Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas

Transcription

1 mn header will be provided by the publisher Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas S ter Horst fdeling Wiskunde, Faculteit der Exacte Wetenschappen, Vrije Universiteit,De Boelelaan 1081a, 1081 HV msterdam, The Netherlands Received xxx, revised xxx, accepted xxx Published online xxx Key words Commutant lifting, Nehari interpolation MSC (000) 470, 4757 The description of all solutions to the relaxed commutant lifting problem in terms of an underlying contraction, obtained earlier in joint work of the author with E Frazho and M Kaashoek, is transformed into a linear fractional Redheffer state space form Under certain additional conditions the coefficient functions in this representation are described explicitly in terms of the original data The main theorem is a generalization of the Redheffer description of all solutions to the classical commutant lifting problem To illustrate the result a relaxed version of the Nehari extension problem is considered, and an explicit Redheffer description of all its solutions is given, assuming that a certain truncated Hankel operator is a strict contraction 1 Introduction The classical commutant lifting theorem, which was obtained by Sz-Nagy-Foias 14 and originated from the work of Sarason 16, has been used to solve, among other things, a large number of metric constraint interpolation and extension problems; see 5 for a recent overview In 6, extending the classical theory, a relaxed commutant lifting problem is introduced, and a particular (so-called central) solution, satisfying a maximum entropy condition, is obtained and used to solve a number of relaxed versions of the classical interpolation problems Descriptions of all solutions to the relaxed commutant lifting problem are given in 8, 13 and 9 The ones in 8 and 9 are in terms of Schur class functions, whereas 13 uses a choice sequences approach The present paper can be seen as an addition to 9, where the description of all solutions (see Theorem 11 below) is given in terms of an underlying contraction The aim of the present paper is to derive a linear fractional Redheffer type description of all solutions which is explicit in terms of the original data For this description some additional conditions on the data are needed The main theorem (see Theorem 1 below) generalizes the corresponding result for the classical commutant lifting problem: Theorem VI61 in 5; see also Theorem XIV71 in 4 To state the problem and results more precisely we first explain in some detail the notation and terminology used in the sequel Throughout capital calligraphic letters denote Hilbert spaces The Hilbert space direct sum of U and Y is denoted by U Y or by U Y n operator is a bounded linear transformation acting between Hilbert spaces With L (U, Y) we denote the set of all operators from U into Y, where we have the standard abbreviation L (U) in case Y = U The identity operator in L (U) is denoted by I U, or just by I when the underlying space is clear from the context By definition, a subspace is a closed linear manifold Let M be a subspace of U Then U M stands for the orthogonal complement of M in U We follow the convention that the symbol Π M denotes the orthogonal projection from U onto M viewed as an operator from U to M, whereas P M stands for the orthogonal projection sterhorst@fewvunl, Phone:

2 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem from U onto M acting as an operator in L (U) With this notation Π M is the canonical embedding of M into U and P M = Π M Π M n operator C in L (U) is referred to as positive definite (notation: C > 0) if C is invertible and positive (ie, Cu, u 0 for each u U) n operator N in L (U, Y) is called left invertible if there exists an operator M in L (Y, U) such that MN = I U In that case M is called a left inverse of N Note that N is left invertible if and only if N N is positive definite For a contraction K the symbol D K denotes the positive square root of I K K, while D K stands for the closure of the range of D C s usual D K and D K are referred to as the defect operator and defect space of K, respectively The unilateral shift on the Hardy space H (Y) is denoted by S Y, and we write E Y for the canonical embedding of Y onto the subspace of constant functions in H (Y), that is, (E Y v)(λ) = v for all λ D and each v Y By definition, the symbol H (U, Y) stands for the class of uniformly bounded analytic functions on the open unit disc D with values in L (U, Y) The set H (U, Y) forms a Banach space with respect to the supremum norm We write S(U, Y) for the closed unit disc in H (U, Y), and refer to S(U, Y) as the Schur class associated with U and Y Functions from S(U, Y) are called Schur class functions Finally, by H (U, Y) we denote the class of all functions F on D with values in L (U, Y) such that for each u in U the map λ F (λ)u defines a function in the Hardy class H (Y) Such a function is automatically analytic on D Obviously, H (U, Y) is properly contained in H (U, Y) Recall from 6 that a lifting data set is a set = {, T, U, R, Q} consisting of five Hilbert space operators The operator is a contraction mapping H into H, the operator U L (K ) is a minimal isometric lifting of the contraction T L (H ), and R and Q are operators from H 0 to H, satisfying the following constraints: T R = Q and R R Q Q (11) Given a lifting data set as above, the relaxed commutant lifting problem is to describe all contractions B from H to K such that Π H B = and U BR = BQ (1) contraction B from H into K satisfying (1) will be called a contractive interpolant for Hence the relaxed commutant lifting problem is to describe all contractive interpolants for the lifting data set Without loss of generality we can, and will, assume that U is the Sz-Nagy-Schäffer isometric lifting of T (4, Section VI3), that is, K is the direct sum of H and the Hardy space H (D T ) and Put U = Then (11) implies that T 0 E DT D T S DT : H H (D T ) H H (D T ) (13) D = (Q Q R R) 1 and D = D H 0 (14) Q D Q = D + R D T R + R D R (15) Indeed, this is the case because for all h H 0 we have D Qh = Qh Qh = D h + Rh T Rh = D h + Rh T Rh + Rh Rh = D h + D T Rh + D Rh With the lifting data set we associate a contraction ω defined by DT R ω : F D T D, ωd Q =, F = D D R QH 0 (16) The relation in (15) guarantees that ω is contractive Moreover, ω is an isometry if and only if D = 0 In terms of the operators from the lifting data set this is equivalent to R R = Q Q The first main theorem from 9 can now be formulated as follows

