On a class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl–Titchmarsh theory

An important representation of the general-type fundamental solutions of the canonical systems corresponding to matrix string equations is established using linear similarity of a certain class of Volterra operators to the squared integration. Explicit fundamental solutions of these canonical systems are also constructed via the GBDT version of Darboux transformation. Examples and applications to dynamical canonical systems are given. Explicit solutions of the dynamical canonical systems are constructed as well. Three appendices are dedicated to the Weyl–Titchmarsh theory for canonical systems, transformation of a subclass of canonical systems into matrix string equations (and of a smaller subclass of canonical systems into matrix Schr¨odinger equations), and a linear similarity problem for Volterra operators.


Introduction
Canonical (spectral canonical) systems have the form w ′ (x, λ) = iλJH(x)w(x, λ), J := 0 where i is the imaginary unit (i 2 = −1), λ is the so called spectral parameter, I p is the p × p (p ∈ N) identity matrix, N stands for the set of positive integer numbers, H(x) is a 2p × 2p matrix valued function (matrix function), and H(x) ≥ 0 (that is, the matrices H(x) are self-adjoint and the eigenvalues of H(x) are nonnegative).Canonical systems are important objects of analysis, being perhaps the most important class of the one-dimensional Hamiltonian systems and including (as subclasses) several classical equations.They have been actively studied in many already classical as well as in various recent works (see, e.g., [1, 8, 13, 15, 21, 33-37, 43, 45, 50, 52] and numerous references therein).We will also consider (and construct explicit solutions) for a more general class w ′ (x, λ) = iλjH(x)w(x, λ), H(x) = H(x) * , j where m 1 , m 2 ∈ N. Here, we set H is an m × m locally integrable matrix function, and H(x) * means the complex conjugate transpose of the matrix H(x).System (1.2) will be called a generalized canonical system and the corresponding matrix function H will be called a generalized Hamiltonian.
In the case m 1 = m 2 =: p, it is easily checked that j and J are unitarily similar: that is (assuming H(x) ≥ 0), system (1.2) is equivalent to (1.1) (see Appendix B for details).We call the system w ′ (x, λ) = iλjH(x)w(x, λ), H(x) ≥ 0 (m 1 = m 2 = p) (1.4) canonical (as well as the equivalent system (1.1)).The matrix function H(x) is called the Hamiltonian of this system.In most works on canonical systems the less complicated 2 × 2 Hamiltonian case (i.e, the case p = 1) is dealt with although the cases with other values of p (p > 1) are equally important.Interesting recent works [11,30,53] on 2 × 2 canonical systems and string equations also contain some useful references.Here, we deal with the case of 2p × 2p Hamiltonians (p ≥ 1).
The normalization condition β ′ (x)jβ(x) * = iI p (1.7) for Hamiltonians of the form (1.6) is essential for the construction of fundamental solutions and solving inverse problems.In Appendix B, we show that matrix Schrödinger equations may be transformed into canonical systems (1.4), (1.6), (1.7) satisfying certain additional condition.There is considerable interest in generalized Schrödinger equations (e.g., in Schrödinger equations with distributional potentials, see some references in [12]).One can say that systems (1.4), (1.6), (1.7) present an important generalization of the of the matrix Schrödinger equations.Canonical systems with Hamiltonians satisfying (1.6), (1.7) were briefly considered in [49,50].However, local boundedness of β ′′ was required there instead of the local squareintegrability of β ′′ , which we require in the next section.In Section 2, we represent the fundamental solutions for this case as the transfer matrix function from [47,49,50].For this purpose, we use the linear similarity of the operator K = iβ(x)j x 0 β(t) * • dt to the operator (2.2) of squared integration as well as the form of the corresponding similarity transformation operator V (see Theorem C.1 and its proof in Appendix C).
The representation of the fundamental solutions in Section 2 is important in itself and (in view of the interconnections between fundamental solutions and Weyl-Titchmarsh functions) it also presents a crucial step in solving the inverse problem to recover canonical system from the Weyl-Titchmarsh function.
