +1 = H n − ,m +1 . If n = 1 then f ∈ H ,m ⇐⇒ (cid:26) A ∗ Af = f,f ∈ ker D A ∗ m ⇐⇒ Af ∈ ker D A ∗ m +1 = H ,m +1 . Similarly A ∗ H n,m = H n +1 ,m − , m ≥ . Therefore relations (3.6) hold true.Let ϕ ∈ H , ψ ∈ H n − ,m +1 . Then A ∗ ψ ∈ H n,m and( AP n,m ϕ, ψ ) = ( P n,m ϕ, A ∗ ψ ) = ( ϕ, A ∗ ψ ) = ( Aϕ, ψ ) = ( P n − ,m +1 Aϕ, ψ ) . Hence(3.12) AP n,m = P n − ,m +1 A. Taking into account (3.6), we get AP n,m Ah = P n − ,m +1 AAh, h ∈ H n,m . This proves (3.10). Since A isometrically maps H n,m onto H n − ,m +1 for n ≥
1, the operators A n − ,m +1 and A n,m are unitarily equivalent, and therefore the operators A n, , A n − , , . . . , A n − k,k , . . . , A ,n are unitarily equivalent.Note that (3.4) and (3.6) yield the equalities(3.13) T k ≥ ker D A kn,m = ker D A ∗ m T T j ≥ ker D A j ! = A m T j ≥ ker D A j ! , T k ≥ ker D A ∗ kn,m = ker D A n T T j ≥ ker D A ∗ j ! = A ∗ n T j ≥ ker D A ∗ j ! . Since A is a completely non-unitary contraction, we get \ k ≥ ker D A kn,m ! \ \ k ≥ ker D A ∗ kn,m ! = { } . It follows that the contractions A n,m are completely non-unitary.Note that H n − ,m +1 ⊂ H n − ,m and H n +1 ,m ⊂ H n,m . Using (3.6) we get A n − ,m +1 P n,m +1 = P n − ,m +1 AP n,m +1 = AP n,m +1 ,A n,m P n +1 ,m = P n,m AP n +1 ,m = AP n +1 ,m . EALIZATIONS OF THE SCHUR ITERATES 11
In particular, it follows that the operators A n,m P n +1 ,m are partial isometries. From (3.12)we obtain AP n,m +1 A = A P n +1 ,m , i.e., A n − ,m +1 P n,m +1 Af = AA n,m P n +1 ,m f for all f ∈ H n,m . Because A is unitary operator from H n,m onto H n − ,m +1 , we get (3.11) and so the operators A n − ,m +1 and A n,m are unitarily equivalent.By induction it can be easily proved that for every k ∈ N holds the equality(3.14) A kn,m f = ( AP n +1 ,m ) k f = AA k − n +1 ,m P n +1 ,m f, f ∈ H n,m . Since A ↾ H n +1 ,m is isometric, relations (3.14) imply ||A kn,m f || = || A k − n +1 ,m P n +1 ,m f || , f ∈ H n,m , k ∈ N . It follows the equivalence of the statements (a) and (b) andker D A kn,m = ker D A k − n +1 ,m = H n + k,m . Similarly, since ( A n,m P n +1 ,m ) ∗ = A ∗ n,m P n,m +1 , we getker D A ∗ kn,m = ker D A ∗ k − n,m +1 = H n,m + k . Thus, relations (3.7) are valid.Now we get that the operators A n,m P n +1 ,m are completely non-unitary. From (3.4) itfollows thatker D A kn,m ∩ ker D A ∗ ln,m = H n + k,m ∩ H n,m + l =ker D A n + k ∩ ker D A ∗ m ∩ ker D A n ∩ ker D A ∗ m + l = ker D A n + k ∩ ker D A ∗ m + l = H n + k,m + l . Hence ( A n,m ) k,l = P n + k,m + l P n,m A ↾ H n + k,m + l = A n + k,m + l . (cid:3) The relation (3.9) yields the following picture for the creation of the operators A n,m : A | | yyyyyyyyy " " EEEEEEEEE A , | | zzzzzzzz " " DDDDDDDD A , | | zzzzzzzz " " DDDDDDDD A , | | zzzzzzzz " " DDDDDDDD A , | | zzzzzzzz " " DDDDDDDD A , | | zzzzzzzz " " DDDDDDDD A , A , A , A , · · · · · · · · · · · · · · · · · · · · · The process terminates on the N -th step if and only ifker D A N = { } ⇐⇒ ker D A N − ∩ ker D A ∗ = { } ⇐⇒ . . . ker D A N − k ∩ ker D A ∗ k = { } ⇐⇒ . . . ker D A ∗ N = { } . Note that from (2.3), (3.7), and (3.13) we get
Proposition 3.2.
Let A be a completely non-unitary contraction. If A does not contain aunilateral shift (co-shift) then the same is true for the operators A n,m and A n,m for all n and m . Conversely, if for some n and m the operator A n,m or A n,m does not contain a unilateralshift (co-shift) then the same is valid for A . Let δ A = dim D A , δ A ∗ = dim D A ∗ be the defect numbers of a completely non-unitarycontraction A . For n = 1 , . . . denote by δ n and δ ∗ n the defect numbers of unitarily equivalentoperators { A n − m,m } nm =0 . From the relations (3.5) it follows that δ n = dim D A ,n = dim (ran ( P ,n D A )) = dim ( D A ⊖ ( D A ∩ D A ∗ n )) ,δ ∗ n = dim D A ∗ n, = dim (ran ( P n, D A ∗ )) = dim ( D A ∗ ⊖ ( D A ∗ ∩ D A n )) . Thus δ A ≥ δ ≥ · · · ≥ δ n ≥ · · · ,δ A ∗ ≥ δ ∗ ≥ · · · ≥ δ ∗ n ≥ · · · . Observe also that δ = dim ( D A ⊖ ( D A ∩ D A ∗ )) , δ ∗ = dim ( D A ∗ ⊖ ( D A ∩ D A ∗ )) , and by induction δ n = dim (cid:16) D A n − , ⊖ ( D A n − , ∩ D A ∗ n − , ) (cid:17) , δ ∗ n = dim (cid:16) D A ∗ n − , ⊖ ( D A n − , ∩ D A ∗ n − , ) (cid:17) . Passive discrete-time linear systems and their transfer functions
Basic definitions.
