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Minimal passive realizations of generalized Schur functions in Pontryagin spaces

Lassi Lilleberg

University of Vaasa, Finland

Operator Theory and Kre˘ ın Spaces, December 19, 2019, Vienna

Operator colligation An operator colligation Σ = (TΣ; X, U, Y; κ) consists of separable Pontryagin spaces X (the state space), U, and Y, and the bounded system operator TΣ ∈ L(X ⊕ U, X ⊕ Y).

Operator colligation An operator colligation Σ = (TΣ; X, U, Y; κ) consists of separable Pontryagin spaces X (the state space), U, and Y, and the bounded system operator TΣ ∈ L(X ⊕ U, X ⊕ Y). The symbol κ is reserved for the finite negative index of the state space, and it is assumed that U and Y have the same negative index.

Operator colligation An operator colligation Σ = (TΣ; X, U, Y; κ) consists of separable Pontryagin spaces X (the state space), U, and Y, and the bounded system operator TΣ ∈ L(X ⊕ U, X ⊕ Y). The symbol κ is reserved for the finite negative index of the state space, and it is assumed that U and Y have the same negative

• index. The operator TΣ has the block representation of the form

TΣ = A B C D

• :

X U

• →

X Y

• ,

where A ∈ L(X) is called as the main operator. If needed, the colligation is written as Σ = (A, B, C, D; X, U, Y; κ).

Passive systems The colligation Σ = (TΣ; X, U, Y; κ) will be called as a system.

Passive systems The colligation Σ = (TΣ; X, U, Y; κ) will be called as a system. When the system operator TΣ is contractive (isometric, co-isometric, unitary), with respect to the indefinite inner product, the corresponding system is called passive (isometric, co-isometric, conservative).

Passive systems The colligation Σ = (TΣ; X, U, Y; κ) will be called as a system. When the system operator TΣ is contractive (isometric, co-isometric, unitary), with respect to the indefinite inner product, the corresponding system is called passive (isometric, co-isometric, conservative). All the systems considered in this talk are passive.

Transfer function The transfer function θΣ of Σ = (A, B, C, D; X, U, Y; κ) is θΣ(z) := D + zC(I − zA)−1B, and it is defined whenever I − zA is invertible.

Transfer function The transfer function θΣ of Σ = (A, B, C, D; X, U, Y; κ) is θΣ(z) := D + zC(I − zA)−1B, and it is defined whenever I − zA is invertible. A system with the transfer function θ is said to be a realization of θ.

Transfer function The transfer function θΣ of Σ = (A, B, C, D; X, U, Y; κ) is θΣ(z) := D + zC(I − zA)−1B, and it is defined whenever I − zA is invertible. A system with the transfer function θ is said to be a realization of θ. Realization problem for an operator valued function θ analytic at the origin is to find a system with the desired minimality or

• ptimality properties such that its transfer function coincides

with θ in a neighbourhood of the origin.

Generalized Schur functions The generalized Schur class Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index, is the set of L(U, Y)-valued functions S(z) holomorphic in a neighbourhood Ω of the origin such that the Schur kernel KS(w, z) = 1 − S(z)S∗(w) 1 − z ¯ w , w, z ∈ Ω, (1) has κ negative squares (κ = 0, 1, 2, . . .).

Generalized Schur functions The generalized Schur class Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index, is the set of L(U, Y)-valued functions S(z) holomorphic in a neighbourhood Ω of the origin such that the Schur kernel KS(w, z) = 1 − S(z)S∗(w) 1 − z ¯ w , w, z ∈ Ω, (1) has κ negative squares (κ = 0, 1, 2, . . .). This means that no Hermitian matrix of the form

• KS(wj, wi)fj, fi
• Y

n

i,j=1 ,

has more than κ negative eigenvalues, and there exists at least

• ne such matrix that has exactly κ negative eigenvalues.

Generalized Schur functions The class S0(U, Y) is written S(U, Y). When U and Y are Hilbert spaces, S(U, Y) is the ordinary Schur class; it consists L(U, Y)-valued functions holomorphic and bounded by one in the unit disc D.

Generalized Schur functions The class S0(U, Y) is written S(U, Y). When U and Y are Hilbert spaces, S(U, Y) is the ordinary Schur class; it consists L(U, Y)-valued functions holomorphic and bounded by one in the unit disc D. Realization problems for ordinary ordinary Schur functions, as well as other properties of passive systems, were studied, for instance, by Arov, Ball and Cohen, de Branges and Rovnyak, Helton and Sz.-Nagy and Foias.