Some basic results and notions on the Weyl-Titchmarsh theory of the general-type canonical systems (1.4) are described in Appendix A. The results are conveniently reformulated in terms of system (1.4) instead of system (1.1), and, what is essentially more important, certain redundant conditions contained in [45, Appendix A] are removed.
In other sections of the paper we study explicit solutions of systems (1.2) with generalized Hamiltonians dj + β(x) * β(x) as well as explicit solutions and corresponding Weyl-Titchmarsh (Weyl) functions of the canonical systems (1.4), (1.6).We note that explicit solutions of Dirac systems and the corresponding Weyl-Titchmarsh theory have been studied sufficiently well (see, e.g., [20,38,45]) but the situation with the systems (1.4), (1.6) is quite different.
Explicit solutions of canonical systems and their properties are of essential theoretical and applied interest.Various versions of Bäcklund-Darboux transformations and related dressing and commutation methods [6,7,16,18,22,27,31,32,54] are fruitful tools in the construction of explicit solutions of linear and integrable nonlinear equations.Bäcklund-Darboux transformations for canonical and dynamical canonical systems, respectively, were constructed in [39] and [42].More precisely, GBDT (generalized Bäcklund-Darboux transformation) was constructed for these systems.It is important that GBDT (see, e.g., [20,27,38,44,45] and references therein) is characterized by the generalized matrix eigenvalues (not necessarily diagonal) and the corresponding generalized eigenfunctions.In Section 3, generalized matrix eigenvalues and generalized eigenfunctions are denoted by A and Λ(x), respectively.
Although GBDT for canonical systems was obtained in [39], a crucial step of constructing the generalized eigenfunctions Λ(x) (which is necessary for constructing explicitly Hamiltonians and fundamental solutions) is done in the present paper.More precisely, the procedure works in the following way.We start with some initial systems (1.2), where initial Hamiltonians H(x) are comparatively simple, and construct explicitly the fundamental solutions and generalized eigenfunctions for these systems.(In particular, some considerations from [40,44] were helpful for this purpose.)Using generalized eigenfunctions, the transformed generalized Hamiltonians and so called Darboux matrices are constructed as well.Recall that Darboux matrix for generalized canonical systems is the matrix function Υ(x, λ) satisfying the equation where H is the initial generalized Hamiltonian and H the transformed one.
In this way, we obtain fundamental solutions W for a wide class of the transformed systems (i.e., systems with the transformed generalized Hamiltonians H(x)).Indeed, it is easy to see that W is expressed via the fundamental solution W of the initial system and the Darboux matrix, namely, Some preliminaries on GBDT for the generalized canonical systems are given, and transformed generalized Hamiltonians and Darboux matrices are constructed in Section 3. Generalized eigenfunctions are constructed explicitly in Section 4. Explicit formulas for fundamental solutions of the initial systems and for the Weyl functions of the transformed canonical systems on the semi-axis [0, ∞) are established in Section 5.It is shown in Section 6 that the second equality in (1.6) (i.e., the equality β(x)jβ(x) * = 0) for the initial matrix function β(x) yields the equalities β(x)j β(x) * = 0 and β(x) ′ j β(x) * = β(x) ′ jβ(x) * for the transformed matrix function β(x).Some interesting examples are treated in Section 7.
There are important connections between spectral and dynamical characteristics as well as between spectral and dynamical systems (see, e.g., [2,5,24,41,42,51] and references therein).In particular, GBDT for the spectral canonical systems (1.4) is closely related to the GBDT for the dynamical canonical system We note that the invertibility of H(x) was assumed for the dynamical canonical system considered in [42], and system (1.8) slightly differs from the one in [42].Dynamical canonical systems are of interest in mechanics and control theory (see, e.g., [23]).The GBDT formula for Y and some explicit examples of H and Y are also discussed in Section 7.
