Boundary Triplets and Canonical Systems of Diﬀerential Equations H.L. Wietsma

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Boundary Triplets and Canonical Systems of Differential Equations

H.L. Wietsma

A Master Thesis

Submitted to the Faculty of Mathematics and Natural Sciences University of Groningen

Groningen, The Netherlands August 2007

Primairy supervisors:

Prof. H.S.V. De Snoo Dr. J. Behrndt

Secondary supervisor:

Prof. A.J. van der Schaft

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iii

Preface and acknowledgements

This report, a partial fulfilment for the requirements for the degree of master of sci- ence, is an illustration of the theory of boundary triplets. This theory was presented in a seminar between April 2006 and April 2007 during a visit of Jussi Behrndt (TU Berlin, Germany).

First and foremost I would like to express my gratitude to my supervisor Henk de Snoo for providing the interesting subject of this thesis and for giving me the opportunity to get involved in doing research. In particular, by giving small sug- estions and lots of articles, perhaps a bit frustrating a first, my understanding of mathematics has grown immensly.

In June 2007 the department of mathematics kindly allowed me to visit Jussi Behrndt in Berlin, who generously made time free to aid me. Thanks to his as- sistence the formal correctness of this thesis was improved significantly.

Finally, I would like to thank Ineke Kruizinga for taking care of the administrative tasks related to this thesis.

Oldeboorn, August 2007, Hendrik Luit Wietsma

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v

Chapter 1: Introduction

This report is concerned with boundary value problems on a compact interval [a, b]

for canonical sytems of the form

Jf (t)^′− [H(t) + λ∆(t)] f(t) = ∆(t)g(t),

where the n× n matrix J satisfies J^∗ = J⁻¹ =−J, the n × n matrix functions H and ∆ are absolutely integrable on [a, b], H satifies H^∗ = H and ∆≥ 0. Further- more it is assumed that a certain definiteness condition holds. Then in the Hilbert space L²_∆((a, b)) (made up of equivalence classes) a minimal and a maximal relation (multivalued operator) Tmin and Tmax associated with the above differential equa- tion are introduced, such that Tmin is symmetric and Tmax = T_min^∗ . The minimal relation is in general not densely defined; it may happen that the relation T_min is indeed multivalued. A treatment of such systems in terms of relations goes back to B.C. Orcutt [31]; for different treatments see for instance [20], [24], [30], and [34].

Orcutt has associated with the system of differential equations a class of self-adjoint boundary value problems of the form

Ay(a) + By(b) = 0 where A and B are n× n matrices satisfying

A(iJ)A^∗ ≥ B(iJ)B^∗, ran A B

=Cⁿ.

These boundary conditions were allowed to be depending on the eigenvalue parameter by H. Langer and B. Textorius [26]. In fact, by means of an associated Q-function and the Kre˘ın-Naimark formula they associated an n× n matrix-valued spectral function to the boundary value problem. These ideas were also considered in a similar way for singular systems on a halfline in [15] and for singular 2× 2 systems in [21], [35] and [36]. A further generalization of the work of Langer and Textorius was given in [14] where also nonstandard boundary conditions (depending on the eigenvalue parameter) were allowed. Nonstandard boundary conditions are

’boundary conditions’ which may involve interior points as in interface conditions and in Stieltjes type boundary conditions. A systematic treatment of such nonstandard boundary-value problems goes back to E.A. Coddington [3] and was continued, for instance, in [4], [5], [6], [14] and [17], for a large class of formally self-adjoint differential equations or systems. For a related, more abstract, framework, see also [27], [28], [29] and [37].

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The present objective is to treat the case of nonstandard boundary conditions for canonical systems of differential equations from the point of view of extension theory via boundary triplets; cf. [19]. For singular Sturm-Liouville equations this was already done in [22] and [23]. Boundary triplets were extensively studied by V.A. Derkach and M.M. Malamud, see [8] and [9] (for the more general notion of boundary relation, see [10], [11], [12] and [13]). Boundary triplets provide a flexible way to describe (abstract) boundary value problems. Moreover, there is a uniquely defined Weyl function (generalizing the Titchmarsh-Weyl coefficient from singular Sturm-Liouville equations) which shows up in the classical Kre˘ın-Naimark formula describing all self-adjoint realizations. This Weyl function is a uniformly strict Nevanlinna functions, which uniquely determines the corresponding boundary value problems (see [25], [9], and the corresponding realizations for boundary relations in [12] and [2]).

The treatment of nonstandard boundary conditions is particularly elegant when boundary triplets are used. In general one has an ’ordinary’ boundary value problem with a symmetric operator or relation S and a self-adjoint extension A, determined by a Weyl function M (λ). In order to obtain nonstandard boundary conditions one takes a symmetric restriction S₁ of A (which need not be an extension of S), which produces a Weyl function M1(λ). The interplay between M (λ) and M1(λ) gives rise to spectral matrices associated with the nonstandard boundary value problems.

The contents of this report are as follows; in Chapter 2 we introduce notation and abstract extension theory of closed symmetric relations in terms of boundary triplets. In Chapter 3 we will study the canonical system of differential equations in detail. In particular, we will show that the maximal and minimal relation, T_max and T_min, are each others adjoint and that T_min is symmetric. These facts allows us to apply the theory of Chapter 2 to canonical systems of differential equations, the obtained results are listed in Chapter 5. In Chapter 6 we give a number of examples of (systems of) differential equations which can be interpreted as canonical systems of differential equations. Finally, to make the report more or less selfcontained, an appendix has been added concerning the existence and uniqueness of solutions of canonical systems of differential equations.

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3

Chapter 2: Preliminairy results

2.1 Notation

LetG and H be Hilbert spaces. By B(G, H) we denote the space of bounded linear operator fromG into H, if H = G we will use the notation B(G). A (linear) relation S fromG into H is a (linear) subspace of the Cartesian product G × H, if H = G we will call S a relation inG. If S is a relation in a finite-dimensional Hilbert space G the dimension of S as a subspace of G² will be called the dimension of S. For two relations S1 and S2 from G into H, their sum is defined as

S₁+ S₂={{f, g1+ g₂} : {f, g1} ∈ S1 and {f, g2} ∈ S2} and their componentwise sum as

S₁+Sˆ ₂ ={{f1+ f₂, g₁+ g₂} : {f1, g₁} ∈ S1 and {f2, g₂} ∈ S2}.

If the componentwise sum S₁+Sˆ ₂ is direct, i.e. if S₁∩ S2 = {0, 0}, we shall write S₁˙b+S2.

The inner product of a Hilbert spaceG will be denoted by (·, ·)G. For f = f₁ . . . fm where fi ∈ G, 1 ≤ i ≤ m, and g = g1 . . . gn

where gi ∈ G, 1 ≤ i ≤ n, we define

(f, g)_G=





(f₁, g₁)_G . . . (fm, g₁)_G ... . .. ... (f₁, g_n)_G . . . (f_m, g_n)_G



 .

By <·, · >_G² we denote the indefinite inner product on G² defined by

<{f, g}, {h, k} >_G²= i [(f, k)_G− (g, h)G] , {f, g}, {h, k} ∈ G².