Let M , N , and H be separable Hilbert spaces. A linear system τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) with bounded linear operators A , B , C , D of the form(4.1) (cid:26) h k +1 = Ah k + Bξ k ,σ k = Ch k + Dξ k , k ≥ , where { h k } ⊂ H , { ξ k } ⊂ M , { σ k } ⊂ N , is called a discrete-time system . The Hilbert spaces M and N are called the input and the output spaces, respectively, and the Hilbert space H iscalled 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. Ifthe linear operator T τ defined by the block form(4.2) T τ = (cid:20) D CB A (cid:21) : M ⊕ H → N ⊕ H is contractive, then the corresponding discrete-time system is said to be passive . If theblock operator matrix T τ is isometric (co-isometric, unitary), then the system is said to be isometric ( co-isometric , conservative ). Isometric and co-isometric systems were studied byL. de Branges and J. Rovnyak (see [21], [22]) and by T. Ando (see [6]), conservative systemshave been investigated by B. Sz.-Nagy and C. Foia¸s (see [38]) and M.S. Brodski˘ı (see [23]).Passive systems have been studied by D.Z. Arov et al (see [11], [12], [13], [14], [15]).The subspaces(4.3) H c := span { A n B M : n = 0 , , . . . } and H o = span { A ∗ n C ∗ N : n = 0 , , . . . } EALIZATIONS OF THE SCHUR ITERATES 13 are said to be the controllable and observable subspaces of the system τ , respectively. Thesystem τ is said to be controllable ( observable ) if H c = H ( H o = H ), and it is called minimal if τ is both controllable and observable. The system τ is said to be simple if H = clos { H c + H o } (the closure of the span). It follows from (4.3) that(4.4) ( H c ) ⊥ = ∞ \ n =0 ker( B ∗ A ∗ n ) , ( H o ) ⊥ = ∞ \ n =0 ker( CA n ) , and therefore there are the following alternative characterizations:(a) τ is controllable ⇐⇒ ∞ T n =0 ker( B ∗ A ∗ n ) = { } ;(b) τ is observable ⇐⇒ ∞ T n =0 ker( CA n ) = { } ;(c) τ is simple ⇐⇒ (cid:18) ∞ T n =0 ker( B ∗ A ∗ n ) (cid:19) ∩ (cid:18) ∞ T n =0 ker( CA n ) (cid:19) = { } . The transfer function (4.5) Θ τ ( λ ) := D + λC ( I H − λA ) − B, λ ∈ D , of the passive system τ belongs to the Schur class S ( M , N ) [11]. Conservative systems arealso called the unitary colligations and their transfer functions are called the characteristicfunctions [23].The examples of conservative systems are given byΣ = (cid:26)(cid:20) − A D A ∗ D A A ∗ (cid:21) ; D A , D A ∗ , H (cid:27) , Σ ∗ = (cid:26)(cid:20) − A ∗ D A D A ∗ A (cid:21) ; D A ∗ , D A , H (cid:27) . The transfer functions of these systemsΦ Σ ( λ ) = (cid:0) − A + λD A ∗ ( I H − λA ∗ ) − D A (cid:1) ↾ D A , λ ∈ D and Φ Σ ∗ ( λ ) = (cid:0) − A ∗ + λD A ( I H − λA ) − D A ∗ (cid:1) ↾ D A ∗ , λ ∈ D are exactly characteristic functions 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 ascontrollable isometric (observable co-isometric, simple conservative, minimal passive); cf.[22], [38], [6] [11], [13], [5]. Moreover, two controllable isometric (observable co-isometric,simple conservative) systems with the same transfer function are unitarily similar: twodiscrete-time systems τ = (cid:26)(cid:20) D C B A (cid:21) ; M , N , H (cid:27) and τ = (cid:26)(cid:20) D C B A (cid:21) ; M , N , H (cid:27) are said to be unitarily similar if there exists a unitary operator U from H onto H suchthat A = U − A U, B = U − B , C = C U ;cf. [21], [22], [6], [23], [5]. However, a result of D.Z. Arov [11] states that two minimal passivesystems τ and τ with the same transfer function Θ( λ ) are only weakly similar , i.e., there is a closed densely defined operator Z : H → H such that Z is invertible, Z − is denselydefined, and ZA f = A Zf, C f = C Zf, f ∈ dom Z, and ZB = B . Defect functions of the Schur class functions.
The following result [38, Proposi-tion V.4.2] is needed in the sequel.
Theorem 4.1.
Let M be a separable Hilbert space and let N ( ξ ) , ξ ∈ T , be an L ( M ) -valuedmeasurable function such that ≤ N ( ξ ) ≤ I M . Then there exist a Hilbert space K and anouter function ϕ ( λ ) ∈ S ( M , K ) satisfying the following conditions: (i) ϕ ∗ ( ξ ) ϕ ( ξ ) ≤ N ( ξ ) a.e. on T ; (ii) if e K is a Hilbert space and e ϕ ( λ ) ∈ S ( M , e K ) is such that e ϕ ∗ ( ξ ) e ϕ ( ξ ) ≤ N ( ξ ) a.e. on T , then e ϕ ∗ ( ξ ) e ϕ ( ξ ) ≤ ϕ ∗ ( ξ ) ϕ ( ξ ) a.e. on T .Moreover, the function ϕ ( λ ) is uniquely defined up to a left constant unitary factor. Assume that Θ( λ ) ∈ S ( M , N ) and denote by ϕ Θ ( ξ ) and ψ Θ ( ξ ), ξ ∈ T the outer functionswhich are solutions of the factorization problem described in Theorem 4.1 for N ( ξ ) = I M − Θ ∗ ( ξ )Θ( ξ ) and N ( ¯ ξ ) = I N − Θ( ¯ ξ )Θ ∗ ( ¯ ξ ), respectively. Clearly, if Θ( λ ) is inner or co-inner, then ϕ Θ = 0 or ψ Θ = 0, respectively. The functions ϕ Θ ( λ ) and ψ Θ ( λ ) are called theright and left defect functions (or the spectral factors ), respectively, associated with Θ( λ );cf. [17], [18], [19], [20], [29]. The following result has been established in [29, Theorem 1.1,Corollary 1] (see also [19, Theorem 3], [20, Theorem 1.5]). Theorem 4.2.
Let Θ( λ ) ∈ S ( M , N ) and let τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) be a simple conservative system with transfer function Θ . Then (1) the functions ϕ Θ ( λ ) and ψ Θ ( λ ) take the form ϕ Θ ( λ ) = P Ω ( I H − λA ) − B,ψ Θ ( λ ) = C ( I H − λA ) − ↾ Ω ∗ , where Ω = ( H o ) ⊥ ⊖ A ( H o ) ⊥ , Ω ∗ = ( H c ) ⊥ ⊖ A ∗ ( H c ) ⊥ and P Ω is the orthogonal projector from H onto Ω ; (2) ϕ Θ ( λ ) = 0 ( ψ Θ ( λ ) = 0 ) if and only if the system τ is observable (controllable). The defect functions play an essential role in the problems of the system theory, in partic-ular, in the problem of similarity and unitary similarity of the minimal passive systems withequal transfer functions [14], [15] and in the problem of optimal and ( ∗ ) optimal realizationsof the Schur function [12], [13].4.3. Parametrization of contractive block-operator matrices.
Let H , K , M and N be Hilbert spaces. The following theorem goes back to [16], [26], [36]; other proofs of thetheorem can be found in [31], [32], [7], [9]. EALIZATIONS OF THE SCHUR ITERATES 15
Theorem 4.3.
Let A ∈ L ( H , K ) , B ∈ L ( M , K ) , C ∈ L ( H , N ) , and D ∈ L ( M , N ) . Theoperator matrix T = (cid:20) D CB A (cid:21) : M ⊕ H → N ⊕ K is a contraction if and only if T is of the form (4.6) T = (cid:20) − KA ∗ M + D K ∗ XD M KD A D A ∗ M A (cid:21) , where A ∈ L ( H , K ) , M ∈ L ( M , D A ∗ ) , K ∈ L ( D A , N ) , and X ∈ L ( D M , D K ∗ ) are contrac-tions, all uniquely determined by T . Furthermore, the following equality holds for all h ∈ M , f ∈ H : (4.7) (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) hf (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) − KA ∗ M + D K ∗ XD M KD A D A ∗ M A (cid:21) (cid:20) hf (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = k D K ( D A f − A ∗ M h ) − K ∗ XD M h k + k D X D M h k . Corollary 4.4.
Let T = (cid:20) − KA ∗ M + D K ∗ XD M KD A D A ∗ M A (cid:21) : M ⊕ H → N ⊕ K be a contraction. Then (1) T is isometric if and only if D K D A = 0 , D X D M = 0 , (2) T is co-isometric if and only if D M ∗ D A ∗ = 0 , D X ∗ D K ∗ = 0 . Note that the relation D Y D Z = 0 for contractions Y and Z means that either Z is anisometry and Y = 0 or D Z = { } and Y is an isometry. From (4.7) we get the followingstatement If T given by (4.6) is unitary then D K ∗ = 0 ⇐⇒ D M = 0 . Let τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) be a conservative system. Then from Corollary 4.4 we get( H c ) ⊥ = \ n ≥ ker( D A ∗ A ∗ n ) = \ n ≥ ker( D A ∗ n ) , ( H o ) ⊥ = \ n ≥ ker( D A A n ) = \ n ≥ ker( D A n ) , (4.8) τ is controllable ⇐⇒ T n ≥ ker( D A ∗ n ) = { } ⇐⇒ the operator A ∗ does not contain a shift ,τ is observable ⇐⇒ T n ≥ ker( D A n ) = { } ⇐⇒ the operator A does not contain a shift . It follows that a conservative system is simple if and only if the state space operator iscompletely non-unitary [23].