3 mn header will be provided by the publisher 3 Theorem 11 Let = {, T, U, R, Q} be a lifting data set with U the Sz-Nagy-Schäffer isometric lifting of T, and let B be an operator from H into H H (D T ) Then B is a contractive interpolant for if and only if B admits a representation of the form B = : H ΓD H H (D T ) where Γ is the operator mapping D into H (D T ) given by, (17) ( Γd)(λ) = Π DT Z(λ)(I λπ D Z(λ)) 1 d (d D, λ D) (18) for some Schur class function Z from S(D, D T D ) with Z(λ) F = ω for each λ D In 9 it was also explained how Z S(D, D T D ) with the additional constraint Z(λ) F = ω for each λ D can be constructed Indeed, for a L (D, D T D )-valued function Z on D it is the case that Z S(D, D T D ) with Z(λ) F = ω for each λ D if and only if Z is given by Z(λ) = ωπ F + D ω Ṽ (λ)π Ker Q D (λ D), (19) for some Ṽ S(Ker Q D, D ω ) Moreover, the functions Z and Ṽ in (19) determine each other uniquely So via (19) the contractive interpolants can be described with Schur class functions from S(Ker Q D, D ω ) without additional constraints For computational matters it is more convenient to derive a description of the form (19) that is explicitly in terms of the operators appearing in the lifting data set To achieve this we need additional constraints on the lifting data set, namely, we will assume that is a strict contraction and R has a left inverse In this case the defect operator D of is positive definite, and from the second condition in (11) we obtain that Q also has a left inverse In particular D Q and D R are left invertible, or equivalently, Q D Q and R D R are positive definite So the extra assumptions allow us to define operators X 1, X, X 3, X 4 and X 5 by X 1 = R(Q D Q) 1 Q D : H H, X = R(R D R) 1 J 1 Π D D T + D Π Ker R 1 R Π Ker R : D D T Ker R H, X 3 = 1 Q Π Ker Q : H Ker Q X 4 = D T R(Q D Q) 1 Q D : H D T, X 5 = Π DT 1 Π D D T : D D T Ker R D T, (110) where Q L (Ker Q ), R L (Ker R ) and L (D D T ) are the positive definite operators: R = Π Ker R D = I + J(R D R) 1 J, with J = Q = Π Ker Q D Π Ker Q, Π Ker R, D D T R : H 0 D D T (111) In Proposition below we show that, via unitary mappings between Ker D Q and Ker Q and between D ω and D D T Ker R, the description of Z in (19) can be transformed into Z(λ) = X 4 D X 1 D 1 + X 5 D X V (λ)x 3 D 1 (λ D), (11) for some V S(Ker Q, D D T Ker R ) Here X 1, X, X 3, X 4 and X 5 are the operators defined in (110) Using some general result on Redheffer products of scattering matrices (cf, Section XIV1 in 4) we arrive at the following description of all contractive interpolants for {, T, U, R, Q} under the additional assumption that is a strict contraction and R has a left inverse

4 4 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Theorem 1 Let = {, T, U, R, Q} be a lifting data set with U the Sz-Nagy-Schäffer isometric lifting of T ssume that is a strict contraction and R has a left inverse Then an operator B mapping H into H H (D T ) is a contractive interpolant for if and only if B admits a representation of the form B = : H Γ H H (D T ) where Γ is the operator mapping H into H (D T ) given by, (113) (Γh)(λ) = Φ (λ)h + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ 1 (λ)h (h H, λ D) (114) for some Schur class function V from S(Ker Q, D D T Ker R ) Moreover, any such function V defines an operator Γ mapping H into H (D T ) via (114) Here Φ 11 and Φ 1 are functions from the Schur classes S(D D T Ker R, Ker Q ) and S(D D T Ker R, D T ), respectively, and Φ 1 and Φ are functions from H (H, Ker Q ) and H (H, D T ), respectively, and these functions are given by Φ 11 (λ) = λx 3 (I λx 1 ) 1 X, Φ 1 (λ) = X 3 (I λx 1 ) 1, Φ 1 (λ) = X 5 + λx 4 (I λx 1 ) 1 X, Φ (λ) = X 4 (I λx 1 ) 1, (λ D) (115) where X 1, X, X 3, X 4 and X 5 are the operators defined in (110) The proof of Theorem 1 is given in Section, where we also derive additional properties of the operator valued functions in (115) (see Corollary 4 below) When taking the zero function for V in Theorem 1 we see that (114) reduces to (Γh)(λ) = Φ (λ)h for h H and λ D The solution obtained in this way is precisely the central solution given in 6 In 6, Proposition 53, it was shown that the spectral radius of X 1 in (110) is strictly less than one if R and Q are such that R λq is left invertible for each λ D In this case the functions Φ 1 and Φ are uniformly bounded on D, that is, Φ 1 and Φ are functions from H (H, Ker Q ) and H (H, D T ), respectively Recall that the classical commutant lifting problem appears when in the lifting data set {, T, U, R, Q} the operator Q is an isometry and R the identity operator in L (H), and thus, in particular, H 0 = H Under these additional conditions Theorem 1 reduces to the first part of Theorem VI61 in 5 (see Corollary 3 below for further details) Moreover, in that case (114) provides a proper parameterization, that is, there exists a unique Schur class function V such that B is given by (113) and (114) In general, for an arbitrary lifting data set formula (114) does not provide a proper parameterization This follows from Theorem 1 in 9 To illustrate Theorem 1 we consider a relaxed version of the operator-valued Nehari extension problem Let N be a positive integer (N > 0) and let F 1, F, be a sequence of operators from U to Y satisfying n=1 F nu < for each u U The relaxed Nehari extension problem considered in this paper is to find all sequences of operators H 0, H 1, from U to Y with the property that n=0 H nu < for each u U, and such that the operator from U N into l (Y) given by the operator matrix representation F F 3 F (N+1) F 1 F F N H 0 F 1 F (N 1) H 1 H 0 F 1 H0 : U N l (Y) (116) has operator norm at most one Here U N is the Hilbert space direct sum of N copies of U, and l (Y) is the Hilbert space of bilateral square summable sequences (y n ) n Z with entries in Y s usual Z stands for the set of