Note that <{·, ·}, {·, ·} >H², is continuous in each of its entries, because the inner product (·, ·)G is continuous in its entries.

Using this indefinite inner product onG² we define the adjoint of a relation S inG, denoted by S^∗, as

S^∗ ={{f, g} ∈ G²: <{f, g}, {h, k} >G²= 0 for all {h, k} ∈ S}.

With this definition of S^∗ we call a relation S symmetric (or self-adjoint) if S ⊆ S^∗ (or S = S^∗). Furthermore, we call a relation S in the spaceG dissipative (or accumulative) if Im (g, f )≤ 0 (or Im (g, f) ≥ 0) for all elements {f, g} ∈ S. A dissipative

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(or accumulative) relation S is called maximal dissipative (or maximal accumulative) if it does not admit a proper dissipative (accumulative) extension in the space G.

Furthermore, for a relation S we will use the notation Nλ(S) = ker (S − λ) and b

N_λ(S) ={{f, λf} : f ∈ N^λ(S)}. Using this notation we define the deficiency index for a relation S, denoted by n_λ(S), as n_λ(S) = dim (N_λ(S^∗)), λ∈ C \ R. For closed symmetric relations the deficiency index is constant onC+ andC₋, see [1]. There- fore the deficiency index of a closed symmetric relation S is characterized by two numbers n₊(S) = n_i(S) and n₋(S) = n_¯i(S), which are called the defect numbers.

By (·, ·) we denote the usual inner product on Cⁿ, i.e. for f, g ∈ Cⁿ (f, g) = g^∗f . OnCⁿ we will use the following norms

||f||∞= max

1≤i≤n|fⁱ|, |f| = Xn

i=1

|fⁱ| and ||f|| = (f^∗f )¹² = Xn i=1

|fⁱ|²

!¹₂

, f ∈ Cⁿ. The space ofC^m×n-valued functions on (a, b) will be denoted byF^m×n((a, b))) and we will use the notationFⁿ((a, b)) =F^n×1((a, b)). By AC^m×n([a, b]) we denote the space of absolutely continuous elements of F^m×n([a, b]). Here f ∈ F^m×n([a, b]) is absolutely continuous on [a, b] if for every ǫ > 0 there exists a δ > 0 such that

Xj i=1

|f(yi)− f(xi)| < ǫ,

for every collection of disjoint segments{[x1, y₁], . . . , [x_j, y_j]}, j ∈ N, of [a, b], which satisfies

Xj i=1

|yⁱ− xⁱ| < δ.

Or, equivalently, f ∈ F^m×n([a, b]) is absolutely continuous if there exists an C^m×n- valued function g∈ L¹([a, b]) such that

f (x) = f (a) + Z x

a

g(t)dt,

i.e. f^′ exists a.e. on [a, b], see [18]. Furthermore, f ∈ AC^m×n_loc (ι), ι a bounded interval inR, if f ∈ AC^m×n([α, β]) for every [α, β]⊆ ι and we will use the notation ACⁿ_loc(ι) = AC^n×1_loc (ι).

Finally, the space of locally absolutely integrableC^m×n-valued functions on (a, b), denoted by L¹_loc((a, b)), consists of all f ∈ F^m×n((a, b)) which satisfy

Z β

α |f(t)|dt < ∞, for every [α, β]⊂ (a, b).

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2.2. Boundary triplets and associated operators 5

2.2 Boundary triplets and associated operators

In this section we will introduce boundary triplets, γ-fields and Weyl functions.

These concepts will be used to investigate closed symmetric and self-adjoint extensions of a closed symmetric relation A in a Hilbert space H. All statements in this section orginate from [1].

Definition 2.2.1. Let A be a closed symmetric relation in the Hilbert space H.

Then {G, Γ0, Γ₁} is called a boundary triplet for A^∗ if (i) G is a Hilbert space;

(ii) The mappings Γ₀, Γ₁ : A^∗ → G are such that Γ =

Γ0

Γ₁

: A^∗ → G² is surjective;

(iii) Γ satisfies the Green’s identity

<{f, g}, {h, k} >H²=< Γ{f, g}, Γ{h, k} >_G² for all{f, g}, {h, k} ∈ A^∗.

Note that Γ satisfies the Green’s identity if and only if

(g, h)H− (f, k)^H= (Γ1{f, g}, Γ0{h, k})G− (Γ0{f, g}, Γ1{h, k})G

for every{f, g}, {h, k} ∈ A^∗.

The following proposition shows how boundary triplets can be used to reduce the problem of finding extensions of the closed symmetric relation A in the ”big” space Hto a problem in the ”smaller” boundary spaceG.

Proposition 2.2.2. Let A be a closed symmetric relation in the Hilbert space H and let {G, Γ0, Γ₁} be a boundary triplet for A^∗. Then Γ induces via

Θ7→ AΘ :={{f, g} ∈ A^∗: Γ{f, g} ∈ Θ}

a bijective correspondance between the set of all closed linear relations in G and the set of closed extensions of A which are restrictions of A^∗. Moreover, (A_Θ)^∗ = A_Θ^∗ and therefore A_Θ is symmetric (or self-adjoint) if and only if Θ is symmetric (or self-adjoint). In particular ker Γ₀ and ker Γ₁ are self-adjoint.

If a closed symmetric relation A has equal defect numbers, then there exists a boundary triplet for A^∗. Therefore Proposition 2.2.2 implies that a closed symmetric relation with equal defect numbers has self-adjoint extensions. The following lemma shows that the converse is also true.

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Lemma 2.2.3. Let A be a closed symmetric relation in the Hilbert space H and let A be a self-adjoint extension of A in H. Then the decomposition˜

A^∗= ˜A ˙b+ bN_λ(A^∗) holds for all λ∈ ρ( ˜A).

The following result shows when surjective mappings Γ0and Γ1between Hilbert spaces are boundary mappings.

Theorem 2.2.4. Let H and G be Hilbert spaces and let T be a linear relation in H.

Assume that Γ =

Γ0

Γ₁

: T → G² is a linear mapping such that the conditions (i) ker Γ₀ contains a self-adjoint relation,

(ii) ran Γ =G²,

(iii) <{f, g}, {h, k} >H²=< Γ{f, g}, Γ{h, k} >G², for all {f, g}, {h, k} ∈ T , are satisfied. Then A := ker Γ is a closed symmetric relation in H and A^∗ = T . Moreover, {G, Γ0, Γ₁} is a boundary triplet for A^∗.

Associated with a boundary triplet {G, Γ0, Γ₁} are the γ-field, denoted by γλ, and Weyl function, denoted by M (λ). The following proposition contains their definition and elementary properties.