In [9] we used Theorem 4.3 for connections between the passive system τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) , its transfer function Θ τ ( λ ), and the characteristic function of A . In particular, an immediateconsequence of (4.6) is the following relation(4.9) Θ τ ( λ ) = K Φ A ∗ ( λ ) M + D K ∗ XD M , λ ∈ D , where Φ A ∗ ( λ ) is the characteristic function of A ∗ .Recall that if Θ( λ ) ∈ S ( H , H ) then there is a uniquely determined decomposition [38,Proposition V.2.1] Θ( λ ) = (cid:20) Θ p ( λ ) 00 Θ u (cid:21) : D Θ(0) ⊕ ker D Θ(0) → D Θ ∗ (0) ⊕ ker D Θ ∗ (0) , where Θ p ( λ ) ∈ S ( D Θ(0) , D Θ ∗ (0) ), Θ p (0) is a pure contraction and Θ u is a unitary constant.The function Θ p ( λ ) is called the pure part of Θ( λ ) (see [17]). If Θ(0) is isometric (co-isometric) then the pure part is of the form Θ p ( λ ) = 0 ∈ S ( { } , D Θ ∗ (0) ) (Θ p ( λ ) = 0 ∈ S ( D Θ(0) , { } )).From (4.6) and (4.9) we get the following statement. Proposition 4.5.
Let τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) be a a simple conservative system and let Θ( λ ) be its transfer function. Then (4.10) dim D A = dim D Θ ∗ (0) = dim( N ⊖ ker C ∗ ) , dim D A ∗ = dim D Θ(0) = dim( M ⊖ ker B ) , and the pure part of Θ coincides with the Sz.-Nagy–Foias characteristic function of A ∗ .In addition1) if Θ(0) is isometric then B = 0 , A is a co-shift of multiplicity dim D Θ ∗ (0) , and thesystem τ is observable;2) if Θ(0) is co-isometric then C = 0 , A is a unilateral shift of multiplicity dim D Θ(0) , andthe system τ is controllable.Proof. According to Theorem 4.3 the operator T = (cid:20) D CB A (cid:21) : M ⊕ H → N ⊕ H takes the form (4.6). Since T is unitary, from (4.12) we get that the operators K ∈ L ( D A , N )and M ∗ ∈ L ( D A ∗ , M ) are isometries and the operator X ∈ L ( D M , D K ∗ ) is unitary. From(4.9) it follows that the pure part of Θ is given byΘ( λ ) ↾ ran M ∗ = K Φ A ∗ ( λ ) M ↾ ran M ∗ : ran M ∗ → ran K. EALIZATIONS OF THE SCHUR ITERATES 17
Thus, the pure part of Θ coincides with Φ A ∗ . Since ran M ∗ = D A ∗ , ran K ∗ = D A , D = Θ(0) = K Φ A ∗ (0) M ∗ = − KA ∗ M ∗ , D ∗ = Θ ∗ (0) = − M AK ∗ , ran K = N ⊖ ker K ∗ = N ⊖ ker C ∗ , ran M ∗ = M ⊖ ker M = M ⊖ ker B, we get (4.10).Suppose D = Θ(0) is an isometry. Then the pure part of Θ is 0 ∈ S ( { } , D D ∗ ). It followsthat M = B = 0 and D A ∗ = { } . Hence, A is co-isometric and since A is a completelynon-unitary contraction, it is a co-shift of multiplicity dim D A = dim D Θ ∗ (0) , and the system τ is observable. Similarly the statement 2) holds. (cid:3) In this paper we will use a parametrization of a contractive block- operator matrix basedon a fixed upper left block D ∈ L ( M , N ). With this aim we reformulate Theorem 4.3 andCorollary 4.4. Theorem 4.6.
The operator matrix T = (cid:20) D CB A (cid:21) : M ⊕ H → N ⊕ K is a contraction if and only if D ∈ L ( M , N ) is a contraction and the entries A , B , and C take the form B = F D D , C = D D ∗ G,A = − F D ∗ G + D F ∗ LD G , (4.11) where the operators F ∈ L ( D D , K ) , G ∈ L ( H , D D ∗ ) and L ∈ L ( D G , D F ∗ ) are contractions.Moreover, operators F, G, and L are uniquely determined. Furthermore,the following equalityholds (4.12) (cid:13)(cid:13)(cid:13)(cid:13) D T (cid:20) hf (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = || D F ( D D h − D ∗ Gf ) − F ∗ LD G f || + || D L D G f || ,h ∈ M , f ∈ H and (4.13) (cid:13)(cid:13)(cid:13)(cid:13) D T ∗ (cid:20) ϕg (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = || D G ∗ ( D D ∗ ϕ − DF ∗ g ) − GL ∗ D F ∗ g || + || D L ∗ D F ∗ g || ,ϕ ∈ N , g ∈ K . (1) the operator T is isometric if and only if D F D D = 0 , D L D G = 0 , (2) the operator T is co-isometric if and only if D G ∗ D D ∗ = 0 , D L ∗ D F ∗ = 0 , (3) if T is unitary then D F ∗ = 0 ⇐⇒ D G = 0 . Let us give connections between the parametrization of a unitary block-operator matrix T given by (4.6) and (4.11). Proposition 4.7.
Let T = (cid:20) − KA ∗ M + D K ∗ XD M KD A D A ∗ M A (cid:21) == (cid:20) D D D ∗ GF D D − F D ∗ G + D F ∗ LD G (cid:21) : M ⊕ H → N ⊕ H be a unitary operator matrix. Then D D = ran M ∗ , D D ∗ = ran K,F ∗ = M ∗ P D A ∗ , G = KP D A ,GF f = KP D A M f, f ∈ D D ,L = A ↾ ker D A . Proof.
Since D = − KA ∗ M + D K ∗ XD M , we have || D D f || = || D A ∗ M f || + || ( D K A ∗ M − K ∗ XD M ) f || + || D X D M f || , f ∈ M , || D D ∗ g || = || D A K ∗ g || + || ( D M ∗ AK ∗ − M X ∗ D K ∗ ) g || + || D X ∗ D K ∗ g || , g ∈ N . By Corollary 4.4 the operators K and M ∗ are isometries and X ∈ L ( D M , D K ∗ ) is unitaryoperator. It follows that || D D f || = || D A ∗ M f || , f ∈ M , || D D ∗ g || = || D A K ∗ g || , g ∈ N . Hence, D D = M ∗ D A ∗ M, D D ∗ = KD A K ∗ . Since K and M ∗ are isometries, we obtain D D = M ∗ D A ∗ M, D D ∗ = KD A K ∗ . It follows that D D = ran M ∗ , D D ∗ = ran K, D A ∗ M = F M ∗ D A ∗ M, and D A K ∗ = G ∗ KD A K ∗ . Therefore,
F M ∗ = I D A ∗ , G ∗ K = I D A . It follows F = M ↾ D D , G ∗ = K ∗ ↾ D D ∗ . Hence, F ∗ = M ∗ P D A ∗ and G = KP D A . In addition D F ∗ = I H − M M ∗ P D A ∗ = P ker D A ∗ , D G = I H − K ∗ KP D A = P ker D A , − F D ∗ G = − F ( − M ∗ AK ∗ + D M X ∗ D K ∗ ) KP D A = AP D A ,A = − F D ∗ G + D F ∗ LD G = AP D A + P ker D A ∗ LP ker D A . On the other hand A = AP D A + AP ker D A . Hence L = A ↾ ker D A . (cid:3) Let D : M → N be a contraction with nonzero defect operators and let Q = (cid:20) GF S (cid:21) : D D ⊕ H → D D ∗ ⊕ K EALIZATIONS OF THE SCHUR ITERATES 19 be a bounded operator. Define the transformation (see[8])(4.14) M D ( Q ) = (cid:20) D − F D ∗ G (cid:21) + (cid:20) D D ∗ I K (cid:21) (cid:20) GF S (cid:21) (cid:20) D D I H (cid:21) . Clearly, the operator T = M D ( Q ) has the following matrix form T = (cid:20) D D D ∗ GF D D S − F D ∗ G (cid:21) : M ⊕ H → N ⊕ K . Proposition 4.8. [8] . Let H , M , N be separable Hilbert spaces and let D : M → N be acontraction with nonzero defect operators. Let Q = (cid:20) GF S (cid:21) : D D ⊕ H → D D ∗ ⊕ H be a boundedoperator. Then (1) T = M D ( Q ) = (cid:20) D CB A (cid:21) : D D ⊕ H → D D ∗ ⊕ H is a contraction if and only if Q is a contraction. T is isometric (co-isometric) ifand only if Q is isometric (co-isometric); (2) holds the relations (4.15) ∞ \ n =0 ker ( B ∗ A ∗ n ) = ∞ \ n =0 ker ( F ∗ S ∗ n ) , ∞ \ n =0 ker ( CA n ) = ∞ \ n =0 ker ( GS n ) . The M¨obius representations
Let T : H → H be a contraction. In [37] and [34] were studied the fractional-lineartransformations of the form(5.1) Z → Q = T + D T ∗ Z ( I D T + T ∗ Z ) − D T = T + D T ∗ ( I D T ∗ + ZT ∗ ) − ZD T defined on the set V T ∗ of all contractions Z ∈ L ( D T , D T ∗ ) such that − ∈ ρ ( T ∗ Z ) . Thefollowing result holds.