As usual, R stands for the real axis, R + = {r : r ∈ R, r ≥ 0}, C stands for the complex plane, the open upper half-plane is denoted by C + , and a means the complex conjugate of a.The notation ℜ(a) stands for the real part of a, and ℑ(a) denotes the imaginary part of a.The notation diag{d 1 , . ..} stands for the diagonal (or block diagonal) matrix with the entries (or blocks) d 1 , . . . on the main diagonal.The space of square-integrable functions on (0, b) (0 < b ≤ ∞) is denoted by L 2 (0, b) and the corresponding space of p-dimensional column vector functions is denoted by L p 2 (0, b).By L p×q 2 (0, b) we denote the class of p × q matrix functions with the entries belonging to L 2 (0, b).The notation I stands for the identity operator.The norm A of the n × n matrix A means the norm of A acting in the space ℓ n 2 of the sequences of length n.The class of bounded operators acting from the Hilbert space H 1 into Hilbert space H 2 is denoted by B(H 1 , H 2 ), and we set B(H) := B(H, H).

General-type fundamental solutions
In this section, we study canonical system (1.4) satisfying conditions (1.6) and (1.7): (2.1) Let us consider the system (1.4), (2.1) on some finite interval [0, T] (T > 0).The linear similarity of the operators K ∈ B L p 2 (0, T) and A ∈ B L p 2 (0, T) , where is essential for us.Here, the operator A is introduced as the squared integration multiplied by −1.(Recall that in the case of Dirac systems the analog of A is the integration multiplied by i.)It is easy to see that (0, T), we have (according to Theorem C.1) K = V AV −1 , which we substitute into (2.3).Multiplying both parts of the derived equality by V −1 from the left and by (V * ) −1 from the right, we obtain the operator identity where Note that Π above is the operator of multiplication by the matrix function Π(x) and the operator V −1 is applied to β (in the expression V −1 β) columnwise.The transfer matrix function corresponding to the so called S-node (i.e., to the triple {A, S, Π} satisfying (2.4)) has the form and was first introduced and studied in [47].We introduce the projectors P ℓ ∈ B L p 2 (0, T), L p 2 (0, ℓ) : Now, we set ) Since V is a triangular operator, V −1 is triangular as wel, and we have Hence, taking into account (2.5) and (2.9) we derive It follows that (2.13) We also have P ℓ A = P ℓ AP * ℓ P ℓ .Thus, multiplying both parts of (2.4) by P ℓ from the left and by P * ℓ from the right (and using (2.9), (2.13), and the last equality in (2.5)) we obtain Clearly w A (ℓ, λ) coincides with w A (T, λ) when ℓ = T.
Remark 2.1 Relations (2.10), (2.13) and (2.14) show that S ℓ and w A (ℓ, λ) may be defined via V ℓ (and β(x) given on [0, ℓ]) precisely in the same way as w A (T, λ) is constructed via V (and β(x) given on [0, T]).Moreover, according to Remark C.2, V ℓ may be constructed in the same way as V , and so w A (ℓ, λ) does not depend on the choice of β(x) for ℓ < x < T and the choice of T ≥ ℓ.In particular, w A (ℓ, λ) is uniquely defined on the semi-axis 0 < ℓ < ∞ for β(x) considered on the semi-axis 0 ≤ x < ∞.
The fundamental solution of the canonical system (1.4),where Hamiltonian has the form (2.1) may be expressed via the transfer functions w A (ℓ, λ) using continuous factorization theorem [50, p. 40] (see also [45,Theorem 1.20] as a more convenient for our purposes presentation).
Theorem 2.2 Let the Hamiltonian of the canonical system (1.4) have the form (2.1), where β(x) is a p × 2p matrix function two times differentiable and such that β ′′ (x) ∈ L p×2p 2 (0, T), if the canonical system is considered on the finite interval [0, β], or the entries of β ′′ (x) are locally square integrable, if the canonical system is considered on [0, ∞).
Then, the fundamental solution W (x, λ) of the canonical system normalized by W (0, λ) = I 2p admits representation is the normalized fundamental solution of the canonical system (1.4) with Hamiltonian where S −1 ℓ is applied to Π ℓ (x) columnwise.Using the second equalities in (2.13) and (2.14), we rewrite (2.16) in the form and the statement of the theorem is proved on [0, T].Taking into account Remark 2.1, we see that the statement of the theorem is valid on [0, ∞) as well.
