Scalar generalized Nevanlinna functions: realizations with block operator matrices

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1 Scalar generalized Nevanlinna functions: realizations with block operator matrices Matthias Langer and Annemarie Luger Abstract. In this paper a concrete realization for a scalar generalized Nevanlinna function q N κ is given using the realizations of the factors in the basic factorization of q. Some cases are discussed in more detail and the representing operators are given as block operator matrices. Mathematics Subject Classification 2). Primary 47A6; Secondary 47A5, 47B32, 47B5. Keywords. Scalar generalized Nevanlinna function, Realization, Block operator matrix.. Introduction It is well known that a generalized Nevanlinna function q N κ for the definition of N κ see Section 2 below) possesses a realization in a Pontryagin space K, [, ] ), and hence can be written as qz) = qz ) + z z )[ I + z z )A z) )v, v ] z ϱa) where A is a self-adjoint relation in K, and with v K and z ϱa). Recently see [DLLuSh] and also [DeHS]), such realizations were constructed based on the basic factorization of q, that is qz) = r # z) q z) rz), where q N is a usual Nevanlinna function, the rational function r collects the generalized poles and zeros of q that are not of positive type, and r # z) := rz ). In these papers the realizations were constructed with the help of a matrix function which was defined using the basic factorization of q and using reproducing kernel space methods. In the present paper, however, we construct a realization in the space K = K +C 2κ, where K is a Hilbert space in which a minimal realization of q The authors gratefully acknowledge the support of the Fond zur Förderung der wissenschaftlichen Forschung FWF, Austria), grant number P554-N5.

2 2 Matthias Langer and Annemarie Luger acts, which can be chosen arbitrarily. The self-adjoint relation A in the realization of q is constructed using the realization of q and Jordan blocks connected with the generalized poles not of positive type of q. In Section 2 preliminaries are put together, in the first two subsections we recall the definitions of realizations and boundary mappings together with its basic properties, as far as we need it. In Subsection 2.3 the canonical realization, which is a particular realization in a reproducing kernel Pontryagin space, is discussed briefly. For a usual Nevanlinna function the connection between its integral representation and the realization is detailed in Subsection 2.4. Furthermore, in Subsection 2.5 we recall the basic factoriazation of a scalar generalized Nevanlinna function. Section 3 is devoted to the main result, Theorem 3., where the realization is given. The block operator representation is discussed in Section 4 and, furthermore, we give an example. 2. Preliminaries By definition a scalar function q : D C C belongs to the generalized Nevanlinna class N κ if it is meromorphic in C \ R, symmetric with respect to the real line, i.e., qz ) = qz), and if the so-called Nevanlinna kernel K q z, w) := qz) qw ) z w z, w D has κ negative squares. This means that for arbitrary numbers N N, and points z,..., z N D C + the matrices Kq z i, z j ) ) N i,j= have at most κ negative eigenvalues, and κ is minimal with this property. Here D is the domain of holomorphy of q, C + is the right half plane, and by we denote the complex conjugate of a complex number as well as the adjoint of an operator or a relation. 2.. Realizations of N κ functions Generalized Nevanlinna functions can also be characterized by their realizations. It is well known see, e.g., [KL]), that a function q belongs to the class N κ if and only if it admits a minimal realization in some Pontryagin space with negative index κ. A realization A, ϕ) for a function q is given by a self-adjoint linear relation A in a Pontryagin space K, [, ] ) and a corresponding defect function ϕz), that is a function ϕ : ϱa) K with the property ϕw) = I + w z)a w) )ϕz), such that for z, w ϱa), z w, the following identity holds qz) qw ) z w = [ϕz), ϕw)].