Theorem 5.1. [34]
Let the T ∈ L ( H , H ) be a contraction and let Z ∈ V T ∗ . Then Q = T + D T ∗ Z ( I D T + T ∗ Z ) − D T is a contraction, (5.2) || D Q f || = || D Z ( I D T + T ∗ Z ) − D T f || , f ∈ H , ran D Q ⊆ ran D T , and ran D Q = ran D T if and only if || Z || < . Moreover, if Q ∈ L ( H , H ) is a contraction and Q = T + D T ∗ XD T , where X ∈ L ( D T , D T ∗ ) then X ∈ V T ∗ , Z = X ( I D T − T ∗ X ) − ∈ V T ∗ , and the operator Q takes the form Q = T + D T ∗ Z ( I D T + T ∗ Z ) − D T . Observe that from (5.1) one can derive the equalities I H − QT ∗ = D T ∗ ( I D T ∗ + ZT ∗ ) − D T ∗ ,Z ↾ ran D T = D T ∗ ( I H − QT ∗ ) − ( Q − T ) D − T . The transformation (5.1) is called in [34] the unitary linear-fractional transformation. It iseasy to see that if || T || < L ( H , H ) belongs to theset V T ∗ and, moreover T + D T ∗ Z ( I D T + T ∗ Z ) − D T = D − T ∗ ( Z + T )( I D T + T ∗ Z ) − D T == D T ∗ ( I D T ∗ + ZT ∗ ) − ( Z + T ) D − T for all Z ∈ L ( H , H ) , || Z || ≤ . Thus, the transformation (5.1) is an operator analog of awell known M¨obius transformation of the complex plane z → z + t tz , | t | ≤ . The next theorem is a version of a more general result established by Yu.L. Shmul’yan in[35].
Theorem 5.2. [35]
Let M and N be Hilbert spaces and let the function Θ( λ ) be from theSchur class S ( M , N ) . Then (1) the linear manifolds ran D Θ( λ ) and ran D Θ ∗ ( λ ) do not depend on λ ∈ D , (2) for arbitrary λ , λ , λ in D the function Θ( λ ) admits the representation Θ( λ ) = Θ( λ ) + D Θ ∗ ( λ ) Ψ( λ ) D Θ( λ ) , where Ψ( λ ) is a holomorphic in D and L (cid:0) D Θ( λ ) , D Θ ∗ ( λ ) (cid:1) -valued function. Now using Theorems 5.1 and 5.2 we get Theorem 1.1. Recall that the representation (1.2)of a function Θ( λ ) ∈ S ( M , N ) is called the M¨obius representation of Θ and the function Z ( λ ) ∈ S ( D Θ(0) , D Θ ∗ (0) ) is called the M¨obius parameter of Θ.The next result established in [8] provides connections between the realizations of Θ( λ )and Z ( λ ) as transfer functions of passive systems. Theorem 5.3. [8] . (1) Let τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) be a passive system and let T = (cid:20) D CB A (cid:21) = (cid:20) D D D ∗ GF D D − F D ∗ G + D F ∗ LD G (cid:21) : M ⊕ H → N ⊕ H . Let Θ( λ ) be the transfer function of τ . Then (a) the M¨obius parameter Z ( λ ) of the function Θ( λ ) is the transfer function of thepassive system ν = (cid:26)(cid:20) GF D F ∗ LD G (cid:21) ; D D , D D ∗ , H (cid:27) ;(b) the system τ isometric (co-isometric) ⇒ the system ν isometric (co-isometric); EALIZATIONS OF THE SCHUR ITERATES 21 (c) the equalities H cν = H cτ , H oν = H oτ hold and hence the system τ is controllable(observable) ⇒ the system ν is controllable (observable), the system τ is simple(minimal) ⇒ the system ν is simple (minimal). (2) Let Θ( λ ) ∈ S ( M , N ) and let Z ( λ ) be the M¨obius parameter of Θ( λ ) . Suppose thatthe transfer function of the linear system ν ′ = (cid:26)(cid:20) GF S (cid:21) ; D Θ(0) , D Θ ∗ (0) , H (cid:27) coincides with Z ( λ ) in a neighborhood of the origin. Then the transfer function ofthe linear system τ ′ = (cid:26)(cid:20) Θ(0) D Θ ∗ (0) GF D
Θ(0) − F Θ ∗ (0) G + S (cid:21) ; M , N , H (cid:27) coincides with Θ( λ ) in a neighborhood of the origin. Moreover (a) the equalities H cτ ′ = H cν ′ , H oτ ′ = H oν ′ hold, and hence the system ν ′ is controllable(observable) ⇒ the system τ ′ is controllable (observable), the system ν ′ is simple ⇒ the system τ ′ is simple (minimal), (b) the system ν ′ is passive ⇒ the system τ ′ is passive (minimal), (c) the system ν ′ isometric (co-isometric) ⇒ the system τ ′ isometric (co-isometric). Corollary 5.4.
1) The equivalences ϕ Θ ( λ ) = 0 ⇐⇒ ϕ Z ( λ ) = 0 ,ψ Θ ( λ ) = 0 ⇐⇒ ψ Z ( λ ) = 0 hold.2) Let || Θ(0) ↾ D Θ(0) || < . Suppose ϕ ( λ ) ∈ S ( M , L ) ( ψ ( λ ) ∈ S ( K , N ) ) and ϕ ∗ ( ξ ) ϕ ( ξ ) = D ξ ) for almost all ξ ∈ T (cid:16) ψ ( ξ ) ψ ∗ ( ξ ) = D ∗ ( ξ ) for almost all ξ ∈ T (cid:17) . Then e ϕ ( λ ) := ϕ ( λ ) D − ( I D Θ(0) + Θ ∗ (0) Z ( λ )) ∈ S ( D Θ(0) , L ) (cid:16) e ψ ( λ ) := ( I D Θ ∗ (0) + Z ( λ )Θ ∗ (0)) D − ∗ (0) P D Θ ∗ (0) ψ ( λ ) ∈ S ( K , D Θ ∗ (0) ) (cid:17) and e ϕ ∗ ( ξ ) e ϕ ( ξ ) = D Z ( ξ ) for almost all ξ ∈ T (cid:16) e ψ ( ξ ) e ψ ∗ ( ξ ) = D Z ∗ ( ξ ) for almost all ξ ∈ T (cid:17) . In particular, Θ( λ ) is inner (co-inner) ⇐⇒ Z ( λ ) is inner (co-inner) . Proof.