Generalized Schur functions The generalized Schur class was first studied by Kre˘ ın and Langer (1972), and after that, for instance, by Alpay, Azizov, Constantinescu, Dijksma, Dritschel, Gheondea, Rovnyak, and de Snoo.

Minimality properties The system Σ = (A, B, C, D; X, U, Y; κ) is said to be controllable (observable, simple) if, respectively,

Minimality properties The system Σ = (A, B, C, D; X, U, Y; κ) is said to be controllable (observable, simple) if, respectively, X c := span {ran AnB : n = 0, 1, . . .} = X X o := span {ran A∗nC∗ : n = 0, 1, . . .} = X X s := span {ran AnB, ran A∗mC∗ : n, m = 0, 1, . . .} = X,

Minimality properties The system Σ = (A, B, C, D; X, U, Y; κ) is said to be controllable (observable, simple) if, respectively, X c := span {ran AnB : n = 0, 1, . . .} = X X o := span {ran A∗nC∗ : n = 0, 1, . . .} = X X s := span {ran AnB, ran A∗mC∗ : n, m = 0, 1, . . .} = X, and minimal if it is both controllable and observable.

Minimality properties The system Σ = (A, B, C, D; X, U, Y; κ) is said to be controllable (observable, simple) if, respectively, X c := span {ran AnB : n = 0, 1, . . .} = X X o := span {ran A∗nC∗ : n = 0, 1, . . .} = X X s := span {ran AnB, ran A∗mC∗ : n, m = 0, 1, . . .} = X, and minimal if it is both controllable and observable. The realization Σ of θ is said to be κ-admissible, if the negative index of the state space coincides with the index of θ.

Minimality properties The system Σ = (A, B, C, D; X, U, Y; κ) is said to be controllable (observable, simple) if, respectively, X c := span {ran AnB : n = 0, 1, . . .} = X X o := span {ran A∗nC∗ : n = 0, 1, . . .} = X X s := span {ran AnB, ran A∗mC∗ : n, m = 0, 1, . . .} = X, and minimal if it is both controllable and observable. The realization Σ of θ is said to be κ-admissible, if the negative index of the state space coincides with the index of θ. If Σ = (A, B, C, D; X, U, Y; κ) is κ-admissible passive realization

• f θ ∈ Sκ(U, Y), then (X c)⊥, (X o)⊥ and (X s)⊥ are always

Hilbert spaces with respect to the inner product of X.

Minimality properties For θ ∈ Sκ(U, Y), there exists a realization Σ = Σ = (TΣ; X, U, Y; κ) of θ and it can be chosen such that it is (i) observable co-isometric or

Minimality properties For θ ∈ Sκ(U, Y), there exists a realization Σ = Σ = (TΣ; X, U, Y; κ) of θ and it can be chosen such that it is (i) observable co-isometric or (ii) controllable isometric or

Minimality properties For θ ∈ Sκ(U, Y), there exists a realization Σ = Σ = (TΣ; X, U, Y; κ) of θ and it can be chosen such that it is (i) observable co-isometric or (ii) controllable isometric or (iii) simple conservative.

Minimality properties For θ ∈ Sκ(U, Y), there exists a realization Σ = Σ = (TΣ; X, U, Y; κ) of θ and it can be chosen such that it is (i) observable co-isometric or (ii) controllable isometric or (iii) simple conservative. Examples of such realizations are the canonical realizations, which can be constructed by using the theory of reproducing kernels and de Branges–Rovnyak spaces.

Minimality properties For θ ∈ Sκ(U, Y), there exists a realization Σ = Σ = (TΣ; X, U, Y; κ) of θ and it can be chosen such that it is (i) observable co-isometric or (ii) controllable isometric or (iii) simple conservative. Examples of such realizations are the canonical realizations, which can be constructed by using the theory of reproducing kernels and de Branges–Rovnyak

• spaces. Any two realizations such that they have the same

property (i), (ii) or (iii), are unitarily similar.

Similarities Realizations Σ1 = (A1, B1, C1, D; X1, U, Y; κ) and Σ2 = (A2, B2, C2, D; X2, U, Y; κ) of θ ∈ Sκ(U, Y) are unitarily similar, if there exists a unitary mapping U : X1 → X2 such that A1 = U−1A2U, B1 = U−1B2, and C1 = C2U,

Similarities Realizations Σ1 = (A1, B1, C1, D; X1, U, Y; κ) and Σ2 = (A2, B2, C2, D; X2, U, Y; κ) of θ ∈ Sκ(U, Y) are unitarily similar, if there exists a unitary mapping U : X1 → X2 such that A1 = U−1A2U, B1 = U−1B2, and C1 = C2U, and weakly similar, it there exists an injective closed densely defined possible unbounded linear operator Z : X1 → X2 with the dense range such that ZA1x = A2Zx, C1x = C2Zx, x ∈ dom(Z), and ZB1 = B2.