3 Scalar generalized Nevanlinna functions: realizations with operator matrices 3 In particular this implies the following representation for the function q: qz) = qz ) + z z )[ I + z z )A z) )v, v ] z ϱa), where z ϱa) \ R is some fixed point and v := ϕz ). The realization is called minimal if the defect elements form a total set in K, that is K = span { I + z z )A z) )v z ϱa) }. In this case D = ϱa) and the realization is unique up to unitary equivalence. Two realizations A, ϕ) and Ã, ϕ) of a function q in spaces K and K, respectively, are said to be unitarily equivalent if there exists a unitary operator Φ : K K such that {f; g} A {Φf); Φg)} Ã and Φϕz)) = ϕz). Sometimes also the triple A, S, ϕ) is called realization, where S is the symmetric restriction of A given by S := { {f; g} A [ g z f, ϕz )] = }, which is independent from the particular choice of the point z ϱa). With this notation one has ϕz) kers z). Note that for unitarly equivalent realizations A, S, ϕ) and Ã, S, ϕ) it follows that 2.2. Boundary mappings {f; g} S {Φf); Φg)} S. In this subsection we recall the notion of boundary mappings as it is contained, e.g., in [De] and [DeM]. Let S be a symmetric relation with equal defect indices in a Pontryagin space K and S its adjoint. The triple H, B, B ) is called a boundary triple for S if H is a Hilbert space with inner product, ) and B, B are linear bounded mappings from S into H such that B B is surjective onto H H and that the following abstract Lagrange or Green) identity holds: [g, f ] [f, g ] = B {f; g}, B {f ; g }) B {f; g}, B {f ; g }) for {f; g}, {f ; g } S. The B i are called boundary mappings. It is easy to show that S = ker B ker B and that A := ker B and ker B are self-adjoint relations. Define the defect subspaces by Ñ z := {{f; g} S g = zf} K K, N z := {f K {f; zf} S } = kers z). Since S = A +Ñz for z ϱa), the mapping B enz is bijective from Ñz onto H, and we can set γz) := B enz ) : H Ñ z, γz) := P γz) : H N z K,

4 4 Matthias Langer and Annemarie Luger where P denotes the projection onto the the first component in K K. The map γz) satisfies the following relation, γw) = I + w z)a w) )γz) 2.) for z, w ϱa). The Titchmarsh Weyl function M is defined by Mz) := B γz) 2.2) for z ϱa), which is an operator function in H. If S is a densely defined operator, then S is also an operator and one can set B i f := B i {f; S f} for f DS ) and i =,. The following proposition shows the connection between boundary mappings of symmetric relations with defect, ) and realizations of scalar N κ functions. Proposition 2.. If S is a symmetric relation in a Pontryagin space with defect, ), then H can be chosen to be C and the Titchmarsh Weyl function is scalar. Set ϕz) := γz) and let A = ker B as above; then A, S, ϕ) is a realization of the Titchmarsh Weyl function. Conversely, let A, S, ϕ) be a realization of an N κ function q. Decompose an element {f; g} S according to S = A +Ñz for some z ϱa)) as follows with {f ; g } A and c C, and define {f; g} = {f ; g } + c{ϕz ); z ϕz )} B {f; g} := c, B {f; g} := cqz ) + [g z f, ϕz )]. Then C, B, B ) is a boundary triple for S with the properties B {ϕz); zϕz)} =, B {ϕz); zϕz)} = qz), 2.3) and A = ker B. Hence the Titchmarsh Weyl function corresponding to C, B, B ) is equal to q. Proof. The first part follows immedeatly from 2.). That B, B in the second part are boundary mappings is a straightforward calculation. The relation {ϕz); zϕz)} = z z ){A z) ϕz ); I +za z)) ϕz )}+{ϕz ); z ϕz )}, where the first term on the right-hand side is in A, implies B {ϕz); zϕz)} = and B {ϕz); zϕz)} = qz ) + z z ) [ I + za z) )ϕz ) z A z) ϕz ), ϕz ) ] = qz ) + z z ) [ ϕz ), I + z z )A z) )ϕz ) ] = qz ) + z z )[ϕz ), ϕz )] = qz), which finishes the proof.