1) Let ϕ Θ ( λ ) = 0 ( ψ Θ ( λ ) = 0) and let τ = (cid:26)(cid:20) D CB A (cid:21) ; M , N , H (cid:27) be a simple con-servative system with transfer function Θ( λ ). By Theorem 4.2 the system τ is observable(controllable). As it is proved above the corresponding system ν with transfer function Z ( λ ) is conservative and observable (controllable). Theorem 4.2 yields that ϕ Z ( λ ) = 0( ψ Z ( λ ) = 0).Conversely. Let ϕ Z ( λ ) = 0 ( ψ Z ( λ ) = 0) and let ν ′ be a simple conservative system withtransfer function Z ( λ ). Again by Theorem 4.2 the system ν ′ is observable (controllable). As it is already proved the corresponding system τ ′ with transfer function Θ( λ ) is conservativeand observable (controllable) as well. Now Theorem 4.2 yields that ϕ Θ ( λ ) = 0 ( ψ Θ ( λ ) = 0).2) Let || Θ(0) ↾ D Θ(0) || <
1. SinceΘ ∗ (0) ↾ D Θ ∗ (0) = (cid:0) Θ(0) ↾ D Θ(0) (cid:1) ∗ , we get || Θ ∗ (0) ↾ D Θ ∗ (0) || <
1. It follows that the operators D Θ(0) ↾ D Θ(0) and D Θ ∗ (0) ↾ D Θ ∗ (0) have bounded inverses. From (5.2) we obtain the relation || D Θ( λ ) D − ( I D Θ(0) + Θ ∗ (0) Z ( λ )) f || = || D Z ( λ ) f || , λ ∈ D , f ∈ D Θ(0) . The non-tangential limits Θ( ξ ) and Z ( ξ ) exist for almost all ξ ∈ T . It follows the relation || D Θ( ξ ) D − ( I D Θ(0) + Θ ∗ (0) Z ( ξ )) f || = || D Z ( ξ ) f || , f ∈ D Θ(0) . for almost all ξ ∈ T . This completes the proof. (cid:3)
Theorem 5.5.
Let A be a completely non-unitary contraction in the Hilbert space H and let Z ( λ ) be the M¨obius parameter of the Sz.Nagy–Foias characteristic function of A . Then Z ( λ ) is the characteristic function of the operator A , = AP ker D A (see (3.2) and (3.3) ). Moreover,the following statements are equivalent (i) the unitary equivalent operators A , and A , are unilateral shifts (co-shifts), (ii) D A ⊂ D A ∗ ( D A ∗ ⊂ D A ), (iii) the M¨obius parameter takes the form Z ( λ ) = λI D A ( Z ∗ (¯ λ ) = λI D A ∗ ) . Proof.
The system Σ = (cid:26)(cid:20) − A D A ∗ D A A ∗ (cid:21) ; D A , D A ∗ , H (cid:27) is conservative and simple and its transfer functionΦ( λ ) = (cid:0) − A + λD A ∗ ( I H − λA ∗ ) − D A (cid:1) ↾ D A is the characteristic function of A . Let F and G ∗ be the embedding of the subspaces D A and D A ∗ into H , respectively. It follows that D F ∗ = P ker D A , D G = P ker D A ∗ . Let L = A ∗ ↾ ker D A ∗ . Then A ∗ = A ∗ P D A ∗ + A ∗ P ker D A ∗ = − F ( − A ∗ ) G + D F ∗ LD G Let Φ( λ ) = Φ(0) + D Φ ∗ (0) Z ( λ )( I + Φ ∗ (0) Z ( λ )) − D Φ(0) , λ ∈ D be the M¨obius representation of the function Φ( λ ). By Theorem 5.3 the system ν = (cid:26)(cid:20) P D A ∗ I D A A ∗ P ker D A ∗ (cid:21) ; D A , D A ∗ , H (cid:27) is conservative and simple and its transfer function is the function Z ( λ ), i.e., Z ( λ ) = λP D A ∗ ( I H − λA ∗ P ker D A ∗ ) − ↾ D A , | λ | < . This function is exactly the Sz.-Nagy–Foias characteristic function of the partial isometry A , = AP ker D A . EALIZATIONS OF THE SCHUR ITERATES 23
Suppose A , = P ker D A A ↾ ker D A is a unilateral shift. Since A ker D A = ker D A ∗ , we haveker D A ∗ ⊂ ker D A . Equivalently D A ⊂ D A ∗ . Hence, P ker D A ∗ ↾ D A = 0 and ( A ∗ P ker D A ∗ ) n ↾ D A = 0 for all n ∈ N . Therefore, Z ( λ ) = λP D A ∗ ↾ D A = λI D A . Conversely, suppose Z ( λ ) = λI D A . Then D A ⊂ D A ∗ ⇒ ker D A ⊃ ker D A ∗ . It follows A ker D A ⊂ ker D A ⇒ A , is isometry . Since the operator A , is completely non-unitary, it is a unilateral shift. (cid:3) Corollary 5.6.
Let A be a completely non-unitary contraction in a separable Hilbert space H and let || A ↾ D A || < ⇐⇒ ran D A = ran D A ) . Then the following statements are equivalent (i) A ∈ C · (respect., A ∈ C · ) , (ii) A , ∈ C · (respect., A , ∈ C · ) .Proof. By (2.4) we have Φ A (0) = − A ↾ D A . Then in accordance with [38], Corollary 5.4, andTheorem 5.5 we get the equivalences A ∈ C · ( C · ) ⇐⇒ Φ A ( λ ) is inner (co-inner) ⇐⇒ Z ( λ ) is inner (co-inner) ⇐⇒ A , ∈ C · ( C · ) . (cid:3) Realizations of the Schur iterates
Realizations of the first Schur iterate.Proposition 6.1.
Let H , L , K be Hilbert spaces and let F ∈ L ( L , H ) , G ∈ L ( H , K ) and L ∈ L ( D G , D F ∗ ) be contractions. Let Z ν ( λ ) be the transfer function of the system (6.1) ν = (cid:26)(cid:20) GF D F ∗ LD G (cid:21) ; L , K , H (cid:27) Then the function Γ( λ ) = λ − Z ν ( λ ) is the transfer function of the passive systems η = (cid:26)(cid:20) GF GD F ∗ LD G F LD G D F ∗ (cid:21) ; L , K , H (cid:27) , η = (cid:26)(cid:20) GF GD F ∗ e LD G F D G D F ∗ e L (cid:21) ; L , K , H (cid:27) , where e L = LP D G .Suppose that the subspaces H ζ = D F ∗ and H ζ = D G are nontrivial. Then the transferfunctions of the passive systems (6.2) ζ = (cid:26)(cid:20) GF GD F ∗ LD G F LD G D F ∗ (cid:21) ; L , K , H ζ (cid:27) , ζ = (cid:26)(cid:20) GF GD F ∗ e LD G F D G D F ∗ e L (cid:21) ; L , K , H ζ (cid:27) are equal to Γ( λ ) . Moreover, for the orthogonal complements to the controllable and observ-able subspaces of the systems ν , ζ , and ζ hold the following relations (6.3) ( H cν ) ⊥ = (cid:0) H cζ (cid:1) ⊥ ∩ ker F ∗ , ( H oν ) ⊥ = (cid:0) H oζ (cid:1) ⊥ ∩ ker G,D G (cid:0) H cζ (cid:1) ⊥ ⊂ ( H cν ) ⊥ , D F ∗ (cid:0) H oζ (cid:1) ⊥ ⊂ ( H oν ) ⊥ . If the operators G ∗ and F are isometries, then (6.4) (cid:0) H oζ (cid:1) ⊥ = ( H oν ) ⊥ ∩ ker F ∗ , (cid:0) H cζ (cid:1) ⊥ = ( H cν ) ⊥ ∩ ker G. Proof.