5 mn header will be provided by the publisher 5 all integers The box in (116) indicates the zero position in l (Y) sequence of operators (H n ) n N from U to Y that forms a solution to the relaxed Nehari problem is referred to as an N-complementary sequence associated with (F n 1 ) n N, or just an N-complementary sequence if no confusion concerning the sequence (F n 1 ) n N can arise Here N stands for the set of nonnegative integers (ie, with zero included) The setup for this problem resembles the way relaxed versions of the Schur, Nevanlinna-Pick and Sarason interpolation problems where formulated in 6 For this relaxed Nehari problem to be solvable it is necessary that the operator given by F 3 F 4 F (N+) F F 3 F (N+1) F 1 F F N : U N l (Y) (117) is contractive Here l (Y) is the Hilbert space of square summable sequences (, y, y 1 ) with entries in Y In Section 3 the relaxed Nehari extension problem is put into a relaxed commutant lifting setting, which yields that the condition that the operator in (117) is contractive is not only necessary but also sufficient The latter can also be seen by repeatedly applying Parrott s lemma (see Corollary IV36 in 4) We use Theorem 1 to give a Redheffer description (in Theorem 3 below) of all N-complementary sequences, under the additional assumption that the operator in (117) is a strict contraction In addition, we specify Theorem 3 for the special case when F n = 0 for n = 1,, (see Corollary 33 below), which leads to an operator-valued version of a result in 1 Redheffer representations and proof of the main theorem The aim in this section is to prove Theorem 1 The main tool to achieve this objective is Proposition 1 given in the next subsection This proposition gives a general scheme for rewriting the description in (18) into one of the type (114) 1 Redheffer type descriptions First we introduce some notation Let C be an operator in L (U) Then C is said to be pointwise stable if C n u converges to zero as n goes to infinity, for each u U In particular, if the spectral radius of C, denoted by r spec (C), is strictly less than 1, then C is pointwise stable Next, fix a function H from H (U, Y) With H we associate an operator Γ H mapping U into H (Y) defined by (Γ H u)(λ) = H(λ)u (λ D, u U) (1) On the other hand, if Γ is any operator from U into H (Y), then there exists a unique function G from H (U, Y) such that Γ = Γ G This function is given by G(λ)u = (Γu)(λ) for u U and λ D Now assume, in addition, that H is from H (U, Y) We associate with H, in the usual way, a multiplication operator M H mapping H (U) into H (Y) defined by (M H f)(λ) = H(λ)f(λ) (λ D, f H (U)) The operator M H is called the multiplication operator defined by H Moreover, the norm of M H is given by M H = sup λ D H(λ) In particular, M H is a contraction if and only if H is from the Schur class S(U, Y) Proposition 1 ssume that Z is a Schur class function from S(U, Y U) given by X4 X5 Z(λ) = + V (λ) X 1 X X Y 3 : U (λ D), () U where V is from the Schur class S(V, W) and X 1 X X = X 3 0 U : U V is a contraction (3) W X 4 X5 Y

6 6 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Define operator-valued functions Φ 11, Φ 1, Φ 1 and Φ by Φ 11 (λ) = λ X 3 (I λ X 1 ) 1 X, Φ 1 (λ) = X 3 (I λ X 1 ) 1, Φ 1 (λ) = X 5 + λ X 4 (I λ X 1 ) 1 X, (λ D) (4) Φ (λ) = X 4 (I λ X 1 ) 1 Then Φ 11 S(W, V), Φ 1 S(W, Y), Φ 1 H (U, V), Φ H (U, Y) and Π Y Z(λ)(I λπ U Z(λ)) 1 = Φ (λ) + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ1 (λ) (λ D) (5) Moreover, the operator M = M Φ11 M Φ1 Γ Φ1 Γ Φ : H (W) U H (V) H (Y) (6) is a contraction which is unitary whenever X in (3) is unitary and X 1 is pointwise stable The formula on the right hand side of (5) is referred to as a linear fractional Redheffer description The term Redheffer comes from scattering theory, see Chapter XIV in 4 Indeed, let Φ 11, Φ 1, Φ 1 and Φ be the functions in (4) where X in (3) is a contraction Consider the Redheffer scattering system p h = M Φ11 Γ Φ1 M Φ1 Γ Φ q u, subject to q = M V p, (7) where V is a Schur class function from S(V, W) Here u is an element from U and the vectors h, p and q are functions from the Hardy spaces H (Y), H (V) and H (W), respectively Solving (7) we obtain that y = Γu, where Γ is the operator from U into H (Y) defined by the function in the right hand side of (5), that is, ( Γu)(λ) = Φ (λ)u + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ1 (λ)u (u U, λ D) Proof of Proposition 1 The first part of Proposition 1, that is, the equality in (5) can be obtained from () by using standard operations of the theory of Redheffer products; see Section XIV1 in 4 Formula (5) can also be checked by multiplication from the right with I λπ U Z(λ) The observations concerning the functions in (4) and the operator in (6) follow from some well known results from standard system theory; see for instance 11 and 7 To be a bit more explicit, one can consider the system { Θ = X 1, X X3,, X 4 0 X5 }, and apply Theorem III104 in 5 and Proposition 17 in 3; the latter concerns time-variant systems and should be specified for the time-invariant case Proof of Theorem 1 Proposition 1 suggests that in order to prove Theorem 1 it suffices to show that a Schur class function Z from S(D, D T D ) with the property that Z(λ) F = ω for each λ D can be expressed as in (), with X 1, X, X3, X4 and X 5 the appropriate operators s mentioned in the introduction, this is done in Proposition below Proposition Let = {, T, U, R, Q} be a lifting data set, and let Z be an L (D, D T D )-valued function on D ssume that is a strict contraction and R is left invertible Then Z S(D, D T D ) and Z(λ) F = ω for each λ D if and only if Z is given by (11) for a V S(Ker Q, D D T Ker R ) Moreover, Z and V define each other uniquely in (11)