Weakly similar systems In general, two minimal passive κ-admissible realizations Σ1 = (A1, B1, C1, D; X1, U, Y; κ) and Σ2 = (A2, B2, C2, D; X2, U, Y; κ) of θ ∈ Sκ(U, Y), are (only) weakly similar.

Weakly similar systems In general, two minimal passive κ-admissible realizations Σ1 = (A1, B1, C1, D; X1, U, Y; κ) and Σ2 = (A2, B2, C2, D; X2, U, Y; κ) of θ ∈ Sκ(U, Y), are (only) weakly similar. Weak similarity does not necessarily preserve the spectral properties of the main operator, or the structural properties (observability etc.) of the system.

Minimal systems and similarity, Problem 1 Cause the limitations of weak similarities, one might want to search conditions, or criterions, which guarantee that all minimal passive κ-admissible realizations of θ are unitarily similar (Problem 1).

Minimal systems and similarity, Problem 1 Cause the limitations of weak similarities, one might want to search conditions, or criterions, which guarantee that all minimal passive κ-admissible realizations of θ are unitarily similar (Problem 1). If U and Y are Hilbert spaces, this happens, for an example, if the boundary values of θ are (co-)isometric almost everywhere on the unit circle T.

Well-behaved systems, Problem 2 Also, one might wonder, that under certain circumstances, is it possible to construct "better", more well-behaved realizations than observable co-isometric (controllable isometric, simple conservative) (Problem 2).

Well-behaved systems, Problem 2 Also, one might wonder, that under certain circumstances, is it possible to construct "better", more well-behaved realizations than observable co-isometric (controllable isometric, simple conservative) (Problem 2). For an example, Arov showed in (1979) that when U and Y are Hilbert spaces and θ ∈ S(U, Y) is bi-inner, simple passive realization of θ are minimal conservative.

Defect functions The presented problems can be partially solved by using generalized defect functions.

Defect functions The presented problems can be partially solved by using generalized defect functions. For an ordinary Schur function θ ∈ S(U, Y) the right and left defect functions are, respectively,

• uter and ∗-outer functions ϕθ ∈ S(U, K) and ψθ ∈ S(H, Y),

where K and H are Hilbert spaces, such that

Defect functions The presented problems can be partially solved by using generalized defect functions. For an ordinary Schur function θ ∈ S(U, Y) the right and left defect functions are, respectively,

• uter and ∗-outer functions ϕθ ∈ S(U, K) and ψθ ∈ S(H, Y),

where K and H are Hilbert spaces, such that ϕ∗

θ(ζ)ϕθ(ζ) ≤ I − θ∗(ζ)θ(ζ),

ψθ(ζ)ψ∗

θ(ζ) ≤ I − θ(ζ)θ∗(ζ)

for a.e. ζ ∈ T,

Defect functions The presented problems can be partially solved by using generalized defect functions. For an ordinary Schur function θ ∈ S(U, Y) the right and left defect functions are, respectively,

• uter and ∗-outer functions ϕθ ∈ S(U, K) and ψθ ∈ S(H, Y),

where K and H are Hilbert spaces, such that ϕ∗

θ(ζ)ϕθ(ζ) ≤ I − θ∗(ζ)θ(ζ),

ψθ(ζ)ψ∗

θ(ζ) ≤ I − θ(ζ)θ∗(ζ)

for a.e. ζ ∈ T, and if functions ϕ ∈ θ(U, K) and ψ ∈ S( H, Y), where K and H are Hilbert space, have corresponding properties, then

• ϕ∗(ζ)

ϕ(ζ) ≤ ϕ∗

θ(ζ)ϕθ(ζ),

• ψ(ζ)

ψ∗(ζ) ≤ ψθ(ζ)ψ∗

θ(ζ)

for a.e. ζ ∈ T.

Defect functions By using defect functions, Arov (1979) proved the existence of an optimal minimal passive realizations of ordinary Schur function θ ∈ S(U, Y).