We have Z ν ( λ ) = λG ( I H − λD F ∗ LD G ) − F. Hence Γ( λ ) = Z ν ( λ ) λ = G ( I H − λD F ∗ LD G ) − F and Γ(0) = GF . It follows thatΓ( λ ) − Γ(0) = G ( I H − λD F ∗ LD G ) − F − GF = λGD F ∗ LD G ( I H − λD F ∗ LD G ) − F = λGD F ∗ ( I H − λLD G D F ∗ ) − LD G F = λGD F ∗ ( I H − λ e LD G D F ∗ ) − e LD G F = λGD F ∗ e L ( I H − λD G D F ∗ e L ) − D G F, (6.5) Γ( λ ) = GF + λGD F ∗ ( I H − λLD G D F ∗ ) − LD G F = GF + λGD F ∗ e L ( I H − λD G D F ∗ e L ) − D G F. The operators K = (cid:20) GF GD F ∗ LD G F LD G D F ∗ (cid:21) : L ⊕ H → K ⊕ H and K = (cid:20) GF GD F ∗ e LD G F D G D F ∗ e L (cid:21) : L ⊕ H → K ⊕ H are contraction. Actually, let f ∈ H and h ∈ L then one can check that (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) fh (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13) K (cid:20) fh (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = || F ∗ f − D F h || L + || D L D G ( D F ∗ f + F h ) || H ≥ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) fh (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13) K (cid:20) fh (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) = || F ∗ e Lf − D F h || L + || D e L f || H ≥ . Thus, the systems η , η , ζ , and ζ are passive and their transfer functions are preciselyΓ( λ ).Since e L ∗ ↾ ker D F ∗ = 0 and F ∗ f = 0 ⇐⇒ D F ∗ f = f , Gh = 0 ⇐⇒ D G h = h , byinduction one can derive the following equalities(6.6) T n ≥ ker ( F ∗ ( D G L ∗ D F ∗ ) n ) = T n ≥ ker (cid:16) F ∗ ( D G e L ∗ ) n ) (cid:17) , T n ≥ ker ( G ( D F ∗ LD G ) n ) = T n ≥ ker (cid:16) G ( D F ∗ e L ) n (cid:17) , T n ≥ ker (cid:16) F ∗ D G e L ∗ ( D F ∗ D G e L ∗ ) n (cid:17) = T n ≥ ker (cid:16) F ∗ ( D G e L ∗ ) n ) (cid:17) , T n ≥ ker (cid:16) GD F ∗ ( e LD G D F ∗ ) n (cid:17) = T n ≥ ker (cid:16) G ( D F ∗ e L ) n (cid:17) , T n ≥ ker (cid:16) F ∗ D G ( e L ∗ D F ∗ D G ) n (cid:17) = T n ≥ ker (cid:16) F ∗ ( D G e L ∗ ) n D G (cid:17) , T n ≥ ker (cid:16) GD F ∗ e L ( D G D F ∗ e L ) n (cid:17) = T n ≥ ker (cid:16) G ( D F ∗ e L ) n (cid:17) . EALIZATIONS OF THE SCHUR ITERATES 25
From (6.6) follow the relations (6.3) and (6.4). (cid:3)
Theorem 6.2.
Let the system τ = (cid:26)(cid:20) D D D ∗ GF D D − F D ∗ G + D F ∗ LD G (cid:21) ; M , N , H (cid:27) be conservative and simple and let Θ( λ ) be its transfer function. Suppose that the first Schuriterate Θ ( λ ) of Θ is non-unitary constant. Then the systems (6.7) ζ = (cid:26)(cid:20) GF GLD G F LD G (cid:21) ; D D , D D ∗ , D F ∗ (cid:27) ,ζ = (cid:26)(cid:20) GF GLD G F D G L (cid:21) ; D D , D D ∗ , D G (cid:27) are conservative and simple and their transfer functions are equal to Θ ( λ ) .Proof. Because the system ν is conservative, the operators F and G ∗ are isometries. SinceΘ ( λ ) is non-unitary constant, from (6.5) it follows that the operator GF is non-unitary.Hence by Theorem 4.6 the subspaces D F ∗ and D G are nontrivial, and the operator L ∈ L ( D G , D F ∗ ) is unitary. In addition, ker F ∗ = D F ∗ , ker G = D G , and the operators D F ∗ and D G are orthogonal projections in H onto ker F ∗ and ker G , respectively. One can directlycheck that the operators (cid:20) GF GLD G F LD G (cid:21) : D D ⊕ D F ∗ → D D ∗ ⊕ D F ∗ , (cid:20) GF GLD G F D G L (cid:21) : D D ⊕ D G → D D ∗ ⊕ D G are unitary. Hence, the systems ζ and ζ are conservative. Relation (6.3) yields in our casethat ( H cν ) ⊥ = (cid:0) H cζ (cid:1) ⊥ , ( H oν ) ⊥ = (cid:0) H oζ (cid:1) ⊥ . Taking into account (6.4) and the simplicity of ν we get that the systems ζ and ζ aresimple. (cid:3) Theorem 6.3.
Let Θ( λ ) ∈ S ( M , N ) , Γ = Θ(0) and let Θ ( λ ) be the first Schur iterate of Θ . Suppose τ = (cid:26)(cid:20) Γ CB A (cid:21) ; M , N , H (cid:27) is a simple conservative system with transfer function Θ . Then the simple conservativesystem ν = (cid:26)(cid:20) D − ∗ CD − A ∗ B AP ker D A (cid:21) , D Γ , D Γ ∗ , H (cid:27) has the transfer function λ Θ ( λ ) while the simple conservative systems (6.8) ζ = (cid:26)(cid:20) D − ∗ C ( D − B ∗ ) ∗ D − ∗ C ↾ ker D A ∗ AP ker D A D − A ∗ B P ker D A ∗ A ↾ ker D A ∗ (cid:21) ; D Γ , D Γ ∗ , ker D A ∗ (cid:27) ,ζ = (cid:26)(cid:20) D − ∗ C ( D − B ∗ ) ∗ D − ∗ CA ↾ ker D A P ker D A D − A ∗ B P ker D A A ↾ ker D A (cid:21) ; D Γ , D Γ ∗ , ker D A (cid:27) have transfer functions Θ ( λ ) . Here the operators D − , D − ∗ , and D − A ∗ are the Moore–Penrosepseudo-inverses. Proof.
Let T = (cid:20) Γ CB A (cid:21) = (cid:20) Γ D Γ ∗ GF D Γ − F Γ ∗ G + D F ∗ LD G (cid:21) == (cid:20) − KA ∗ M + D K ∗ XD M KD A D A ∗ M A (cid:21) : M ⊕ H → N ⊕ H . Then G = D − ∗ C , F ∗ = D − B ∗ , F = M ↾ D Γ , M = D − A ∗ B . According to Proposition 4.7 wehave D F ∗ = P ker D A ∗ , D G = P ker D A , L = A ↾ ker D A . Hence GF = D − ∗ C ( D − B ∗ ) ∗ , D G D F ∗ L = P ker D A A ↾ ker D A ,D G F = P ker D A M = P ker D A D − A ∗ B, GD F ∗ L = D − ∗ CP D A A ↾ ker D A ,LD G ↾ ker D A ∗ = AP ker D A ↾ ker D A ∗ , LD G F = AP ker D A D − A ∗ B. Note that if f ∈ ker D A ∗ then AP ker D A f = P ker D A ∗ AP ker D A f = P ker D A ∗ Af − P ker D A ∗ AP D A f = P ker D A ∗ Af.
Now the statement of theorem follows from Theorem 5.3 and Theorem 6.2. (cid:3)
Remark 6.4.