7 mn header will be provided by the publisher 7 P r o o f We already observed in the introduction that Z S(D, D T D ) satisfies the equality Z(λ) F = ω for each λ D if and only if there exists a Schur class function Ṽ from S(Ker Q D, D ω ) such that Z is given by (19), and that in this case Ṽ in (19) is uniquely determined by Z We first show how the operators appearing in (19) can be expressed in terms of the original data in case < 1 and R is left invertible Recall that < 1 and R being left invertible imply that D Q and D R are left invertible, or equivalently, that Q D Q and R D R are positive definite So left inverses of D Q and D R are given by L D Q = (Q D Q) 1 Q D and L D R = (R D R) 1 R D, respectively The proof of Proposition consists of four parts In Part 1 we derive a formula for ωπ F explicitly in the original data Put Y = 1 Π D T 1 JL D R DT D D : T 0 Π Ker R D D Ker R, D where and J are the operators in (111) In the second part we prove that Y Y = Dω and Ker Y = {0} In Part 3 we show that Im D 1 Π Ker Q = Ker Q D and Im D 1 Π Ker R = Ker R D With these identities we obtain that P Ker Q D = D 1 Π Ker Q 1 Q Π Ker Q D 1 and P Ker R D = D 1 Π Ker R 1 R Π Ker R D 1, (8) where Q and R are the operators in (111) The results from the Parts 1 to 3 are then combined in Part 4 to complete the proof of Proposition Part 1 Since D F = Ker Q D, we have L D Q D F = 0 So the definition of ω in (16) yields that DT R DT R(Q ωπ F = L D R D Q = D Q) 1 Q D X D R(Q D Q) 1 Q = 4 D 1 D D X (9) 1 Part We first observe that the inverse of is given by 1 = I J(Q D Q) 1 J This follows from the fact that J intertwines with Q D Q(R D R) 1 and a straightforward computation These observations imply that L D R J 1 JL D R is equal to P Im D R D R(Q D Q) 1 R D Hence Y Π DT 1 Y = Π D Π T DT 1 JL D R L D R J 1 P Ker R D + L D R J 1 = IDT 0 0 I D Π D T DT R D R JL D R (Q DQ) 1 R D T R D = D ω The last identity follows with the first formula for ωπ F in (9) and the fact that L D QL D Q = (Q D Q) 1 Next we show that Ker Y = {0} Since Y Ker R D = Π Ker R D and Y (D D T ) D Im D R = D (D Ker R D ), it suffices to show that Ker (Y D D T ) = {0} To see that this is the case observe that R D L D R = I So Ker L D R = {0} Moreover, we have Ker D = {0}, when D is seen as an operator on D Therefore Ker Y Π D D T = Ker DT 1 L D R J 1 IDT 0 ΠDT = Ker 0 L D R J 1 = 1 ΠDT Ker J = 1 0 IDT Ker D R D T = {0} Part 3 Let N be an operator from H 0 to H that has a left inverse In particular, this holds for Q and R Then D 1 Π Ker N has a left inverse, and thus N = Π Ker N D Π Ker N is positive definite Since D 1 (D N) = N, we see that D 1 Im D N = Im N This implies that Im D 1 Π Ker N = D 1 Ker N = Ker N D Then we obtain in the same way as for D R in Part that P Ker N D = P Im D 1 Π Ker N = D 1 Π Ker N 1 N Π Ker N D 1

8 8 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Filling in Q and R for N gives the desired results Part 4 The identities Y Y = D ω and Ker Y = {0} derived in Part show that there exists a unitary operator φ ω mapping D ω onto D D T Ker R D such that Y φ ω = D ω Furthermore, from (8) we obtain that there exist unitary operators φ Q and φ R mapping Ker Q and Ker R onto Ker Q D and Ker R D, respectively, such that φ Q 1 Q ID D φ = T 0 0 φ R Π ker Q D 1 = Π Ker Q D and φ R 1 R ΠD D T φ ω : D Π Ker R D φ ω ω Then φ is unitary and with the definitions of φ and φ R we obtain that X 5 φ D X = Y φ ω = D ω Π ker R D 1 D D T Ker R = Π Ker R D Now put Moreover, we have φ Q X 3 D 1 = φ Q 1 Q Π ker Q D 1 = Π Ker Q D Finally, the fact that φ Q and φ are unitary implies that a function V is from the Schur class S(Ker Q, D D T Ker R D ) if and only if V (λ) = φ Ṽ (λ)φ Q (λ D) (10) for some Schur class function Ṽ from S(Ker Q D, D ω ), and V and Ṽ in (10) define each other uniquely Therefore we obtain with the statement in the first paragraph of the proof that Z S(D, D T D ) and Z(λ) F = ω for each λ D if and only if there exists a V S(Ker Q, D D T Ker R D ) such that Z(λ) = ωπ F + D ω φ V (λ)φ QΠ Ker Q D X = 4 D 1 X D X 1 D V (λ)x D X 3 D 1 and V and Z define each other uniquely (λ D), Proof of Theorem 1 Let B be an operator from H into H (D T ) ssume that B is given by (17) with Γ the operator defined by (18), where Z is a function from the Schur class S(D, D T D ) with Z(λ) F = ω for each λ D Let V be the Schur class function from S(Ker Q, D D T Ker R D ) such that Z is given by (11) Put Then X 1 = D X 1 D 1, X = D X, X3 = X 3 D 1, X 3 0 X 4 X5 = X 1 X = X 3 D 1 0 X 4 D 1 X 5 D X 1 D 1 D X φ Q 0 ΠD F 0 0 I DT D ωπ F X4 = X 4 D 1 and X5 = X 5 (11) D ω ID 0 0 φ (1) Here φ Q and φ are the unitary operators constructed in Part 4 of the proof of Proposition Since ω is a contraction, we see that the operator ΠD F 0 D Ker Q : D (13) ωπ F D ω D ω D T D is a contraction fter rearranging rows in (1) we obtain from the fact that ψ Q and ψ are unitary, that X in (3) is a contraction Let Φ 11, Φ 1, Φ 1 and Φ be the functions defined in (4) and let Φ 11, Φ 1, Φ 1 and Φ be the functions defined in (115) Using that (I λ X 1 ) 1 = D (I λx 1 ) 1 D 1 we obtain from the relations in (11) that for each λ D Φ 11 (λ) = Φ 11 (λ), Φ1 (λ) = Φ 1 (λ)d 1, Φ 1 (λ) = Φ 1 (λ) and Φ (λ) = Φ (λ)d 1 (14)