Proposition 2.2.5. Let {G, Γ0, Γ₁} be a boundary triplet for a closed symmetric relation A in the Hilbert space H. Then the γ-field γλ and Weyl function M (λ) are defined as

γλ = π₁(Γ₀|Nbλ(A^∗))⁻¹ and M (λ) = Γ₁(Γ₀|Nbλ(A^∗))⁻¹, λ∈ ρ(A0),

where π₁ is the projection defined by π₁ : {f, g} 7→ f and A0 = ker Γ₀. Further- more, we define ˆγλ as ˆγλ = (Γ0|_N_b_λ_(A∗))⁻¹. The γ-field and Weyl function have the following properties

(i) γ_λ ∈ B(G, H) for every λ ∈ ρ(A0);

(ii) For all λ, µ∈ ρ(A0) the operators γλ and γµ are connected via γλ = (IH+ (λ− µ)(A0− λ)⁻¹)γµ; (iii) For all λ∈ ρ(A0) the operator γ_λ^∗ ∈ B(H, G) satisfies

γ_λ^∗h = Γ₁{(A0− ¯λ)⁻¹h, (IH+ ¯λ(A₀− ¯λ)⁻¹h}, h∈ H;

(iv) M (λ)∈ B(G) for every λ ∈ ρ(A0);

(v) M (λ)Γ₀{fλ, λf_λ} = Γ1{fλ, λf_λ} for every fλ ∈ Nλ(A^∗) and λ∈ ρ(A0);

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2.2. Boundary triplets and associated operators 7

(vi) λ7→ M(λ) is holomorphic on ρ(A0);

(vii) For all λ, µ∈ ρ(A0) the relation

M (λ)− M(µ)^∗ = (λ− ¯µ)γ_µ^∗γλ

holds. In particular M (λ)^∗ = M (¯λ) and ^{Im M (λ)}_{Im λ} is uniformly positive for λ∈ ρ(A0).

Note that the γ-field and Weyl function are well-defined functions by Lemma 2.2.3 with ˜A = A₀. Since a function Q(λ) is a (uniform strict) Nevanlinna function if Q(¯λ) = Q(λ)^∗, λ7→ Q(λ) is holomorphic on C \ R and ^{Im M (λ)}_{Im λ} is (uniformly positive) nonnegative, the above proposition shows that the Weyl function M (λ) is an uniformly strict Nevanlinna function. The converse also holds, i.e. each uniformly strict Nevanlinna function can be realized as the Weyl funtion of a boundary triplet, see [2].

Using the γ-field and Weyl function we are able to give an explicit expression for the resolvent of any extension A_Θ of A.

Proposition 2.2.6. For any closed linear relation Θ in G let AΘ be defined as in Proposition 2.2.2. Then, with A₀ = ker Γ₀,

(AΘ− λ)⁻¹ = (A0− λ)⁻¹− γλ(M (λ) + Θ)⁻¹γ_λ^∗_¯, for λ∈ ρ(A0).

Boundary triplets for closed symmetric relations are non-unique. The following proposition shows which transformations conserve boundary mappings and how the γ-field and Weyl function change under the transformation.

Proposition 2.2.7. Let A be a closed symmetric relation in the Hilbert space H and let{G, Γ0, Γ₁} be a boundary triplet for A^∗ with associated γ-field γ_λ and Weyl function M (λ). Then the following implications hold

(i) If ˜G is a Hilbert space and

W =

W11 W12

W₂₁ W₂₂

an operator from G² onto ˜G², which satisfies W^∗

0 −iIG˜

iI_G_˜ 0

W =

0 −iIG

iI_G 0

. Then{ ˜G, ˜Γ0, ˜Γ₁}, where

˜Γ₀ Γ˜₁

:= W

Γ₀ Γ₁

,

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is another boundary triplet for A^∗. Furthermore, the γ-field and Weyl function associated with { ˜G, ˜Γ0, ˜Γ1}, λ ∈ ρ(ker Γ0)∩ ρ(ker ˜Γ0), are

˜

γλ = γλ(W₁₁+ W₁₂M (λ))⁻¹ and

M (λ) = (W˜ ₂₁+ W₂₂M (λ))(W₁₁+ W₁₂M (λ))⁻¹.

(ii) Each two boundary triplets {G, Γ0, Γ1} and { ˜G, ˜Γ0, ˜Γ1} for A^∗ are related by the surjective operator W defined as

W ={{Γ{f, g}, ˜Γ{f, g}} : {f, g} ∈ A^∗}.

2.3 The Kre˘ın-Naimark formula

In the previous section we have seen that all self-adjoint extensions of the closed symmetric relation A in a Hilbert space H are characterized by self-adjoint relations in the boundary space, see Proposition 2.2.2. In this section we will look at the families of extensions of A induced by self-adjoint exit-space extensions of A. Here a self-adjoint exit space extension is a self-adjoint relation ˜A in the space H⊕ K, where the exit space K is a Hilbert space, if ˜A restricted to H is an extension of A.

In particular, we will show that each of these families of extensions can be identified with a function of a certain class.

An exit space extension will be called minimal if there is no nontrivial decomposition of H⊕ K into H1⊕ H2, such that ˜A∩ H²1 is self-adjoint. Finally, self-adjoint relations in H which are extension of A will henceforth be called canonical self-adjoint extensions of A to dinstinguish them from self-adjoint exit-space extension.

For a self-adjoint (exit space) extension ˜A the corresponding ˘Straus family T (λ), λ∈ C \ R, of extension of A is defined as

T (λ) ={{π^Hf, πHg} : {f, g} ∈ ˜A, g− λf ∈ H}, T (∞) = {{π^Hf, πHg} : {f, g} ∈ ˜A, f ∈ H}.

The following proposition gives an equivalent characterization for ˘Straus extensions, see [16].

Proposition 2.3.1. The family of relations T (λ), λ∈ C \ R, in the Hilbert space H corresponding to a self-adjoint extension ˜A in H⊕K of the closed symmetric relation A in H is called a ˘Straus family if and only if

(i) T (λ) is maximal accumulative (maximal dissipative) for λ∈ C+ (λ∈ C−);

(ii) T (¯λ) = T (λ)^∗ for λ∈ C \ R;

(iii) The B(H)-valued function λ7→ (T (λ) + λ)⁻¹ is holomorphic on C \ R.

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2.3. The Kre˘ın-Naimark formula 9

Between the minimal self-adjoint extensions and ˘Straus families there exist (up to unitary equivalence) an one-to-one correspondence, see [16].

Another object related to self-adjoint extensions of closed symmetric relations is the generalized resolvent. With ˜A a self-adjoint extension of A, the generalized resolvent of A corresponding to ˜A is defined as

R(λ) = πH( ˜A− λ)⁻¹|^H, λ∈ C \ R,

where πH: H⊕ K → H is the projection defined by π^H:{f, g} 7→ f. The following proposition gives an equivalent characterization for generalized resolvents, see [16].

Proposition 2.3.2. The B(H)-valued function λ7→ R(λ) is a generalized resolvent of the closed symmetric relation A in the Hilbert space H corresponding to a self- adjoint extension ˜A in H⊕ K if and only if

(i) (A− λ)⁻¹ ⊂ R(λ);

(ii) For all λ∈ C \ R

R(λ)− R(λ)^∗

λ− ¯λ − R(λ)^∗R(λ)≥ 0;

(iii) R(¯λ) = R(λ)^∗;

(iv) The B(H)-valued function λ7→ (R(λ) + λ)⁻¹ is holomorphic on C \ R.

The correspondence between ˘Straus families, corresponding to minimal self- adjoint extensions of A, and generalized resolvents of A, corresponding to minimal self-adjoint extensions of A, given by

(2.1) R(λ) = (T (λ)− λ)⁻¹, λ∈ C \ R, is easily seen to be one-to-one.