Since F ∗ = D − B ∗ , we get F = (cid:0) D − B ∗ (cid:1) ∗ ∈ L ( D Γ , H ) . Hence D − A ∗ B ↾ D Γ = (cid:0) D − B ∗ (cid:1) ∗ . Using the Hilbert spaces and operators defined by (3.1) and (3.2) , we get P ker D A D − A ∗ B ↾ D Γ = P , D − A ∗ B ↾ D Γ = (cid:0) D − ( B ∗ ↾ H , ) (cid:1) ∗ ∈ L ( D Γ , H , ) . In addition D − ∗ C ( D − B ∗ ) ∗ = Γ ∈ L ( D Γ , D Γ ∗ ) . So, (6.9) ζ = (cid:26)(cid:20) Γ D − ∗ CA (cid:0) D − ( B ∗ ↾ H , ) (cid:1) ∗ A , (cid:21) ; D Γ , D Γ ∗ , H , (cid:27) ,ζ = (cid:26)(cid:20) Γ D − ∗ CA (cid:0) D − ( B ∗ ↾ H , ) (cid:1) ∗ A , (cid:21) ; D Γ , D Γ ∗ , H , (cid:27) . It follows that ran (cid:16) D − ∗ C ↾ H , (cid:17) ⊂ ran D Γ ∗ , ran (cid:0) D − B ∗ ↾ H , (cid:1) ⊂ ran D Γ Schur iterates of the characteristic function.Theorem 6.5.
Let A be a completely non-unitary contraction in a separable Hilbert space H . Assume ker D A = { } and let the contractions A n,m be defined by (3.1) and (3.2) . Thenthe characteristic functions of the operators A n, , A n − , , . . . , A n − m,m , . . . A ,n − , A ,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 { A n − k,k } nk =0 is EALIZATIONS OF THE SCHUR ITERATES 27 (1) a unilateral shift (co-shift) if and only if the n -th Schur parameter Γ n of Φ is isometric(co-isometric), (2) the orthogonal sum of a unilateral shift and co-shift if and only if (6.10) D Γ n − = { } , D Γ ∗ n − = { } and Γ m = 0 for all m ≥ n. Each subspace from the set { H n − k,k } nk =0 is trivial if and only if Γ n is unitary.Proof. We will prove by induction. The systemΣ = (cid:26)(cid:20) − A D A ∗ D A A ∗ (cid:21) ; D A , D A ∗ , H (cid:27) is conservative and simple and its transfer function Φ( λ ) is Sz.-Nagy–Foias characteristicfunction of A . As in Theorem 5.5, let F and G ∗ be the embedding of the subspaces D A and D A ∗ into H , respectively. Then D F ∗ = P ker D A = P , , D G = P ker D A ∗ = P , , and L = A ∗ ↾ ker D A ∗ ∈ L ( D A ∗ , D A ) is unitary operator. The system ν = (cid:26)(cid:20) P D A ∗ I D A A ∗ P ker D A ∗ (cid:21) ; D A , D A ∗ , H (cid:27) is conservative and simple and its transfer function Z ( λ ) is the M¨obius parameter of Φ( λ ).Constructing the systems given by (6.7) in Theorem 6.2 we get ζ = (cid:26)(cid:20) P D A ∗ ↾ D A P D A ∗ ↾ ker D A A ∗ P ker D A ∗ ↾ D A A ∗ P ker D A ∗ ↾ ker D A (cid:21) ; D A , D A ∗ , ker D A (cid:27) and ζ = (cid:26)(cid:20) P D A ∗ ↾ D A P D A ∗ A ∗ ↾ ker D A ∗ P ker D A ∗ ↾ D A P ker D A ∗ A ∗ ↾ ker D A ∗ (cid:21) ; D A , D A ∗ , ker D A ∗ (cid:27) . By Theorem 6.2 the systems ζ and ζ are conservative and simple and their transfer functionsare exactly the first Schur iterate Φ ( λ ) of Φ( λ ). Note (see (3.1) and (3.2)) that A ∗ P ker D A ∗ ↾ ker D A = A ∗ , , P ker D A ∗ A ∗ ↾ ker D A ∗ = A ∗ , . Applying Proposition 4.5 we get that the pure part of Φ ( λ ) coincides with the characteristicfunctions of the operators A , and A , .By Theorem 3.1 completely non-unitary contractions { A n − k,k } nk =0 are unitarily equivalent.Assume that their characteristic functions coincide with the pure part of the n -th Schuriterate Φ n ( λ ) of Φ. The first Schur iterate of Φ n is the function Φ n +1 ( λ ). As is alreadyproved above the pure part of Φ n +1 coincides with the characteristic function of the operators( A n − k,k ) , and ( A n − k,k ) , . From (3.9) it follows( A n − k,k ) , = A n +1 − k,k , ( A n − k,k ) , = A n − k,k +1 = A n +1 − ( k +1) ,k +1 . Thus, characteristic functions of the unitarily equivalent completely non-unitary contractions { A n +1 − k,k } n +1 k =0 coincide with Φ n +1 .Note that the M¨obius parameter of the n − n − is λ Φ n ( λ ) and byTheorem 5.5 this function coincides with the characteristic function of the operator A n, = A n, P ker D An, . Applying Theorem 5.5 once again, we get that A n, is a unilateral shift if andonly if Γ n is a isometry.The function Φ ∗ (¯ λ ) is the characteristic function of the operator A ∗ and its Schur param-eters are adjoint to the corresponding Schur parameters of Φ. In addition if B = A ∗ then B n,m = A ∗ m,n . Therefore, A ∗ ,n is a unilateral shift if and only if Γ ∗ n is isometric. But A ∗ ,n isunuitarily equivalent to A ∗ n, . Hence, A n, is a co-shift if and only if Γ n is a co-isometry.It follows that Γ n is a unitary if and only if A n, is a unilateral shift and co-shift in H n, ⇐⇒ H n, = { } .Condition (6.10) holds true if and only if Φ n is identically equal zero. This is equivalentto the condition that A n, (as well and A n − , , A n − , , . . . A ,n ) is the orthogonal sum of ashift and co-shift. (cid:3) Remark 6.6.
It is proved that Γ n is isometry ⇐⇒ ker D A n +1 = ker D A n ⇐⇒ ker D A n ∩ ker D A ∗ = ker D A n − ∩ ker D A ∗ ⇐⇒ . . . ⇐⇒ ker D A n +1 − k ∩ ker D A ∗ k = ker D A n − k ∩ ker D A ∗ k ⇐⇒ . . . ⇐⇒ ker D A ∗ n ⊂ ker D A ;Γ ∗ n is isometry ⇐⇒ ker D A ∗ ⊂ ker D A n ⇐⇒ ker D A n − ∩ ker D A ∗ = ker D A n − ∩ ker D A ∗ ⇐⇒ . . . ⇐⇒ ker D A n − k ∩ ker D A ∗ k +1 = ker D A n − k ∩ ker D A ∗ k ⇐⇒ . . . ⇐⇒ ker D A ∗ n +1 = ker D A ∗ n ;(6.10) ⇐⇒ ker D A n = T l ≥ ker D A l ! ⊕ T l ≥ ker D A ∗ l ! ,P ker D An A T l ≥ ker D A ∗ l ! ⊂ T l ≥ ker D A ∗ l ! . Conservative realizations of the Schur iterates.Theorem 6.7.