9 mn header will be provided by the publisher 9 pplying Proposition 1 we see that Φ 11 and Φ 1 are Schur class functions from S(D D T Ker R, Ker Q ) and S(D D T Ker R, D T ), respectively, and that Φ 1 and Φ are functions from H (H, Ker Q ) and H (H, D T ), respectively Moreover, we have for each h H and each λ D that ( ΓD h)(λ) = Π DT Z(λ)(I λπ D Z(λ))D h = Φ (λ)d h + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ1 (λ)d h = Φ (λ)h + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ 1 (λ)h = (Γh)(λ), where Γ is the operator defined in (114) with V the Schur class function determined by (11) 3 Corollaries We conclude this section with two corollaries The first specifies Theorem 1 for the classical commutant lifting setting, cf, Theorem VI61 in 5, and follows from Theorem 1 with some easy computations that we leave to the reader Corollary 3 Let {, T, U, R, Q} be a lifting data set where U is the Sz-Nagy-Schäffer isometric lifting of T ssume that Q is an isometry in L (H), R = I H and is a strict contraction Then Ker R = {0}, D = {0} and the operator-valued functions Φ 11, Φ 1, Φ 1 and Φ in (115) are functions from S(D T, Ker Q ), H (H, Ker Q ), S(D T, D T ) and H (H, D T ), respectively, and they are given by Φ 11 (λ) = λ 1 Q Π Ker Q (I λt ) 1 D D T 1, Φ 1 (λ) = 1 Q Π Ker Q (I λt ) 1, Φ 1 (λ) = 1 D T (I λt ) 1 D D T 1, Φ (λ) = D T (I λt ) 1 T, (λ D) where T L (H), Q L (Ker Q ) and L (D T ) are given by T = D Q Q D, Q = Π Ker Q D 1 Π Ker Q and = I + D T D D T Next we obtain some properties of the operators corresponding to the functions in (115) Corollary 4 Let = {, T, U, R, Q} be a lifting data set with U the Sz-Nagy-Schäffer isometric lifting of T ssume that is a strict contraction and R has a left inverse Let Φ 11, Φ 1, Φ 1 and Φ be the functions defined by (115) and (110) Put M = 0 M Φ11 M Φ1 Γ Φ1 Γ Φ H : (D D T Ker R ) H H H (Ker Q ) H (D T ) Then M is a contraction, and M is an isometry if R R = Q Q and X 1 is pointwise stable (15) P r o o f Let X 1, X, X 3, X 4 and X 5 be the operators in (110) and X 1, X, X3, X4 and X 5 the operators given by (11) Define Φ 11, Φ 1, Φ 1 and Φ by (4) We saw in the proof of Theorem 1 that the results from Proposition 1 can be applied for this choice of X 1, X, X3, X4 and X 5 Then the operator M in (6) is a contraction which is unitary in case X in (3) is unitary and X 1 is pointwise stable Since X 1 and X 1 are similar (D X 1 = X 1 D and D > 0), we see that X1 is pointwise stable if and only if X 1 is pointwise stable Moreover, the computation in (1) and the remark below (13) show that X is unitary if and only if the operator in (13) is unitary This is precisely the case if and only if ω is an isometry In other words, X in (3) is unitary if and only if R R = Q Q We conclude that the operator M in (6) is a contraction which is unitary if X 1 is pointwise stable and R R = Q Q In terms of operators the identities in (14) become M Φ11 = M Φ11, Γ Φ1 = Γ Φ11 D 1, M Φ1 = M Φ1 and Γ Φ11 = Γ Φ11 D 1 (16)

10 10 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Next define a contraction by = 0 H : (W) H H, where W = D D T Ker R The defect operator of is given by IH D = (W) 0 H : (W) H (W) 0 D H D Then the identities in (16) show that 0 M = = M Φ11 M Φ1 Γ Φ1 Γ Φ 0 D M Φ11 Γ Φ1 D M Φ1 Γ Φ = IH 0 = MD 0 M D It is well known that the operator D is an isometry Therefore we obtain that M is a contraction which is isometric in case R R = Q Q and X 1 is pointwise stable 3 The relaxed Nehari extension problem In this section we apply Theorem 1 to obtain a description of all N-complementary sequences for the relaxed Nehari extension problem formulated in the last but one paragraph of the introduction 31 The relaxed Nehari problem in the relaxed commutant lifting setting Throughout this subsection N is a positive integer and F 1, F, is a sequence of operators from U to Y satisfying n=1 F nu < for each u U We refer to the operator from U N into l (Y) given by (117) as the N-truncated Hankel operator defined by the sequence (F n 1 ) n N Our first remark is that an operator from U N into l (Y) is an N-truncated Hankel operator if and only if satisfies the intertwining relation T R = Q, where T is the shift operator in L (l (Y)) given by T =, (31) I Y I Y 0 and R and Q are the operators from U N 1 to U N defined by IU N 1 R = : U N 1 U N 1 and Q = 0 U 0 I U N 1 : U N 1 U U N 1 (3) Indeed, one easily checks that for an operator from U N into l (Y), say = i,j j=1,,n i=,, 1, we have T R = Q if and only if i,j = i 1 j+1 for appropriate indices i and j, that is, T R = Q is equivalent to being an N-truncated Hankel operator We shall also need the bilateral forward shift V in L (l (Y)) which is given by V = 0 I I I in L (l (Y)) (33)