5 Scalar generalized Nevanlinna functions: realizations with operator matrices 5 In the second part of the proposition one could equivalently define for an element B {f; g} := c + c 2, B {f; g} := c qz ) + c 2 qz ) {f; g} = {f ; g } + c {ϕz ); zϕz )} + c 2 {ϕz ); z ϕz )} in S with {f ; g } S, c, c 2 C, and z ϱa) according to von Neumann s decomposition. We say that the boundary mappings B, B are compatible with a realization A, S, ϕ) of an N κ function q if 2.3) holds. If A, S, ϕ) and Ã, S, ϕ) are unitarily equivalent realizations of a function q N κ, where the unitary operator Φ gives the equivalence, and B, B and B, B, respectively, are boundary mappings as in Proposition 2., then B i {f; g} = B i {Φf); Φg)} for i =, The canonical realization In the proof of the main theorem we will make use of reproducing kernel spaces even for some matrix-valued generalized Nevanlinna functions and also of a particular realization in this space. Let Q : D C n n be a matrix-valued generalized Nevanlinna function, Q Nκ n n, that is, it is meromorphic in C \ R, symmetric with respect to the real line, i.e., Qz ) = Qz), and the kernel K Q z, w) := Qz) Qw ) z w z, w D 2.4) has κ negative squares. This means that for N N, points z,..., z N D C +, and vectors x,..., x N C n the matrices KQ z i, z j )x i, x j ) ) N i,j= have at most κ negative eigenvalues, and κ is minimal with this property. By LQ) denote the reproducing kernel Pontryagin space associated with the function Q: this is the closed linear span of the kernel functions K Q, z)c for z D and c C n with respect to the norm that corresponds to the inner product KQ, z)c, K Q, w)d := K Q w, z)c, d ) C n, which has κ negative squares. The elements of this space are functions, which are holomorphic on the domain of holomorphy of Q. Now we restrict ourselves again to the case of scalar generalized Nevanlinna functions. The following theorem describes the so-called canonical realization. For a detailed discussion see [DLLuSh2] and the references given there. Proposition 2.2. Let the function q N κ be given. Define the self-adjoint linear relation A q by A q := { {f; g} Lq) 2 c C : gζ) ζfζ) c },

6 6 Matthias Langer and Annemarie Luger and the symmetry S q by S q := { {f; g} Lq) 2 gζ) ζfζ) }, and set ϕ q z) := K q, z ). Then the triple A q, S q, ϕ q ) forms a minimal realization for q. Moreover, the adjoint of S q is given by S q := { {f; g} Lq) 2 c, d C : gζ) ζfζ) c d qζ) } 2.5) and boundary mappings that are compatible with the realization are given by where c and d are as in 2.5). B {f; g} := d, B {f; g} := c, 2.6) Note that whenever we refer to the canonical realization, we use the function q as subscript Realizations of N functions In this section we recall realizations for N functions which are connected with their integral representation qz) = a + bz + t z t ) + t 2 dσt), where a R, b and σ is a measure with dσt)/ + t2 ) <. We list the relations A, S and S in the space K and the defect function ϕ such that A, S, ϕ) is a realization of the function q above. Moreover, it is not difficult to determine boundary mappings that are compatible with the realization. Parts of this list where mentioned in [LLu]. We have to consider two cases.. b = : K = L 2 σ A = { {f; g} gt) = tft) } S = { {f; g} A ft)dσt) = } S = { {f; g} c C : gt) = tft) c } ϕz) = ϕz, t) = t z B {f; g} = c where c is such that gt) = tft) c) t ) B {f; g} = ac + ft) c + t 2 dσt)

7 Scalar generalized Nevanlinna functions: realizations with operator matrices 7 2. b > : K = L 2 σ C, [ f ξ), g η)] = [f, g]l 2 + bξη σ A = { { f ξ) ; g ) } η } gt) = tft), ξ = S = { { f ξ) ; g ) η } A bη + ft)dσt) = } S = { { f ξ) ; g ) } η } gt) = tft) ξ ) ϕz) = ϕz, t) = t z B { f ξ) ; g η) } = ξ B { f ξ) ; g ) t ) η } = aξ + bη + ft) ξ + t 2 dσt) If the measure σ is infinite and b = which is equivalent to the fact that S is densely defined), then S is an operator and the boundary mappings depend only on the first component. Hence one can set B i f := B i {f; S f} for f DS ) and i =,. If the measure σ is finite and b = which is equivalent to the fact that S is not densely defined but A is an operator), then the function q has also a representation as a u-resolvent, i.e., qz) = s + [A z) u, u] with s R, some self-adjoint operator A in a Hilbert space K and an element u K. A corresponding realization A, S, ϕ) and boundary mappings are given by S = { {f; g} A f, u) = } S = { {f; g} f DA), c C : g = Af cu } ϕz) = A z) u B {f; g} = c B {f; g} = sc + f, u) where c is such that g = Af cu) Here for instance one can take K = L 2 σ, A the multiplication operator by the independent variable, and u = Basic factorization A point α C { } is called a generalized pole of the function q N κ if it is an eigenvalue of the relation A in some minimal realization of q. Of particular interest are those generalized poles that are not of positive type, that is the corresponding eigenvector of the self-adjoint relation is not a positive element in K. Its degree of non-positivity ν α is the dimension of a maximal non-positive invariant subspace of the root space of A at α. A point β C { } is called a generalized zero of q not of positive type with degree of non-positivity κ β ) if it is a generalized pole of