Remark 2.3 The operators S ℓ satisfying (2.14) are so called structured operators.The study of the structured operators in inverse problems takes roots in the seminal note [28] by M.G.Krein and was developed by L.A. Sakhnovich in [48][49][50].

GBDT: Darboux matrices for generalized canonical systems
Let us consider systems (1.2) on finite or semi-infinite intervals.Without loss of generality, we choose either the intervals We also fix an initial generalized Hamiltonian H(x) = H(x) * .Given an initial m × m generalized Hamiltonian H(x), each GBDT is (as usual) determined by some n × n matrices A and S(0) = S(0) * (n ∈ N) and by an n × m matrix Λ(0) which satisfy the matrix identity Taking into account the initial values Λ(0) and S(0) (and using the matrix A and the matrix function H(x)) we introduce matrix functions Λ(x) and S(x) = S(x) * via the equations: It is easy to see that (3.1) and (3.2) yield [39] the identity Remark 3.1 We note that, similar to the case of the general-type fundamental solutions in Section 2, we also use operator identities and transfer matrix function in Lev Sakhnovich form in our GBDT constructions.However, instead of the infinite-dimensional operators in Section 2, identities (3.1) and (3.3) are written for matrices.Here, we use calligraphic letter A and S instead A and S in Section 2 (and the notation Λ instead of Π) for the elements of the S-node (of the triple {A, S, Λ}).The so called Darboux matrix from Darboux transformations is represented in GBDT (for each x ) as the transfer matrix function.More precisely, we will show that in the points of invertibility of S(x) (for the case of the generalized canonical system) the Darboux matrix is expressed via (see (3.11)).The dependence of S, Λ and w A on x is of basic importance and greatly differs from the dependence of S ℓ , Π ℓ and w A (ℓ, λ) on ℓ in Section 2.
According to [39], w A (x, λ) satisfies the equation Note that (3.5) follows directly from (3.2)-(3.4).Moreover, (3.3) yields (see [39] or [45, (1.88)]): Relation (3.8) shows that, under conditions det((A − λI n ) = 0 and det(A * − λI n ) = 0, w A is invertible and Further we assume that det A = 0, ( and so w A (x, 0) is well defined (in the points of invertibility of S(x)).We note that (3.5) yields ) Thus, one can see that j H is linear similar to jH.Moreover, in view of (3.8) we can rewrite (3.13) in the form According to (3.14), the equality H(x) = H(x) * is valid.Hence, H(x) is the transformed generalized Hamiltonian of the transformed generalized canonical system Clearly, H ≥ 0 if H ≥ 0, and H > 0 if H > 0. Therefore, in the case of an initial canonical system, the transformed system is also canonical.By virtue of (3.12), a fundamental solution W of the transformed system is given by the formula where W is a fundamental solution of the initial system.
Remark 3.3 In view of (3.12) (or (3.16)) the matrix function v(x, λ) is the so called Darboux matrix of the generalized canonical system.According to (3.7), (3.8) and (3.11), the representation of v(x, λ) in terms of Λ(x) and S(x) may be simplified.Namely, we have (3.17) 4 Explicit solutions of the transformed generalized canonical systems Consider the case, where the initial generalized Hamiltonian H(x) has the form Here, β(x) is an m 1 × m matrix function, α is an m 1 × m 2 matrix function and In view of (4.1) and (4.2), we have Recall that the matrix function Λ(x) is determined by Λ(0) and by the system We construct generalized eigenfunction Λ(x) in the case (4.1) explicitly.