11 mn header will be provided by the publisher 11 Let l +(Y) be the subspace of l (Y) consisting of the sequences that have the zero vector on all entries with strictly negative index Identifying l (Y) with the subspace of l (Y) consisting of all sequences that have the zero vector on all entries with a non-negative index, we see that l (Y) = l (Y) l +(Y), and V partitions as V = T 0 X S : l (Y) l +(Y) l (Y) l +(Y) (34) Here T is given by (31), the operator S is the unilateral forward shift in L (l +(Y)), and X is the operator from l (Y) into l +(Y) given by X (, y 3, y, y 1 ) = (y 1, 0, 0, ) ((, y 3, y, y 1 ) l (Y)) Since V is unitary, (34) shows that V is an isometric lifting of T s is easily seen this lifting is also minimal We are now ready to state the main result of this subsection Proposition 31 Let N be a positive integer and F 1, F, a sequence of operators from U to Y satisfying n=1 F nu < for each u U ssume that the N-truncated Hankel operator defined by (F n 1 ) n N is a contraction Then = {, T, V, R, Q}, with T, V, R and Q defined by (31), (33) and (3), is a lifting data set, and there exists a contractive interpolant for if and only if there exists an N-complementary sequence associated with (F n 1 ) n N More precisely, an operator B from U N to l (Y) is a contractive interpolant for if and only if B is an operator of the form given in (116) with (H n ) n N an N-complementary sequence associated with (F n 1 ) n N P r o o f We already know that T R = Q and that V is a minimal isometric lifting of T Since R and Q are both isometries, we have R R = Q Q So the constraints (11) are fulfilled, and hence is a lifting data set Now let B be an operator from U N into l (Y) Using similar arguments as in the first paragraph of this subsection we obtain that B satisfies V BR = BQ if and only if B = H H 3 H (N+1) H 1 H H N H 0 H 1 H (N 1) H 1 H 0 H (N ) H H 1 H (N 3) : U N l (Y) (35) for some sequence of operators, H, H 1, H 0, H 1, H, from U into Y satisfying n= H nu < for each u U Moreover, if B is given by (35), then Π B l (Y) = holds if and only if H k = F k for k = 1,, 3, Henceforth, we obtain that B is a contractive interpolant for if and only if B is equal to the operator in (116) with (H n ) n N an N-complementary sequence associated with (F n 1 ) n N In order to apply Theorem 1 we need to establish a relation between the contractive interpolants for the lifting data set = {, T, V, R, Q} in Proposition 31 and those for = {, T, U, R, Q}, with U being the Sz-Nagy-Schäffer isometric lifting of T For that purpose we use the Fourier transform F Y from l +(Y) to the Hardy space H (Y), that is, F Y is the unitary operator mapping l +(Y) onto H (Y) defined by (F Y (y n ) n N )(λ) = λ n y n ((y n ) n N l +(Y), λ D) (36) n=0 Using F Y, let Ψ be the unitary operator from l (Y) = l (Y) l +(Y) into l (Y) H (Y) given by Il 0 l Ψ = (Y) : (Y) l 0 F Y l (Y) +(Y) H (37) (Y)

12 1 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Since D T = Y and the bilateral shift S in L (l +(Y)) and the bilateral shift S Y in L (H (Y)) are related via F Y S = S Y F Y, we see from (34) and (13) that Ψ intertwines the bilateral shift V with the Sz-Nagy-Schäffer isometric lifting U of T, that is, ΨV = U Ψ Moreover, we have that Ψf = f for each sequence f in l (Y) Using the properties of Ψ found above, a straightforward computation shows that B is a contractive interpolant for if and only if B = Ψ B for some contractive interpolant B for Finally, observe that a sequence of operators H 0, H 1, in L (U, Y) has the property that n=0 H nu < for each u U if and only if it is the sequence of Taylor coefficients at zero of a function in H (U, Y) So, alternatively, we seek functions H in H (U, Y) such that the operator in (116) is a contraction, where H n is the n th Taylor coefficient of H at zero 3 The solution to the relaxed Nehari problem In the previous subsection we saw how the solution to the relaxed commutant lifting problem can be applied to obtain all N-complementary sequences for the relaxed Nehari extension problem Two important operators in the description of all contractive interpolants in Theorem 1 are the defect operators of the contractions T and From the definition of T in (31) we immediately see that D T = Y and D T = 0 0 I Y : l (Y) Y (38) The defect operator of the N-truncated Hankel operator is the positive square root of Λ 1,1 Λ 1,N { D = I in L (U N n=i ), where Λ i,j = F nf n if i = j, Λ N,1 Λ n=i F nf n+i j if i j N,N (39) In what follows we assume the N-truncated Hankel operator to be a strict contraction, or equivalently, we assume that the defect operator D is positive definite We also use the entries Λ i,j in the N N operator matrix representation of D, D = Λ 1,1 Λ 1,N Λ N,1 Λ N,N in L (U N ) (310) The fact that D is positive definite implies that D and Λ n,n in (310) are also positive definite for n = 1,, N In particular, this is true for n = 1 and n = N Moreover, we have that the N 1 by N 1 left upper block matrix operator of D in (39) is positive definite Therefore there exists a unique solution G1 G N 1 to the equation Λ 1,1 Λ 1,N 1 Λ N 1,1 Λ N 1,N 1 G 1 G N 1 = F 1 F N+1 (311) description of all N-complementary sequences associated with (F n 1 ) n N under the assumption that the N-truncated Hankel operator for (F n 1 ) n N is a strict contraction is given in the next theorem Theorem 3 Let N be a positive integer and F 1, F, a sequence of operators from U to Y satisfying n=1 F nu < for each u U ssume that the N-truncated Hankel operator defined by (F n 1 ) n N is a strict contraction Let H be a function from H (U, Y) Then the Taylor coefficients (H n ) n N of H at zero form an N-complementary sequence associated with (F n 1 ) n N if and only if there exists a Schur class function V from S(U, Y U) such that H(λ) = ˆΦ (λ) + ˆΦ 1 (λ)v (λ)(i ˆΦ 11 (λ)v (λ)) 1 ˆΦ1 (λ) (λ D) (31)