Let Θ( λ ) ∈ S ( M , N ) and let τ = (cid:26)(cid:20) Γ CB A (cid:21) ; M , N , H (cid:27) be a simple conservative realization of Θ . Then the Schur parameters { Γ n } n ≥ of Θ can becalculated as follows (6.11) Γ = D − ∗ C (cid:0) D − B ∗ (cid:1) ∗ , Γ = D − ∗ D − ∗ CA (cid:0) D − D − ( B ∗ ↾ H , ) (cid:1) ∗ , . . . , Γ n = D − ∗ n − · · · D − ∗ CA n − (cid:16) D − n − · · · D − ( B ∗ ↾ H n − , ) (cid:17) ∗ , . . . . Here the operator (cid:16) D − n − · · · D − ( B ∗ ↾ H n − , ) (cid:17) ∗ ∈ L ( D Γ n − , H n − , ) is the adjoint to the operator D − n − · · · D − ( B ∗ ↾ H n − , ) ∈ L ( H n − , , D Γ n − ) , and ran (cid:16) D − n − · · · D − ( B ∗ ↾ H n, ) (cid:17) ⊂ ran D Γ n , ran (cid:16) D − ∗ n − · · · D − ∗ ( C ↾ H ,n ) (cid:17) ⊂ ran D Γ ∗ n EALIZATIONS OF THE SCHUR ITERATES 29 for every n ≥ . Moreover, for each n ≥ the unitarily equivalent simple conservativesystems (6.12) τ ( k ) n = (" Γ n D − ∗ n − · · · D − ∗ ( CA n − k ) A k (cid:16) D − n − · · · D − ( B ∗ ↾ H n, ) (cid:17) ∗ A n − k,k ; D Γ n − , D Γ ∗ n − , H n − k,k ) ,k = 0 , , . . . , n are realizations of the n -th Schur iterate Θ n of Θ . Here the operator B n = (cid:16) D − n − · · · D − ( B ∗ ↾ H n, ) (cid:17) ∗ ∈ L ( D Γ n − , H n, ) is the adjoint to the operator D − n − · · · D − ( B ∗ ↾ H n, ) ∈ L ( H n, , D Γ n − ) . Proof.
We will prove by induction. For n = 1 it is already established (see Remark 6.4, (6.8),and (6.9)) that Γ = D − ∗ C (cid:0) D − B ∗ (cid:1) ∗ and the systems τ (0)1 = (cid:26)(cid:20) Γ D − ∗ ( CA ) (cid:0) D − ( B ∗ ↾ H , ) (cid:1) ∗ A , (cid:21) ; D Γ , D Γ ∗ , H , (cid:27) and τ (1)1 = (cid:26)(cid:20) Γ D − ∗ ( C ) A (cid:0) D − ( B ∗ ↾ H , ) (cid:1) ∗ A , (cid:21) ; D Γ , D Γ ∗ , H , (cid:27) are conservative and simple realizations of Θ . Suppose τ (0) m = (" Γ m D − ∗ m − · · · D − ∗ ( CA m ) (cid:16) D − m − · · · D − ( B ∗ ↾ H m, ) (cid:17) ∗ A m, ; D Γ m − , D Γ ∗ m − , H m, ) is a simple conservative realization of Θ m . Then B m = (cid:16) D − m − · · · D − ( B ∗ ↾ H m, ) (cid:17) ∗ ∈ L ( D Γ m − , H m, ) ,C m = D − ∗ m − · · · D − ∗ ( CA m ) ∈ L ( H m, , D Γ ∗ m − ) , A m, ∈ L ( H m, , H m, ) . Hence B ∗ m = D − m − · · · D − ( B ∗ ↾ H m, ) ∈ L ( H m, , D Γ m − ) . The first Schur iterate of Θ m ( λ ) is the function Θ m +1 ( λ ) ∈ S ( D Γ m , D Γ ∗ m ) and the first Schurparameter of Θ m is Γ m +1 . From (3.4) and (3.9) it follows thatker D A m, = H m +1 , , ( A m, ) , = A m +1 , ∈ L ( H m +1 , , H m +1 , ) . Hence by (6.8), and (6.9)Γ m +1 = D − ∗ m C m (cid:0) D − m B ∗ m (cid:1) ∗ = D − ∗ m · · · D − ∗ CA m (cid:0) D − m · · · D − ( B ∗ ↾ H m, ) (cid:1) ∗ and the system τ (0) m +1 = (cid:26)(cid:20) Γ m +1 D − ∗ m · · · D − ∗ ( CA m +1 ) (cid:0) D − m · · · D − ( B ∗ ↾ H m +1 , ) (cid:1) ∗ A m +1 , (cid:21) ; D Γ m , D Γ ∗ m , H m +1 , (cid:27) is a simple conservative realization of Θ m +1 . From Proposition 6.3 it follows that the system τ ( k ) m +1 = (cid:26)(cid:20) Γ m D − ∗ m · · · D − ∗ ( CA m +1 − k ) A k (cid:0) D − m · · · D − ( B ∗ ↾ H m +1 , ) (cid:1) ∗ A m +1 − k,k (cid:21) ; D Γ m , D Γ ∗ m , H m +1 − k,k (cid:27) is unitarily equivalent to the system τ (0) m +1 for k = 1 , . . . , m + 1 and hence have transferfunctions equal to Θ m +1 . This completes the proof. (cid:3) Let us make a few remarks which follow from (4.9), Proposition 4.5, and Theorem 6.5.If D Γ N = 0 and D Γ ∗ N = 0 then D Γ n = 0, Γ ∗ n = 0 ∈ L ( D Γ ∗ N , { } ), D Γ ∗ n = D Γ ∗ N , and H ,n = H ,N for n ≥ N . The unitarily equivalent observable conservative systems τ ( k ) N = (cid:26)(cid:20) Γ N D − ∗ N − · · · D − ∗ ( CA N − k )0 A N − k,k (cid:21) ; D Γ N − , D Γ ∗ N − , H N − k,k (cid:27) , k = 0 , , . . . , N have transfer functions Θ N ( λ ) = Γ N and the operators A N − k,k are unitarily equivalent co-shifts of multiplicity dim D Γ ∗ N , the Schur iterates Θ n are null operators from L ( { } , D Γ ∗ N ) for n ≥ N + 1 and are transfer functions of the conservative observable system τ N +1 = (cid:26)(cid:20) D − ∗ N − · · · D − ∗ C A ,N (cid:21) ; { } , D Γ ∗ N , H ,N (cid:27) . If D Γ ∗ N = 0 and D Γ N = 0 then D Γ ∗ n = 0, D Γ n = D Γ N , and Γ n = 0 ∈ L ( D Γ N , { } ), H n, = H N, for n ≥ N . The unitarily equivalent controllable conservative systems τ ( k ) N = (" Γ N A k (cid:16) D − N − · · · D − ( B ∗ ↾ H N, ) (cid:17) ∗ A N − k,k ; D Γ N − , D Γ ∗ N − , H N − k,k ) have transfer functions Θ N ( λ ) = Γ N and the operators A N − k,k are unitarily equivalent unilat-eral shifts of multiplicity dim D Γ N , the Schur iterates Θ n are null operators from L ( D Γ N , { } )for n ≥ N + 1 and are transfer functions of the conservative controllable system τ N +1 = (cid:26)(cid:20) (cid:0) D − N · · · D − ( B ∗ ↾ H N +1 , ) (cid:1) ∗ A N, (cid:21) ; D Γ N , { } , H N, (cid:27) . References [1] D. Alpay, T. Azizov, A. Dijksma, and H. Langer, The Schur algorithm for generalized Schur functios.I. Coisometric realizations, Oper. Theory Adv. Appl., 129 (2001), 1–36.[2] D. Alpay, T. Azizov, A. Dijksma, H. Langer, and G. Wanjala, The Schur algorithm for generalizedSchur functions. II. Jordan chains and transformations of characteristic functions. Monatsh. Math.138 (2003), no. 1, 1–29.[3] D. Alpay, T. Azizov, A. Dijksma, H. Langer, and G. Wanjala, The Schur algorithm for generalizedSchur functions. IV. Unitary realizations. Oper. Theory Adv. Appl., 149 (2004), 23–45.[4] D. Alpay, A. Dijksma, and H. Langer, The transformation of Issai Schur and related topics in indefinitesetting, Oper. Theory Adv. Appl., 176 (2007), 1–98.[5] D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo,
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Department of Mathematical Analysis, East Ukrainian National University, KvartalMolodyozhny 20-A, Lugansk 91034, Ukraine
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