P r o o f.Using (4.1)-(4.6)we derive It follows from (4.5) that According to (4.6), we also have It is easy to see that one can set Q = 0 (in the Proposition 4.1) in the case c = 0.A more interesting case, where c = 0 and (4.7) holds, is generated by the matrices A and Q of the form (4.12) Then, Q satisfying (4.7) may be constructed explicitly and has the form where D is a block diagonal matrix with the blocks of the same orders as the corresponding Jordan blocks of J .Moreover, the blocks of D are upper triangular Toeplitz matrices (or scalars if the corresponding blocks of J are scalars).If z is the eigenvalue of some block of J , then the entries on the main diagonal of the corresponding block of D equal √ z (and one can fix any of the two possible values of √ z for this main diagonal).
Given generalized eigenfunction Λ(x), one can construct (explicitly) the fundamental solution W (x, λ) of the transformed generalized canonical system using relations (3.4), (3.11), (3.16) and the second equality in (3.2).We note that an explicit expression for W (x, λ), which we need for this purpose, is constructed similar to the way it is done in Proposition 5.1.
5 The case of the spectral canonical systems Here, β is the corresponding transformation of β.Setting we obtain a class of canonical systems Further in the text we normalize the fundamental solutions W and W of the systems (1.2) and (3.15), respectively, setting W (0, λ) = W (0, λ) = I 2p . (5.5) We write down the Hamiltonian H(x) given by (4.1), (4.2), and (5.3) in the form Proposition 5.1 The fundamental solution of the canonical system (1.4),where ℑ(λ) = 0, the Hamiltonian H is given by (5.6) and c = 0, has the form (5.8) P r o o f.It is easy to see that E is invertible (one may consider, for instance, the linear span of the rows of E, which coincides with C 2p ).Moreover, using the equality we have (5.9) It follows that (5.10) Relations (5.7) and (5.10) yield and for W (x, λ) of the form (5.11) we immediately obtain Taking into account (5.6) and (5.12), we see that W given by (5.7), (5.8) is, indeed, the normalized fundamental solution of the canonical system described in the proposition.
Assume that for some λ = λ 0 ∈ C + we have (5.20) and also have We will show (by contradiction) that this is impossible for sufficiently large values of ℑ z 1 (λ 0 ) .Indeed, since (5.20) implies On the other hand, taking into account that z 2 (λ) = −z 1 (λ) we similar to (5.15) derive Next, we should consider g(x, λ) = β(x)w A (x, λ)e −icxj in a more detailed way, and we note that according to (3.2), (3.4), (4.1), (4.2), (4.5), and (4.6) the entries g ik of g admit representation x ℓs e hsx P (λ) x ns e ζsx . (5.25) where P and P s are polynomials, and N 1 , P s , ℓ s and h s depend on i, k.Moreover, similar to (5.19) one can show that (excluding isolated points λ) we have where E is the "rational part" of E 2 .It follows that g(x, λ)E 2 (λ)f = 0. (5.26) Taking into account (5.24)-(5.26),we see that (5.23) (and so (5.22)) does not hold for sufficiently large values of ℑ z 1 (λ 0 ) .Since (5.22) does not hold for sufficiently large values of ℑ z 1 (λ 0 ) (excluding, may be, isolated points), there is an open domain in C + , where ϕ(λ) (given by (5.16)) is uniquely defined via (5.20).Thus, each Weyl function of our system coincides with ϕ(λ) in this domain (see the Definition A.2 of the Weyl functions).Recall that Weyl functions are holomorphic in C + .Hence, the Weyl function of our system is unique (and its existence follows from Proposition A.1).We see that the Weyl function exists, is unique and coincides with ϕ(λ) in some domain.Therefore, ϕ(λ) given by (5.16) is the Weyl function and admits holomorphic continuation in all C + .Remark 5.3 In Proposition 5.1 and Theorem 5.2, we assume that c = 0.The constructions are much simpler when c = 0.In particular, we recall that βjβ * ≡ 0 (see (4.3)).Moreover, K given in (5.6) equals β * β as c = 0. Hence, KjK = 0. Therefore, using (5.6) we have: for the case c = 0.