13 mn header will be provided by the publisher 13 Moreover, for any function V from S(U, Y U) the formula (31) defines a function H from H (U, Y) Here ˆΦ 11 and ˆΦ 1 are Schur class functions from S(Y U, U) and S(Y U, Y), respectively, and ˆΦ 1 and ˆΦ are functions from H (U, U) and H (U, Y), respectively, and these functions are given by ˆΦ 11 (λ) = λ(λ 1,1 ) 1 E N (I U N λt state) 1 G (I + F G ) 1 C, ˆΦ 1 (λ) = (Λ 1,1 ) 1 λ(λ 1,1 ) 1 E N (I U N λt state) 1 C 1, ˆΦ 1 (λ) = (I + F G ) 1 F C F (IU N λt state ) 1 G (I + F G ) 1 C, (313) ˆΦ (λ) = F (I U N λt state ) 1 C 1, where T state L (U N ), E N : U U N, C 1 : U U N, C : U U N, F : U N Y and G : U N Y are the operators given by Λ,1 (Λ 1,1 ) 1 I 0 0 I Λ 3,1 (Λ 1,1 ) 1 0 I 0 T state = 0, E N =, Λ N,1 (Λ 1,1 ) 1 0 I (314) Λ,1 Λ 1,N C 1 = (Λ 1,1 ) 1, C = (Λ N,N ) 1, Λ N,1 0 F = F 1 F N Λ N,N and G = G 1 G N 1 0, with Λ m,n for m, n = 1,, N as in (310), and G 1 G N 1 the solution to the equation (311) Furthermore, we have r spec (T state ) < 1, and the operator ˆM defined by ˆM = 0 Γ MˆΦ11 MˆΦ1 ΓˆΦ1 ΓˆΦ H : (Y U) l (Y) H U (U), (315) H (Y) is isometric, where Γ is given by Γ = F F 1 : l (Y) U Observe that the state operator T state in (314) is close to a companion operator To be precise, let E be the flip over operator in L (U N ) given by E(u 1, u,, u N ) = (u N, u N 1,, u 1 ) ((u 1, u,, u N ) U N ) Then E is unitary and ET state E is precisely the second companion operator corresponding to the operator-valued polynomial K(λ) = λ N 1 k=0 λk Λ N k,1, see Chapter 14 in 1 Note that the leading coefficient Λ 1,1 of K is positive definite s a computational remark, note that to obtain the solution G 1 G N 1 to the equation (311) it suffices to compute the inverse of the operator Λ N,N in (310) Indeed, with a standard Schur complement type of argument we see that the inverse of the N 1 by N 1 left upper corner of D in (39) is given by Λ 1,1 Λ 1,N 1 Λ 1,N (Λ N,1 ) (Λ N,N ) 1 Λ N 1,1 Λ N 1,N 1 Λ N 1,N (Λ N,N 1 ) Furthermore, if U is finite dimensional, then all computations involve finite matrices only In this sense the relaxed Nehari problem is very different from the classical Nehari problem

14 14 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Proof of Theorem 3 Since R in (3) is an isometry, we have that the lifting data set = {, T, U, R, Q}, with U the Sz-Nagy-Schäffer isometric lifting of T in (31) and R and Q as in (3), has the property that is a strict contraction (by assumption) and R is left invertible So the description of all contractive interpolants in Theorem 1 can be applied to this particular lifting data set Since also Q is an isometry, we have R R = Q Q Hence D = 0 and D = {0}, where D and D are given by (14) We already made the identification between D T and Y Moreover, we have Ker Q = U {0} N 1 U N and Ker R = {0} N 1 U U N So both Ker Q and Ker R can be identified with U In that case the projections Π Ker Q and Π Ker R are given by Π Ker Q = I U 0 0 : U N U and Π Ker R = 0 0 I U : U N U (316) In particular, Π Ker Q = E N, with E N the operator in (314) Now let H be a function from H (U, Y) with Taylor coefficients (H n ) n N at zero Then the operator Γ H from U into H (Y) defined in (1) is given by Γ H = F Y Π l + (Y) BE N = Π H (Y)Ψ BE N, where B is the operator in (116), and F Y and Ψ are the unitary operators defined in (36) and (37), respectively ccording to Proposition 31 and the last but one paragraph of the previous subsection we have that (H n ) n N is an N- complementary sequence associated with (F n 1 ) n N if and only if Ψ B is a contractive interpolant for Moreover, all contractive interpolants for are obtained in this way Then Theorem 1 and the above analysis of the spaces Ker Q, Ker R, D and D T imply that (H n ) n N is an N-complementary sequence if and only if there exists a Schur class function V from S(U, Y U) such that H(λ) = Φ (λ)e N + Φ 1 (λ)v (λ)(i Φ 11 (λ)v (λ)) 1 Φ 1 (λ)e N (λ D), where Φ 11, Φ 1, Φ 1 and Φ are the functions defined in Theorem 1 Put for each λ D ˆΦ 11 (λ) = Φ 11 (λ), ˆΦ1 (λ) = Φ 1 (λ)e N, ˆΦ1 (λ) = Φ 1 (λ) and ˆΦ (λ) = Φ (λ)e N (317) Then we obtain from Theorem 1 that ˆΦ 11 S(Y U, U), ˆΦ 1 H (U, U), ˆΦ 1 S(Y U, Y) and ˆΦ H (U, Y) Note that = Γ ΓˆΦ1 Φ1 E N, = Γ ΓˆΦ Φ E N and Γ = E N Hence the operator ˆM in (315) is obtained from M in (15) after multiplication from the right by IH (Y U) 0 : 0 E N H (Y U) U H (Y U) U N Since R R = Q Q, we obtain from Corollary 4 that M, and thus ˆM, is an isometry if X 1 in (110) is pointwise stable This proves to be the case In fact, we have r spec (X 1 ) < 1 To see this, note that R λq is left invertible for each λ in D, and apply Proposition 53 in 6 (see also the remark in the third paragraph after Theorem 1) So to complete the proof it remains to show that the functions ˆΦ 11, ˆΦ 1, ˆΦ 1 and ˆΦ defined in (317) can also be written as in (313), and that r spec (T state ) < 1 The latter immediately shows that ˆΦ 1 and ˆΦ are functions in H (U, U) and H (U, Y), respectively From the formulas for Π Ker Q and Π Ker R in (316), Π DT in (38) and D in (310) one easily checks that we have the identities: Q = Λ 1,1, R = Λ N,N, = I +F G, F = D T and G = D T R(R D R) 1 R, where Q, R and are given by (111) and F and G are the operators in (314) Let X 1, X, X 3, X 4 and X 5 be the operators defined in (110) Then we immediately obtain that X 3 = (Λ 1,1 ) 1 E N, X 4 = F X 1 and X 5 = (I + F G ) 1 ΠY (318) Writing out ˆΦ 11, ˆΦ 1, ˆΦ 1 and ˆΦ in terms of X 1, X, X 3, X 4 and X 5 with the identities for X 3, X 4 and X 5 in (318) we get ˆΦ 11 (λ) = λ(λ 1,1 ) 1 E N (I λx 1) 1 X, ˆΦ 1 (λ) = (Λ 1,1 ) 1 + λ(λ 1,1 ) 1 E N (I λx 1) 1 X 1 E N, ˆΦ 1 (λ) = X 5 + λf X 1 (I λx 1 ) 1 X = X 5 F X + F (I λx 1 ) 1 X, (λ D) (319) ˆΦ (λ) = F (I λx 1 ) 1 X 1 E N