Generalized resolvents can also be characterized by Nevanlinna families via the Kre˘ın-Naimark formula, see [1].

Theorem 2.3.3. Let A be a closed symmetric relation in the Hilbert space H and let {G, Γ0, Γ₁} be a boundary triplet for A^∗ with associated γ-field γ_λ and Weyl function M (λ). Then, with A0= ker Γ0, the Kre˘ın-Naimark formula

(2.2) R(λ) = (A₀− λ)⁻¹− γ^λ (M (λ) + τ (λ))⁻¹γ_¯_λ^∗, λ∈ C \ R,

establishes an one-to-one correspondence between the generalized resolvents R(λ) of A and Nevanlinna families τ (λ) in G.

Here the definition of Nevanlinna families, which is a generalization of Nevan- linna functions, is given below, see [2].

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Definition 2.3.4. A family Q(λ), λ∈ C \ R, of linear relations in the Hilbert space G is said to be a Nevanlinna family in G if

(i) Q(λ) is maximal dissipative (maximal accumulative) for λ∈ C+ (λ∈ C−);

(ii) Q(¯λ) = Q(λ)^∗;

(iii) For some, and hence for all, ν ∈ C+ (ν ∈ C−) the B(G)-valued function λ7→ (Q(λ) + ν)⁻¹ is holomorphic on C+ (C₋).

Let Q(λ), λ ∈ C \ R, be a Nevanlinna family in G and let µ ∈ C+, then the B(G)-valued functions

(2.3) λ7→ A(λ) =

(Q(λ) + µ)⁻¹, λ∈ C+, (Q(λ) + ¯µ)⁻¹, λ∈ C−, and

(2.4) λ7→ B(λ) =

I− µ(Q(λ) + µ)⁻¹, λ∈ C+, I− ¯µ(Q(λ) + ¯µ)⁻¹, λ∈ C−,

are well-defined functions by (iii) of the above definition. Furthermore, the B(G)- valued functions A(λ) and B(λ) are symmetric, meaning that A(λ) = A(¯λ)^∗ and B(λ) = B(¯λ)^∗ for λ ∈ C \ R. Using the above defined functions Q(λ), λ ∈ C \ R, can be written as

Q(λ) ={{A(λ)g, B(λ)g} : g ∈ G}, λ∈ C \ R.

We conclude that a Nevanlinna family can be represented by functions A(λ) and B(λ) which satify A(λ) = A(¯λ)^∗ and B(λ) = B(¯λ)^∗. This gives cause for the following definition, see [2].

Definition 2.3.5. A pair {Φ(λ), Ψ(λ)} of B(G)-valued functions is said to be a Nevanlinna pair in the Hilbert space G if

(i) (Im λ)Im (Ψ(λ)Φ(λ)^∗)≥ 0 for all λ ∈ C \ R;

(ii) Ψ(λ)Φ(¯λ)^∗= Φ(λ)Ψ(¯λ)^∗;

(iii) (Ψ(λ) + νΦ(λ))⁻¹ ∈ B(G) for λ, ν ∈ C±.

For a Nevanlinna family Q(λ) {−B(λ), A(λ)}, where A(λ) and B(λ) are given by (2.3) and (2.4), is a symmetric Nevanlinna pair.

Conversely, for a Nevanlinna pair{Φ(λ), Ψ(λ)} Q(λ) defined as Q(λ) ={{−Ψ(¯λ)^∗g, Φ(¯λ)^∗g} : g ∈ G}

is a Nevanlinna family. In particular, if the Nevanlinna pair{Φ(λ), Ψ(λ)} is symmetric, i.e. Φ(λ) and Ψ(λ) are symmetric, Q(λ) is given by

Q(λ) ={{−Ψ(λ)g, Φ(λ)g} : g ∈ G}.

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2.3. The Kre˘ın-Naimark formula 11

Using the above results we will rewrite the term (M (λ) + τ (λ))⁻¹, occuring in the Kre˘ın-Naimark formula, see (2.2), using a Nevanlinna pair representation of τ (λ).

I.e. with τ (λ) ={{−B(λ)g, A(λ)g} : g ∈ G}

M (λ) + τ (λ) ={{−B(λ)g, [A(λ) − M(λ)B(λ)] g} : g ∈ G}, from which it follows that

(M (λ) + τ (λ))⁻¹={{[M(λ)B(λ) − A(λ)] g, B(λ)g} : g ∈ G}

= B(λ) (M (λ)B(λ)− A(λ))⁻¹. (2.5)

Here the last equality holds for all λ∈ C \ R, see [25].

Since the generalized resolvents are in one-to-one correspondence with ˘Straus families via (2.1), the resolvent in the Kre˘ın-Naimark formula is the resolvent of a ˘Straus family. The following proposition gives a characterization of that ˘Straus family.

Proposition 2.3.6. Let T (λ), λ∈ C \ R, be the ˘Straus family corresponding to the generalized resolvent in (2.2), then

T (λ) ={{f, g} ∈ A^∗: Γ{f, g} ∈ −τ(λ)}, λ∈ C \ R.

Proof. In this proof {−B(λ), A(λ)} is any Nevanlinna pair representation of the Nevanlinna family τ (λ), λ∈ C \ R. By (2.2) and (2.5)

(T (λ)− λ)⁻¹ ={{h, (A0− λ)⁻¹h− γ^λB(λ)(M (λ)B(λ)− A(λ))⁻¹γ_λ^∗_¯h} : h ∈ H}.

Therefore with X(λ) = (M (λ)B(λ)− A(λ))⁻¹γ_λ^∗_¯

T (λ)− λ = {{(A0− λ)⁻¹h− γλB(λ)X(λ)h, h} : h ∈ H}

from which it follows that T (λ) is given by

T (λ) ={{(A0− λ)⁻¹h− γλB(λ)X(λ)h, (IH+ λ(A0− λ)⁻¹)h− λγλB(λ)X(λ)h} : h ∈ H}.

Now apply the boundary mappings to an element in T (λ). With h ∈ H we have that

Γ{(A0− λ)⁻¹h− γλB(λ)X(λ)h, (IH+ λ(A0− λ)⁻¹)h− λγλB(λ)X(λ)h}

=Γ{(A0− λ)⁻¹h, (IH+ λ(A₀− λ)⁻¹)h} + ΓˆγλB(λ)X(λ)h

={0, γ¯λ^∗h} + {B(λ)X(λ)h, M(λ)B(λ)X(λ)h} = {B(λ)X(λ)h, A(λ)X(λ)h}.

Here in the first step we used the linearity of the boundary mappings on A^∗. In the second step we used the fact that {(A0 − λ)⁻¹h, (IH+ λ(A₀ − λ)⁻¹)h} ∈ A0 and Proposition 2.2.5 (iii) for the first term and Proposition 2.2.5 (v) combined with the definition of ˆγ_λ, see also Proposition 2.2.5, for the second term.

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As a consequence of the above proposition define A_{−τ (λ)} as (2.6) A_{−τ (λ)} ={{f, g} ∈ A^∗ : Γ{f, g} ∈ −τ(λ)}.