Matrix string equation
Consider again the case of the initial canonical systems where H(x) = β(x) * β(x) and β(x) are p × 2p matrix functions.According to (3.14), the transformed Hamiltonians (of the GBDT-transformed canonical systems (3.15)) have the form When the matrix functions β(x) have the form presented in (4.1) (and (4.2), (5.3) hold), our assertions below show (in view of Appendix B) that the considered transformed canonical systems correspond to a special subclass of string equations.Thus, our explicit formulas may be transferred for the case of string equations as explained in Remark B.3.
Recall that according to the second equalities in (4.1), (4.2), and (4.7), we have In order to define the sign of the square root above, we assume that ℑ(Q) > 0. By virtue of (4.5), (4.6)) and ( 7.1), we obtain where f 1 and f 2 are scalars and aΛ is written down more conveniently than Λ.It follows from (3.3) and ( 7.3) that and the requirement S(0) > 0 takes the form Relations (3.4), (5.2) and (7.1)-( 7.3) yield According to (3.17) and (7.1), the corresponding Darboux matrix is given by the formula Formulas (5.7), (5.14) and ( 7.3), (7.7) give explicitly fundamental solutions of the canonical systems with β of the form (7.6).In view of (5.16) and (7.7), the Weyl functions ϕ of such canonical systems on [0, ∞) have the form: ) ) (7.13) In this case, we have Hence, formulas (4.5), (4.6) and simple calculations yield In view of (7.16), the required matrix identity (3.1) may be written in the form where S ik are the entries of S. Hence, we cannot choose an arbitrary entry a in A but demand f + gqξ −1 = 0 and choose a and S 22 (0) satisfying the following conditions (which is always possible): Next, we choose S 12 (0) (and so S 21 (0) = S 12 (0)) such that (7.17) holds, and we choose such S 11 (0) > 0 that S(0) > 0.
Since ξ ∈ R, we cannot use (3.3) in order to recover S(x) from Λ(x) and construct S(x) in a different way.It follows from (4.1), (7.12) and (7.16) that Therefore, the second equality in (3.2) yields Using (7.19), we rewrite the equality (5.2) for β (transformed β) in the form where S(x), A and Λ(x) are given in (7.21), (7.12) and (7.16), respectively.Finally, the Darboux matrix v(x, λ) is expressed via Λ(x) and S(x) in (3.17), and the expression for the corresponding fundamental solution W follows from (5.14) and Remark 5.3.
In this way, explicit expressions for Λ and S in Examples 7.1 and 7.2 give us explicit expessions for Y (x, t).Moreover, it is immediate from (7.12) that e itA in (7.26) takes under assumptions of Example 7.2 a simple form e itA = e itξ I 2 + ita 0 1 0 0 .
By L 2 (H) we denote the space of vector functions on R + with the scalar product Proposition A.1 Let H(x) (x ≥ 0) be the Hamiltonian of a canonical system.Then, there is ϕ(λ), which satisfies (A.19) for this system.If (A. 19) holds, the columns of W (x, λ) P r o o f.We already proved that r>0 N (r) is non-empty.Moreover, in view of (A.14), for any ϕ satisfying (A.19) and any r > 0 we have Taking into account (A.2) and (A.27), we derive and (A.26) follows.
Proposition A.1 implies that Weyl function always exists.
B Canonical systems and matrix string and Schrödinger equations: interconnections 1.In view of (1.3), canonical systems (1.4) with Hamiltonians H(x) of the form (1.6) may be transformed into systems (1.1) with Hamiltonians H: using the transformation Clearly, the inverse transformation works as well, that is, systems (1.4), (1.6) and systems (B.1) are equivalent.

C On linear similarity to squared integration
We will consider similarity transformations of linear integral operators K in L p 2 (0, T) (0 < T < ∞): where β(x) is a p × 2p matrix function and Recall that the operator A is introduced in (2.2).The class of operators K = x 0 K(x, t) • dt, which are linear similar to A above, was studied (for the case of the scalar kernel function K(x, t)) in the essential for our considerations paper [46].Here, we study an important special subclass (C.1) of such operators under reduced smoothness conditions on K(x, t).We include the matrix case (i.e., the case p > 1) and present a complete proof of the similarity result.