Defect functions By using defect functions, Arov (1979) proved the existence of an optimal minimal passive realizations of ordinary Schur function θ ∈ S(U, Y). Later, Arov et all. (1997) constructed (∗-)optimal minimal passive realizations geometrically, without using the defect functions.

Defect functions By using defect functions, Arov (1979) proved the existence of an optimal minimal passive realizations of ordinary Schur function θ ∈ S(U, Y). Later, Arov et all. (1997) constructed (∗-)optimal minimal passive realizations geometrically, without using the defect functions. This construction can be applied with the appropriate changes for the generalized Schur class Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index, to obtain (∗-)optimal minimal passive realizations.

Defect functions In general, if θ ∈ Sκ(U, Y) and Σ = (A, B, C, D; X, U, Y; κ) is a passive κ-admissible realization of θ, there exists a Hilbert space H and L(U, K)-valued function ϕ holomorphic at the

• rigin such that

Defect functions In general, if θ ∈ Sκ(U, Y) and Σ = (A, B, C, D; X, U, Y; κ) is a passive κ-admissible realization of θ, there exists a Hilbert space H and L(U, K)-valued function ϕ holomorphic at the

• rigin such that

I − θ∗(w)θ(z) = (1 − z ¯ w)F ∗(w)F(z) + ϕ∗(w)ϕ(z), where F(z) = (I − zA)−1B, holds for all z and w in a sufficiently small symmetric neighbourhood or the origin.

Defect functions I − θ∗(w)θ(z) = (1 − z ¯ w)F ∗(w)F(z) + ϕ∗(w)ϕ(z) I − θ∗(w)θ(z) = (1 − z ¯ w)F ′∗(w)F ′(z) + ϕ′∗(w)ϕ′(z).

Defect functions I − θ∗(w)θ(z) = (1 − z ¯ w)F ∗(w)F(z) + ϕ∗(w)ϕ(z) I − θ∗(w)θ(z) = (1 − z ¯ w)F ′∗(w)F ′(z) + ϕ′∗(w)ϕ′(z). If Σ = (A, B, C, D; X, U, Y; κ) and Σ′ = (A′, B′, C′, D; X ′, U, Y; κ) are κ-admissible realizations of θ such that Σ′ is optimal minimal,

Defect functions I − θ∗(w)θ(z) = (1 − z ¯ w)F ∗(w)F(z) + ϕ∗(w)ϕ(z) I − θ∗(w)θ(z) = (1 − z ¯ w)F ′∗(w)F ′(z) + ϕ′∗(w)ϕ′(z). If Σ = (A, B, C, D; X, U, Y; κ) and Σ′ = (A′, B′, C′, D; X ′, U, Y; κ) are κ-admissible realizations of θ such that Σ′ is optimal minimal, it holds F ′∗(z)F ′(z) ≤ F ∗(z)F(z) and ϕ∗(z)ϕ(z) ≤ ϕ′∗(z)ϕ′(z). The function ϕ′ := ϕθ is the generalized right defect function of θ.

Defect functions I − θ∗(w)θ(z) = (1 − z ¯ w)F ∗(w)F(z) + ϕ∗(w)ϕ(z) I − θ∗(w)θ(z) = (1 − z ¯ w)F ′∗(w)F ′(z) + ϕ′∗(w)ϕ′(z). If Σ = (A, B, C, D; X, U, Y; κ) and Σ′ = (A′, B′, C′, D; X ′, U, Y; κ) are κ-admissible realizations of θ such that Σ′ is optimal minimal, it holds F ′∗(z)F ′(z) ≤ F ∗(z)F(z) and ϕ∗(z)ϕ(z) ≤ ϕ′∗(z)ϕ′(z). The function ϕ′ := ϕθ is the generalized right defect function of θ. By using similar kind of construction and a ∗-optimal realization, the generalized left defect function ψθ of θ can be constructed.

The main result Theorem Let θ ∈ Sκ(U, Y) and let ϕθ and ψθ be the generalized right and left defect functions of θ, with values in L(U, K) and L(H, Y), respectively.

The main result Theorem Let θ ∈ Sκ(U, Y) and let ϕθ and ψθ be the generalized right and left defect functions of θ, with values in L(U, K) and L(H, Y),

• respectively. Then, all minimal passive κ-admissible

realizations of θ are unitarily similar if and only if there exists an L(H, K)-valued function χθ holomorphic at the origin such that Θ := θ ψθ ϕθ χθ

• ∈ Sκ

U H

• ,

Y K

• .