Note that this notation is an extension of the notation in Proposition 2.2.2 to Nevanlinna families. Then using this notation and Proposition 2.3.6 the Kre˘ın- Naimark formula can be written as

(2.7) (A_{−τ (λ)}− λ)⁻¹= (A₀− λ)⁻¹− γ^λ (M (λ) + τ (λ))⁻¹γ_λ^∗_¯, λ∈ C \ R.

Using the (Nevanlinna) pair respresentation of τ (λ) and the following lemma, see [1], we will determine an alternative expression for A_{−τ (λ)}.

Lemma 2.3.7. Let A be a relation in a Hilbert space, then ker A^∗= (ran A)^⊥.

Proposition 2.3.8. Assume that τ (λ) = {{−B(λ)g, A(λ)g} : g ∈ G}, λ ∈ C \ R, then A_τ(λ)∗ can be written as

A_τ(λ)∗ ={{f, g} ∈ A^∗: A^∗(λ) B^∗(λ)

Γ{f, g} = 0}

for λ∈ C \ R. In particular, if τ(λ) is a Nevanlinna family and {A(λ), B(λ)} is a symmetric Nevanlinna pair representation, then A_{−τ (λ)} can be written as

A_{−τ (λ)}={{f, g} ∈ A^∗ : A(λ) −B(λ)

Γ{f, g} = 0}

for λ∈ C \ R.

Proof. By definition

A_τ(λ)∗ ={{f, g} ∈ A^∗ : Γ{f, g} ∈ τ(λ)^∗},

thus to prove the first part of the proposition we need to show that Γ{f, g} ∈ τ(λ)^∗ if and only if Γ{f, g} ∈ ker A(λ)^∗ B(λ)^∗

.

By definition Γ{f, g} ∈ τ(λ)^∗ if and only if < {Γ0{f, g}, Γ1{f, g}}, {h, k} >_G²= 0 for every {h, k} ∈ τ(λ). Or, equivalently, Γ{f, g} ∈ τ(λ)^∗ if and only if <

{Γ0{f, g}, Γ1{f, g}}, {−B(λ)g, A(λ)g} >_G²= 0 for every g ∈ G. Thus Γ{f, g} ∈ τ (λ)^∗ if and only if

i

({Γ0{f, g}, A(λ)g)G+ ({Γ1{f, g}, B(λ)g)G

= i

Γ{f, g},

A(λ) B(λ)

g

G²

= 0 for every g ∈ G. From the above calculation it follows that Γ{f, g} ∈ τ(λ)^∗ if and only if Γ{f, g} ∈ ran

A(λ) B(λ)

⊥

, which proves the first statement of the proposition by Lemma 2.3.7.

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2.4. Spectral theory 13

If τ (λ) is a Nevanlinna family and {A(λ), B(λ)} is a symmetric Nevanlinna pair representation, then using the first part of the proposition

A_{−τ (λ)} = A_{−τ (¯}_λ)∗ ={{f, g} ∈ A^∗: A^∗(¯λ) −B^∗(¯λ)

Γ{f, g}}

={{f, g} ∈ A^∗ : A(λ) −B(λ)

Γ{f, g}}, which proves the second part of the proposition.

2.4 Spectral theory

Given a closed symmetric relation A with boundary triplet{Cⁿ, Γ₀, Γ₁} the Kre˘ın- Naimark formula, see (2.7), allows us to determine the structure of the resolvent of any extension A_{−τ (λ)}, see (2.6), of A. In this section we will show how we can use the information on the resolvent of A_{−τ (λ)} to obtain information on the extension A_{−τ (λ)} itself.

For any self-adjoint relation Ã in a Hilbert space G mul ( Ã) = dom ( Ã)^⊥. Therefore with G₀ = mul ( Ã) we can decompose Ã into a self-adjoint operator and a multivalued part as Ã = Ão⊕ Ã_∞, where Ão = π_G⊥

0

A˜|G^⊥₀ and Ã_∞= 0G₀ × mul ( Ã). For the self-adjoint operator Ão we can use the spectral theorem, see [32].

Theorem 2.4.1. Let S be a self-adjoint operator in H, then there exists a family E(l), l ∈ R, of orthogonal projection operators on H, called the resolution of the identity, satisfying

(i) E(l) is increasing, i.e. E(l₂)− E(l1) is nonnegative for l₁ ≤ l2; (ii) E(l) is right-continuous, i.e. E(l)u→ E(l)0u as l₀< l→ l0, u∈ H;

(iii) E(l)→

0 as l→ −∞, I_H as l→ ∞;

(iv) p(S) =R

Rp(l)dE(l) for any polynomial p(l).

With Eo(l), l ∈ R, the resolution of the identity associated with ˜A₀ it follows from the boundedness of l7→ (l − λ)⁻¹, l∈ R and λ ∈ C \ R, that

( ˜Ao− λ)⁻¹ = Z

R

dEo(l)

l− λ , λ∈ C \ R, but then

( ˜A− λ)⁻¹= Z

R

dE(l)

l− λ, λ∈ C \ R, where

E(l)u = E_o(l)π_G⊥

0u⊕ 0^G0.

If we assume that the self-adjoint relation ˜A in the space G = H⊕ K is an extension of a closed symmetric relation A in the space H, then, see the previous section, the

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compression of the resolvent of ˜A is a generalized resolvent, denoted by R(λ), of A, thus

R(λ) = πH( ˜A− λ)⁻¹|^H= Z

R

πHdE(l)|^H

l− λ , λ∈ C \ R.

The family F (l) = πHE(l)|^H, l∈ R, is a so-called generalized resolution of the identity. Conversely, given the resolvent of a closed symmetric relation the corresponding generalized resolution of the identity can be obtained via the Stieltjes-Livˇsic inversion formula, see [6]. For continuity points ν, µ of F the Stieltjes-Livˇsic inversion formula says that

(2.8) ([F (ν)− F (µ)] u, u)^H= lim

ǫ→0

1 π

Z ν µ

Im (R(ξ + iǫ)u, u)Hdξ, u∈ H.

If we now assume that R(λ) has the following special form (2.9) R(λ) = G_λ+ Y_λQ(λ)Y_¯_λ^∗,

where Q(λ)∈ B(Cⁿ) is a Nevanlinna function, Gλ ∈ B(H) is entire in λ and satisfies Gλ¯ = G^∗_λ. Finally, Y_λ ∈ B(Cⁿ, H) is analytic on C \ R and Yλc ∈ Nλ(A^∗) for all c∈ Cⁿ. Then the Stieltjes-Livˇsic inversion formula implies the following result, see [6].

Theorem 2.4.2. Under the assumptions made in this section the following statements hold

(i) [F (ν)− F (µ)] f =Rν

µY_λdρ(λ)Y_λ^∗f for all f ∈ H;

(ii) Y_λ_¯^∗f ∈ L²ρ for all f ∈ H;

(iii) F (∞)f =R

RYλdρ(λ)Y_λ^∗f , where the integral converges in norm in H.