15 mn header will be provided by the publisher 15 Therefore, to complete the proof, it suffices to show that X 1 = T state and X = G (I + F G ) 1 C (30) Indeed, assume that (30) holds Then C 1 = X 1 E N and X 5 F X = (I + F G ) F G (I + F G ) 1 F C = (I + F G )(I + F G ) 1 F C = (I + F G ) 1 F C This computation combined with the formulas for ˆΦ 11, ˆΦ 1, ˆΦ 1 and ˆΦ found in (319) show that (313) holds, and also that r spec (T state ) = r spec (X 1 ) < 1 We will now prove (30), starting with the first identity First we claim that X 1 can be written as X 1 = RQ + R(Q D Q) 1 Q D P Ker Q To see that this is the case, observe that both Q and (Q D Q) 1 Q D are left inverses of Q Certainly all left inverses of Q are equal on Im Q Since Q (U N Im Q) = Q Ker Q = 0, we obtain that (Q D Q) 1 Q D = Q + (Q D Q) 1 Q D P Ker Q, which proves our claim Note that RQ is the backward shift on U N This implies that T state = RQ + C 1 EN Using that P Ker Q = E N EN and Π Ker R C 1 = 0 we obtain that the first identity in (30) holds if R C 1 = (Q D Q) 1 Q D E N This identity follows from the next computation: Λ 0,1 Q DQR C 1 Λ 1,1 = Q DQ = Q D Λ,1 Λ N,1 Λ N,1 = Q D D E N Q D E N Λ 1,1 = Q E N Q D E N Λ 1,1 = Q D E N Λ 1,1 To see that the identity for X in (30) holds, note that X can be written as X = R(R D R) 1 R D T 1 D Π 1 Ker R Y : U N R U Earlier we obtained that = I + F G and R(R D R) 1 R D T = G This shows that X Y = G (I + F G ) 1 The fact that X U = C follows from the definition of Π Ker R in (316), the formula for D in (310) and the identity R = Λ N,N Thus the second identity in (30) holds s an illustration of Theorem 3 we consider the relaxed Nehari problem with F 1 = F = = 0 That is, we seek a description of all functions H from H (U, Y) with Fourier coefficients (H n ) n N at zero such that the operator H H 1 H H H 1 H 0 0 H N 1 H N H N 3 H 0 : U N l +(Y) (31) is a contraction By associating with each function H from H (U, Y) the norm of the operator in (31) we induce a Banach space structure on the set H (U, Y) This Banach space appears in 6, in the contexts of certain interpolation problems, and for the case that U and Y are finite dimensional, in 1 and By specifying Theorem 3 for the case that F 1 = F = = 0 we obtain the following description of all functions H from H (U, Y) for which the operator in (31) is contractive

16 16 S ter Horst: Relaxed commutant lifting and a relaxed Nehari problem Corollary 33 Let H be a function from H (U, Y) with Fourier coefficients (H n ) n N at zero Then the operator in (31) is a contraction if and only if there exists a Schur class function V from S(U, Y U) such that H(λ) = Π Y V (λ)(i λ N Π U V (λ)) 1 (λ D) (3) Corollary 33 is an operator-valued version of a result from 1 The case N = 1 appears as a corollary in 8 and is fundamental in the proof of the first main theorem in 9 more general result is obtained in 10 Proof of Corollary 33 Note that the N-truncated Hankel operator for the sequence F 1 = F = = 0 is the zero operator from U N to l (Y) In particular, is a strict contraction So we can apply the result of Theorem 3 to this Nehari data The fact that = 0 implies that D = D 1 = I U N, which leads to F = G = 0, C 1 = 0, C = 0 0 I and T = RQ is the backward shift operator in L (U N ) With these identities it is not difficult to compute that ˆΦ 11 (λ) = λ N Π U, ˆΦ1 (λ) = I ˆΦ1 (λ) = Π Y and ˆΦ (λ) = 0 (λ D) Hence (31) reduces to (3) cknowledgements The author thanks Prof M Kaashoek and Prof E Frazho for their many useful comments and suggestions The author is grateful to the referee for his/her suggestions to restructure the introduction References 1 D lpay, V Bolotnikov, and Ph Loubaton, On tangential H interpolation with second order norm constraints, Integral Equations and Operator Theory 4 (1996), pp D lpay, V Bolotnikov, and Ph Loubaton, On interpolation for Hardy functions in a certain class of domains under moment type constraints, Houston Journal of Mathematics, No 3 (1997), Vol 3, pp DR Pik, Block lower triangular operators and optimal contractive systems, PhD Thesis, Vrije Universiteit, msterdam, C Foias and E Frazho, The Commutant Lifting pproach to Interpolation Problems, OT 44, Birkhäuser-Verlag, Basel, C Foias, E Frazho, I Gohberg and M Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems, OT 100, Birkhäuser-Verlag, Basel, C Foias, E Frazho, and M Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations and Operator Theory, 4 (00), pp B Francis, course in H control theory, Lecture Notes in Control and Information Sciences 88, Springer-Verlag, Berlin, E Frazho, S ter Horst, and M Kaashoek, Coupling and relaxed commutant lifting, Integral Equations and Operator Theory, 54 (006), pp E Frazho, S ter Horst, and M Kaashoek, ll solutions to the relaxed commutant lifting problem, cta Sci Math (Szegged) 7 (006), pp E Frazho, S ter Horst, and M Kaashoek, Relaxed commutant lifting: an equivalent version and a new application, submitted 11 P Fuhrmann, Linear systems and operators in Hilbert space, McGraw-Hill International Book Co, New York, P Lancaster, and M Tismenetsky, The theory of matrices Second edition, cademic Press, Inc, Orlando, FL, WS Li and D Timotin, The relaxed intertwining lifting in the coupling approach, Integral Equations and Operator theory 54 (006), pp B Sz-Nagy and C Foias, Dilation des commutants d opérateurs, C R cad Sci Paris, série, 66 (1968), pp B Sz-Nagy and C Foias, Harmonic nalysis of Operators on Hilbert Space, North Holland Publishing Co, msterdam-budapest, D Sarason, Generalized interpolation in H, Trans merican Math Soc, 17 (1967), pp