Theorem C.1 Let operator K be given by the first equality in (C.1), and let β(x) satisfy the second and third equalities in (C.1).Assume that β(x) is two times differentiable and the entries of β ′′ (x) are square-integrable, that is, β ′′ (x) ∈ L p×2p 2 (0, T).Then, K is linear similar to A : where u(x) is a p × p matrix function, which is unitary (i.e., u * = u −1 ) and absolutely continuous on [0, T], and (C.4) P r o o f.In the proof, we construct an operator V , which satisfies theorem's conditions.This V is closely related to transformation operators in inverse spectral and scattering theories.
Step 1. Together with K, we consider the operators: The operator K has a semi-separable kernel, and so (see, e.g., [19,Section IX.2]) the matrix function R in (C.5) has the form where the 2p × 2p matrix function u 1 is the normalized fundamental solution of the system Introduce the p × p matrix function g(x) by the equalities where the operator (I − K) on the right-hand side of (C.9) is applied columnwise to the p × p matrix function above.Further in the proof, we study the matrix function y(x, z) := (I − z 2 K) −1 g(x).
11)(where A and Q are 2r × 2r matrices, A 12 and Q 12 are some r × r matrices) or by the block diagonal matrices with the blocks of the same form as the matrices on the right-hand sides of the equalities in(4.11).The next immediate corollary of [44, Proposition B.1] (and its proof) deals with the case c = 0. Corollary 4.2 Let c = 0, let det(2A + cI n ) = 0, and let E be the similarity transformation matrix and J Jordan normal form in the representation c(2A + cI n ) = EJ E −1 .

u 7 ( 0 (t − x) t 0 F
x)u(s) * h 1 (s)ds.(C.45)Here, u 4 is given in (C.24) and the following transformation is used:x (t, s) • dsdt = x)u(s) * h 1 (s)ds h 2 (t)u(t) • dt − x)u(s) * h 1 (s)ds h 2 (t)u(t) • dt.Since B is a triangular operator and the integral part of B has a semiseparable kernel, it easily follows (see, e.g.,[19, Section IX.2]) that B is invertible and B −1 is a bounded operator.(In fact, the integral part of B is a Volterra operator from Hilbert-Schmidt class and B −1 − I is again a triangular Volterra operator with a semi-separable kernel.)Thus, we rewrite (C.43) asy 1 = (I − z 2 B −1 A) −1 B −1 I p .(C.47)Now, it is easy to see that y 1 is unique.Recall that this unique solution admits representation (C.3), and so, taking into account (C.22), we obtainy(x, z) = V cos(zx)I p , (C.48)where V is given by the second equality in (C.3).One easily checks that(I − z 2 A) −1 I p = cos(zx)I p .(C.49)In view of (C.10), (C.48), and (C.49), we have(I − z 2 K) −1 g = V (I − z 2 A) −1 I p .(C.50)Presenting the resolvents in both parts of (C.50) as series, we rewrite (C.50) in the form K n g = V A n I p .In particular, setting n = 0, we derive g = V I p .The substitution g = V I p into K n g = V A n I p yieldsK n V I p = V A n I p (n ≥ 0).(C.51)It follows that (KV )A n I p = K(V A n I p ) = K n+1 V I p = V A n+1 I p = (V A)A n I p .(C.52)One can easily see (using, e.g., Weierstrass approximation theorem) that the closed linear span of the columns of the matrix functions A n I p (n ≥ 0) coincides with L p 2 (0, T).Therefore, (C.52) implies KV = V A, and (C.3) follows.The required properties of u and V have already been proved.Remark C.2 It is important for the study of the canonical systems on the semi-axis [0, ∞) that, according to (C.26)-(C.28),(C.30), and (C.41), the matrix function V(x, ζ) in the domain 0 ≤ ζ ≤ x ≤ ℓ is uniquely determined by β(x) on [0, ℓ] (and does not depend on the choice of β(x) for ℓ < x < T and the choice of T ≥ ℓ).