Here the operator valued function ρ(λ) defined as ρ(λ) = lim

ǫ→0+

1 π

Z λ 0

Im Q(ξ + iǫ)dξ

is non-decreasing and of bounded variation on any finite subinterval ofR. Note that the operator F (∞) is given by

F (∞) : (H ∩ G^⊥0)⊕ (H ∩ G0)→ (H ∩ G^⊥0)⊕ (H ∩ G0), {f1, f₂} 7→ {f1, 0}.

Therefore Theorem 2.4.2 (iii) gives a decomposition for f ∈ H ∩ G^⊥0. 2.5 Intermediate extensions

Proposition 2.2.2 shows that all closed symmetric (or self-adjoint) extensions A_Θof a closed symmetric relation A with boundary triplet{Cⁿ, Γ₀, Γ₁} are characterized by closed symmetric (or self-adjoint) relations Θ in the boundary space Cⁿ. In this section we will determine a boundary triplet for the adjoint A_Θ^∗ of the closed

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2.5. Intermediate extensions 15

extension A_Θ of A.

To determine the boundary triplet we will first give a characterization of the symmetric (or self-adjoint) relations Θ in the boundary space. Thereafter we will use a specific representation of Θ to construct an auxilliary boundary triplet for A^∗. In the final step we will use this auxilliary boundary triplet to construct a boundary triplet for A_Θ.

Proposition 2.5.1. Let A be a closed symmetric relation and let Θ be defined as Θ = {{−D^∗g, C^∗g}, g ∈ Cⁿ} for matrices C and D in Cⁿ. Then the following statements hold

(i) Θ is symmetric if and only if CD^∗= DC^∗;

(ii) Θ is self-adjoint if and only if Θ is symmetric and ran C D

=Cⁿ. Moreover, if Θ is self-adjoint its dimension is n.

Proof. Recall that with Θ as in statement of the proposition Θ^∗ = ker C D , see Proposition 2.3.8. Let {f, g} ∈ Θ, then {f, g} = {−D^∗x, C^∗x}, x ∈ Cⁿ and

C D

−D^∗x C^∗x

= [−CD^∗+ DC^∗] x, which shows that Θ⊂ Θ^∗ if and only if CD^∗= DC^∗.

Since we have already proven (i) to prove (ii) we only need to show that a symmetric relation Θ is sel-adjoint if and only if ran C D

=Cⁿ. Assume that ran C D

=Cⁿ, then on the one hand the dimension of ker C D , i.e. Θ^∗, is n. On the other hand using Lemma 2.3.7 and the stated assumption we have that ker C D∗

= ran C D⊥

= {0}. But if ker C D∗

= {0}, then also ker C^∗∩ ker D^∗ = {0}, i.e. Θ has dimension n. Thus Θ and Θ^∗ have equal dimension and since we assumed that Θ⊂ Θ^∗, we conclude that Θ = Θ^∗.

Next we prove the converse implication, i.e. we prove that if Θ is self-adjoint then ran C D

=Cⁿ. Assume the contrary, i.e. let 06= x ∈ ran C D⊥

, then by the previous reasoning x∈ ker C^∗∩ ker D^∗, from which it follows that the dimension of Θ is strictly smaller than n. On the other hand the dimension of ker C D

, i.e. Θ^∗, is clearly strictly bigger than n, because ran C D

( Cⁿ, a contradiction which proves (ii).

For any invertible operator T in Cⁿ the symmetric relation Θ in Cⁿ can be written as

Θ ={{−D^∗g, C^∗g}, g ∈ Cⁿ} = {{−D^∗T^∗g, C^∗T^∗g}, g ∈ Cⁿ}

={{−(T D)^∗g, (T C)^∗g}, g ∈ Cⁿ}.

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Therefore it is no restriction to assume that the representation {{−D^∗g, C^∗g}, g ∈ Cⁿ} of Θ is such that

C D

= ˜C D˜ 0 0

, where C˜ D˜

has full rank and its row are orthonormal. In the remainder of this section we will assume that Θ is in this specific form.

If the dimension of Θ, denoted by k, is strictly smaller than n, then to the above matrix there corresponds a closed symmetric extension A_Θ of A by Proposition 2.2.2. Let e^∗_α₁, . . . , e^∗_α_n−k, eα1 ∈ C²ⁿ and αi ∈ {0, . . . , n}, be orthogonal to the rows of C D

, such eα1 exist because the dimension of Θ is smaller than n, and define

C˘ D˘

=



 e^∗_α₁

... e^∗_α_n−k



 .

Then

Cˆ Dˆ

= ˜C D˜ C˘ D˘

is such that ˆΘ ={{− ˆD^∗g, ˆC^∗g}, g ∈ Cⁿ} is self-adjoint and the rows of ˆC Dˆ are orthonormal. Now define the matrix W as

W = ˆC ˆD

− ˆD Cˆ

,

then

W W^∗ = ˆC ˆD

− ˆD Cˆ

ˆC^∗ − ˆD^∗ Dˆ^∗ Cˆ^∗

= ˆC ˆC^∗+ ˆD ˆD^∗ − ˆC ˆD^∗+ ˆD ˆC^∗

− ˆD ˆC^∗+ ˆC ˆD^∗ D ˆˆD^∗+ ˆC ˆC^∗

=

In 0 0 I_n

,

which shows that W is unitairy, because W is a finite dimensional linear operator.

Moreover,

W

0 −In

In 0

W^∗= ˆC ˆD

− ˆD Cˆ

0 −In

In 0

ˆC^∗ − ˆD^∗ Dˆ^∗ Cˆ^∗

= ˆD − ˆC Cˆ Dˆ

ˆC^∗ − ˆD^∗ Dˆ^∗ Cˆ^∗

= ˆD ˆC^∗− ˆC ˆD^∗ − ˆD ˆD^∗− ˆC ˆC^∗ C ˆˆC^∗+ ˆD ˆD^∗ − ˆC ˆD^∗+ ˆD ˆC^∗

=

0 −Iⁿ I_n 0

.

Therefore

C D

Γ{f, g} = ˜C D˜ 0 0

W^∗W Γ{f, g} =

I_k 0 0 0

ΓW{f, g}, (2.10)

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2.5. Intermediate extensions 17

where Γ_W is a boundary mapping for A^∗ by Proposition 2.2.7. Its associated γ- field and Weyl function will be denoted by γλ,W and MW(λ). We recall that by Proposition 2.3.8 we can write A_Θ^∗ as

AΘ^∗={{f, g} ∈ A^∗ : C D

Γ{f, g} = 0}.

If we combine the above formula for A_Θ∗ with (2.10) we obtain the following characterization of AΘ^∗

(2.11) A_Θ∗ ={{f, g} ∈ A^∗ : [Γ_W{f, g}]i = 0, 1≤ i ≤ k}.

Using the above representation for A_Θ^∗ straight-forward calculations show that A_Θ is given by

(2.12) A_Θ={{f, g} ∈ A^∗ : [ΓW{f, g}]ⁱ = 0, 1≤ i ≤ n, n + k < i ≤ 2n}.

Finally, define the n× (n − k) matrix R as

R = e_k+1 . . . e_n ,

then R^∗R = I_n−k. Then with the introduced notation we have the following result.