The main result Theorem Let θ ∈ Sκ(U, Y) and let ϕθ and ψθ be the generalized right and left defect functions of θ, with values in L(U, K) and L(H, Y),

• respectively. Then, all minimal passive κ-admissible

realizations of θ are unitarily similar if and only if there exists an L(H, K)-valued function χθ holomorphic at the origin such that Θ := θ ψθ ϕθ χθ

• ∈ Sκ

U H

• ,

Y K

• .

For an ordinary Schur functions, this result is due to D.Z. Arov and M.A. Nudel’man (2000, 2002).

Scattering suboperator Θ := θ ψθ ϕθ χθ

Scattering suboperator Θ := θ ψθ ϕθ χθ

• For an ordinary Schur function, it is always possible to choose

χθ from the unit ball of L∞(H, K) such that Θ belongs to the unit ball of L∞ U H

• ,

Y K

• .

Scattering suboperator Θ := θ ψθ ϕθ χθ

• For an ordinary Schur function, it is always possible to choose

χθ from the unit ball of L∞(H, K) such that Θ belongs to the unit ball of L∞ U H

• ,

Y K

• . The function χθ is called as a

scattering suboperator, and it was extensively studied, for instance, by Boiko and Dubovoy (2011, 2014).

Structural properties and defect functions Problem 2 presented earlier demands criterions that when there exists realizations of θ ∈ Sκ(U, Y) with better structural properties than the canonical realizations a priori give.

Structural properties and defect functions Problem 2 presented earlier demands criterions that when there exists realizations of θ ∈ Sκ(U, Y) with better structural properties than the canonical realizations a priori give. The solution in some sense can also be obtained by using the generalized defect functions.

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent:

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0);

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0); (b) all κ-admissible controllable (observable) passive realizations of θ are minimal isometric (co-isometric);

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0); (b) all κ-admissible controllable (observable) passive realizations of θ are minimal isometric (co-isometric); (c) there exists an observable (controllable) conservative realization of θ;

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0); (b) all κ-admissible controllable (observable) passive realizations of θ are minimal isometric (co-isometric); (c) there exists an observable (controllable) conservative realization of θ; (d) all simple conservative realization of θ are observable (controllable);

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0); (b) all κ-admissible controllable (observable) passive realizations of θ are minimal isometric (co-isometric); (c) there exists an observable (controllable) conservative realization of θ; (d) all simple conservative realization of θ are observable (controllable); (e) all observable co-isometric (controllable isometric) realizations of θ are conservative.

Structural properties and defect functions Theorem Let θ ∈ Sκ(U, Y), where U and Y are Pontryagin spaces with the same negative index. The following statements are equivalent: (a) ϕθ ≡ 0 (ψθ ≡ 0); (b) all κ-admissible controllable (observable) passive realizations of θ are minimal isometric (co-isometric); (c) there exists an observable (controllable) conservative realization of θ; (d) all simple conservative realization of θ are observable (controllable); (e) all observable co-isometric (controllable isometric) realizations of θ are conservative.

Structural properties and defect functions If U and Y are Hilbert spaces, ϕθ ≡ 0 means that the boundary values of θ ∈ Sκ(U, Y) are "almost" isometric.

Structural properties and defect functions If U and Y are Hilbert spaces, ϕθ ≡ 0 means that the boundary values of θ ∈ Sκ(U, Y) are "almost" isometric. Indeed, then the

• nly function ϕ ∈ Sκ′(U, H), for any κ′, which satisfy

ϕ∗(ζ)ϕ(ζ) ≤ I − θ∗(ζ)θ(ζ) for a.e ζ ∈ T is ϕ ≡ 0.

Structural properties and defect functions If U and Y are Hilbert spaces, ϕθ ≡ 0 means that the boundary values of θ ∈ Sκ(U, Y) are "almost" isometric. Indeed, then the

• nly function ϕ ∈ Sκ′(U, H), for any κ′, which satisfy

ϕ∗(ζ)ϕ(ζ) ≤ I − θ∗(ζ)θ(ζ) for a.e ζ ∈ T is ϕ ≡ 0. Note that this does not guarantee that the boundary values of θ are isometric a.e.

References The talk was based on the following articles of the author, mainly with the second one. All the mentioned references can be found from the references listed therein.

• L. Lilleberg, Isometric discrete-time systems with

Pontryagin state space, Complex Anal. Oper. Theory (2019). https://doi.org/10.1007/s11785-019-00930-1

• L. Lilleberg, Minimal passive realizations of generalized

Schur functions in Pontryagin spaces, arXiv:1910.11053 (2019).