Proposition 2.5.2. Let A be a closed symmetric relation with defect numbers (n, n) and let Θ be a symmetric relation in Cⁿ of dimension k. Furthermore, let R, ΓW, γλ,W and MW(λ) be as above. Then A_Θ is a closed symmetric relation with defect numbers (n− k, n − k) and its adjoint is AΘ^∗. A boundary triplet{n − k, ΓΘ,0, Γ_Θ,1} for AΘ^∗ is given by

(2.13)

Γ_Θ,0{f, g} := R^∗Γ_W,0{f, g} and ΓΘ,1{f, g} := R^∗Γ_W,1{f, g}, {f, g} ∈ AΘ^∗. Furthermore, A_Θ,0 := ker Γ_Θ,0 = ker Γ_W,0 is a self-adjoint extension of A_Θ. The γ-field and Weyl function associated with the boundary mappings in (2.13) are

γ_λ,Θ= γλ,WR and M_Θ(λ) = R^∗MW(λ)R, for λ∈ ρ(AΘ,0).

Proof. To prove the first part of this statement we use Theorem 2.2.4. Since A_Θ,0 = ker Γ_W,0, A_Θ,0 is self-adjoint by Proposition 2.2.2. The surjectivity of Γ_Θ is a direct consequence of the surjectivity of Γ_W combined with the fact that ran R^∗ =C^n−k. Finally, note that for any{f, g} ∈ AΘ^∗and z∈ Cⁿ(Γ_W,0{f, g}, z) = (R^∗Γ_W,0{f, g}, R^∗z}) = (ΓΘ,0{f, g}, R^∗z}), because [ΓW,0{f, g}]i = 0, 1≤ i ≤ k, for {f, g} ∈ AΘ^∗, see (2.11). With this observation the Green’s identity for Γ_Θ is a direct consequence of the Green’s identity for ΓW.

Finally, we show that γ_λ,Θ and M_Θ(λ) are the γ-field and Weyl function associated with the boundary mappings (2.13). Let g∈ C^n−k, then

Γ_W,0γˆ_λ,Θg = Γ_W,0ˆγλ,WRg = Rg =

0 g

,

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which shows that γ_λ,Θg∈ dom (AΘ^∗), see (2.11). Multiplying the above result from the left with R^∗ shows that ΓΘ,0γˆ_λ,Θg = g for all g ∈ C^n−k. Combined these two result prove that γ_λ,Θ is the γ-field. M_Θ(λ) is by definition, see Proposition 2.2.5, given by

M_Θ(λ) ={{ΓΘ,0ˆγ_λ,Θg, Γ_Θ,1γˆ_λ,Θg} : g ∈ C^n−k}

={{g, R^∗Γ_W,1ˆγ_λ,WRg} : g ∈ C^n−k}

={{g, R^∗M (λ)Rg} : g ∈ C^n−k} = R^∗M (λ)R.

2.6 Finite-dimensional graph restrictions

In this section we will investigate finite-dimensional graph restrictions S of a given a closed symmetric relation A in H with finite and equal defect numbers, in particular we will determine a boundary tiplet for S^∗. Here a relation S is a finite-dimensional graph restriction of A if there exist a finite-dimensional subspace Z of H² such that S = A∩ Z^∗.

To determine a boundary triplet for S^∗ we will start by determining the structure of S^∗. Thereafter we will show that it is no restriction to assume that A is self-adjoint and Z symmetric. Finally, using the specific structure of S^∗ and Z we will be able to write down a boundary triplet for S^∗.

Proposition 2.6.1. Let A be a closed symmetric relation in the Hilbert space H and Z be a finite-dimensional subspace of H² such that A^∗∩ Z = {0, 0}. Then the adjoint of S = A∩ Z^∗ is given by

S^∗= A^∗˙b+Z.

Proof. Note that we only need to prove that A^∗+Z = Sb ^∗, because by assumption A^∗ and Z are disjoint as sets. We start by proving that A^∗+Zb ⊆ S^∗. Let {f, g} ∈ A^∗+Z, i.e.b {f, g} = {f0, g₀} + {σ, τ}, where {f0, g₀} ∈ A^∗ and {σ, τ} ∈ Z. For an arbitrary element {h, k} of S = A ∩ Z^∗ we have that

(g, h)− (f, k) = (g0+ τ, h)− (f0+ σ, k) = [(g₀, h)− (f0, k)] + [(τ, h)− (σ, k)] = 0, which proves that A^∗+Zb ⊆ S^∗.

Next we will prove that S^∗ ⊆ A^∗+Z by proving the equivalent inclusion (Ab ^∗+Z)b ^∗ ⊆ S = A∩ Z^∗. These two statements are equivalent because S and A^∗+Z are closed.b Let{f, g} ∈ (A^∗+Z)b ^∗ and let{h, k} be an arbitrary element of A^∗+Z be arbitrary.b Then {h, k} = {h0, k0} + {σ, τ}, where {h0, k0} ∈ A^∗ and {σ, τ} ∈ Z and we have that

0 = (g, h)− (f, k) = (g, h0+ σ)− (f, k0+ τ ) = [(g, h₀)− (f, k0)] + [(g, σ)− (f, τ)] .

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2.6. Finite-dimensional graph restrictions 19

Since the above equality holds for all {h, k} ∈ A^∗+Z,b {h0, k₀} = {0, 0} shows that {f, g} ∈ Z^∗ and {σ, τ} = {0, 0} shows that {f, g} ∈ A^∗∗ = A. We conclude that {f, g} ∈ A ∩ Z^∗ = S, which proves the inclusion S^∗ ⊆ A^∗+Z and thereby theb proposition.

Henceforth we will only consider finite-dimensional graph restrictions of self- adjoint relations. This is no restriction as we will proceed to show. A closed symmetric relation A with equal and finite defect numbers has self-adjoint extensions, see Proposition 2.2.2, let Ã be one such a self-adjoint extension. By Lemma 2.2.3 A^∗ = Ã ˙b+ bN_λ(A^∗) for each λ ∈ ρ( Ã), thus if we investigate graph restrictions of A by a finite-dimensional subspace Z, we can equivalently investigate the graph restriction of Ã by Z∪ bN_λ(A^∗), λ∈ ρ( Ã).

Two finite-dimensional subspace Z and ˜Z of H² are called equivalent (with respect to ˜A) if

A˜∩ Z = {0, 0} = ˜A∩ ˜Z and A b˜+Z = ˜A b+ ˜Z.

The following result shows that it is no restriction to assume that Z is symmetric, see [22].

Lemma 2.6.2. Let ˜A be a self-adjoint relation in a Hilbert space H and let Z be a n-dimensional subspace of H² satisfying ˜A ∩ Z = {0, 0}. Then there exists a n-dimensional symmetric subspace ˜Z of H², which is equivalent to Z.

For a n-dimensional symmetric subspace Z of H², ˜A∩ Z = {0, 0}, fix a basis {{ϕ1, ψ₁}, . . . , {ϕⁿ, ψn}}. Then each element of ˜A b+Z can uniquely be written as

{f, g} = {f0, g₀} + {ϕ, ψ}c, {f0, g₀} ∈ ˜A and c∈ Cⁿ. Here ϕ = ϕ1 . . . ϕn

, ψ = ψ1 . . . ψn

and{ϕ, ψ}c = {ϕc, ψc}. The following theorem gives a boundary triplet for ˜A ˙b+Z and its associated γ-field and Weyl function, see [22].

Theorem 2.6.3. Let Ã be a self-adjoint relation in a Hilbert space H. Assume that Z is as above, then S = Ã∩ Z^∗ is a closed symmetric relation with defect numbers (n, n) and S^∗ = Ã ˙b+Z. A boundary triplet {Cⁿ, Γ₀, Γ₁} for S^∗ is given by

Γ₀{f, g} = c and Γ1{f, g} =< {f0, g₀}, {ϕ, ψ} >H² .

Here {f, g} = {f0, g0} + {ϕ, ψ}c ∈ S^∗, where {f0, g0} ∈ ˜A and {ϕ, ψ}c ∈ Z, c ∈ Cⁿ. Futhermore, ker Γ₀ = ˜A, ker Γ₁ = S b+Z and the γ-field and Weyl function associated with the above boundary triplet are

γλ= ϕ + ( ˜A− λ)⁻¹(λϕ− ψ) and

M (λ) = (γλ(·), ¯λϕ − ψ)^H for λ∈ ρ( ˜A). Here γλ(·) = γλe₁ . . . γλen

.

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Chapter 3: General results concerning a system of differential equations

3.1 A system of differential equations

Consider the following system of differential equations inCⁿ (3.1) Jf^′(t)− [H(t) + λ∆(t)] f(t) = ∆(t)g(t), t∈ (a, b),

where−∞ ≤ a < b ≤ ∞. Assume that the above system of differential equations satisfies the following conditions

(C1) J is a n× n matrix such that J^∗ = J⁻¹=−J;

(C2) H(t) and ∆(t) are locally absolutely integrableC^n×n-valued functions on (a, b) such that H(t) = H(t)^∗ and ∆(t) = ∆(t)^∗ a.e. on (a, b).

The endpoint a (or b) is called regular if a (or b) is finite and ∆ and H are absolutely integrable up to a (or b). An endpoint which is not regular is called singular.

The following result will be fundamental for our investigation, a proof of the statement can be found in the appendix, see [7].

Theorem 3.1.1. Let g ∈ Fⁿ((a, b)) be such that ∆g ∈ L¹loc((a, b)) and let ξ be an analytic function with values in Cⁿ. Then the initial value problem

Jf^′(t)− [λ∆(t) + H(t)] f(t) = ∆(t)g(t), f (τ ) = ξ(λ), λ∈ C and t, τ ∈ (a, b), has an unique locally absolutely continuous solution, which depends analytically on λ. If the system is regular at a (or b) and ∆g is absolutely integrable up to a (or b), then τ = a (or τ = b) is allowed and the solution is locally absolutely continuous on [a, b) (or (a, b]).

Corollary 3.1.2. For each c ∈ (a, b) there exists an unique linear operator Yλ : Cⁿ→ AC^locⁿ ((a, b)), z7→ Yλ(·)z, such that

(i) For all z∈ Cⁿ Yλ(·)z is the unique locally absolutely continuous solution of (3.2) Jf^′(t) = [H(t) + λ∆(t)] f (t), f (c) = z and t∈ (a, b);

(ii) Y_λ(·)z is entire in λ for all z ∈ Cⁿ.

Y_λ(·) ∈ AC^n×nloc ((a, b)), the matrix representation of Y_λ, will be called the fundamental matrix of (3.1). If the endpoint a (or b) is regular the operator Yλ exist for c∈ [a, b) (or c ∈ (a, b]). Finally, we define ˆY_λ by ˆY_λz ={Yλz, λY_λz} for z ∈ Cⁿ.

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3.1. A system of differential equations 21

The fundamental matrix Y_λ(·) has the following elementary properties.

Lemma 3.1.3. Let Y_λ(·) be the fundamental matrix of (3.1), see Corollary 3.1.2.

Then the following statements hold (i) Yl(t)^∗JYλ(t)− J = (λ − l)Rt

c Yl(s)^∗∆(s)Yλ(s)ds, l, λ∈ C and t ∈ (a, b);

(ii) Yλ¯(t)^∗JY_λ(t) = J = Y_λ(t)JY¯λ(t)^∗, λ∈ C and t ∈ (a, b);

(iii) Y_λ(t) is invertible for λ ∈ C and t ∈ (a, b).

Proof. To prove (i) note that by definition Y_λ(·) satisfies (3.3) JY_λ^′(s) = [H(s) + λ∆(s)] Y_λ(s), s∈ (a, b),

with initial condition Yλ(c) = In. Therefore using partial integration for absolutely continuous functions, see [18], we have that

λ Z t

c

Y_l(s)^∗∆(s)Y_λ(s)ds = Z t

c

Y_l(s)^∗JY_λ^′(s)ds− Z t

c

Y_l(s)^∗H(s)Y_λ(s)ds

=[Yl(s)^∗JYλ(s)]^t_c− Z t

c

Y_l^′(s)^∗JYλ(s)ds− Z t

c

Yl(s)^∗H(s)^∗Yλ(s)ds

=Yl(t)^∗JYλ(t)− J + Z t

c

JY_l^′(s)− H(s)Yl(s)_∗

Yλ(s)ds

=Y_l(t)^∗JY_λ(t)− J + Z t

c

[l∆(s)Y_l(s)]^∗Y_λ(s)ds

=Y_l(t)^∗JY_λ(t)− J + ¯l Z t

c

Y_l(s)^∗∆(s)Y_λ(s)ds,

which proves (i). The first equality of (ii) follows directly from (i) by taking l as ¯λ.

That equality indicates that Yλ(t) has a trivial kernel for every t∈ (a, b), because J is invertible. We conclude that Y_λ(t), t∈ (a, b), is invertible proving (iii). The second equality of (ii) is obtained by multiplying the first equality of (ii) by Yλ(t)J from the left by which, after rearrangement, the following expression is obtained

[Yλ(t)JYλ¯(t)^∗− In] JYλ(t) = 0.

Since J and Y_λ(t), t ∈ (a, b), are invertible the above equality proves the second equality of (ii).

The fundamental matrix Y_λ(·) can be used to make the solutions of the system of differential equations (3.1) explicit.

Proposition 3.1.4. Let Yλ(·) be the fundamental matrix. Then for all g ∈ Fⁿ((a, b)) for which

(Gλg)(t) = Yλ(t)J 2

Z b a

sgn (s− t)Yλ^¯(s)^∗∆(s)g(s)ds exists for all t∈ (a, b) and all λ ∈ C, the following statements hold

Boundary Triplets and Canonical Systems of Diﬀerential Equations H.L. Wietsma

Boundary Triplets and Canonical Systems of Differential Equations

H.L. Wietsma

Primairy supervisors:

Prof. H.S.V. De Snoo Dr. J. Behrndt

Secondary supervisor:

Prof. A.J. van der Schaft

Preface and acknowledgements

Contents

Chapter 1: Introduction

Chapter 2: Preliminairy results

Chapter 3: General results concerning a system of differential equations