8 8 Matthias Langer and Annemarie Luger qz) := qz) not of positive type with degree of non-positivity κ β). Note that if q N κ then also q N κ. In [DLLuSh] and also [DeHS] it was shown that every function q N κ admits a basic factorization: let the points α i, i =, 2,..., l, β j, j =, 2,..., k, respectively) be the generalized poles zeros, respectively) of q in C + R that are not of positive type, denote by ν i κ j, respectively) the degree of non-positivity of α i β j, respectively), and define Then there exists a function q N such that where r # z) := rz ). Note that if rz) := z β ) κ... z β k ) κ k z α )ν... z α. 2.7) l )ν l qz) = r # z)q z)rz), 2.8) τ := κ κ k ν ν l ) 2.9) is positive negative, respectively), then is a generalized pole zero, respectively) of q which is not of positive type and with degree of non-positivity τ. Since cannot be a generalized zero and a generalized pole at the same time, we have κ = max {κ κ k, ν ν l }. 3. The realization Let q N κ be given by its basic factorization 2.8): qz) = r # z) q z) rz). Write the rational function r as partial fractional decomposition k j= rz) = z β j) l κj l i= z = r i z) 3.) α i )νi with the functions ν ν i r z)= σ j z j σ ij and r i z)= z α for i =,..., l. i )j j= Here we assume that σ iνi for i > and σ ν if ν >. Note that κ = l ν i. Let A, S, ϕ ) be a minimal realization of the Nevanlinna function q in the Hilbert space K, [, ] ) and denote by B S, and B S, corresponding boundary mappings that are compatible with A, S, ϕ ), i.e., j= i= B S,{ϕ z); zϕ z)} =, B S,{ϕ z); zϕ z)} = q z). i=

9 Scalar generalized Nevanlinna functions: realizations with operator matrices 9 Using the above ingredients we will define a space K, relations A and S, and a function ϕz) such that A, A, ϕ) is a realization of q. To this end let us first introduce some notations. Let K := K [+]C κ +C κ ) be the Pontryagin space with the inner product [, ] given by the Gram operator G := I K I C κ. I C κ Here + denotes a direct sum and [+] a direct sum that is even orthogonal with respect to the indefinite inner product [, ]. In the following a vector h C κ will be decomposed according to 3.) as h = h h... h l ) where hi C νi for i =,,..., l. Moreover, set e :=,,..., ), a vector of suitable size, σ i := σ i,..., σ iνi ), and e z) = e z) := e i z) := z α i z α i) 2. z α i) ν i z.. z ν, and e i z) := z α i z α i )2. z α i )ν i for i =,..., l. Furthermore, denote by Jα) for α C a lower Jordan block of suitable size, α Jα) := α, 3.2) and by G i the matrix σ i σ i2... σ iνi σ i2 G i :=.... σ iνi..., which is related to r i z).

10 Matthias Langer and Annemarie Luger Theorem 3.. Let q N κ be given and let the notations be as above. Define the relation S in K as follows: f h ; F H S k K if and only if there exist constants c, c 2, c 3, c 4 C such that and {f ; F } S with B S,{f ; F } = c 3 and B S,{f ; F } = c, 3.3) h = J)H + c e, 3.4) H i = Jα i )h i + c e for i =,..., l, 3.5) k = J) K + c 4 σ, 3.6) K i = Jα i ) k i + c 4 σ i for i =,..., l, 3.7) ν j= σ j H j = K = l ν i σij h ij + c 2 c σ 3.8) i= j= l k i + c 3 c 4 σ, 3.9) i= where K = if ν =. Define the relations A and S as the restrictions of S to elements for which c 4 = and c 2 = c 4 =, respectively. Furthermore, denote by ϕz) the function rz)ϕ z) rz)q z)e i z) ) l ϕz) := G i e i z)) l i= Then the triple A, S, ϕ ) is a realization for q. Boundary mappings that are compatible with this realization are given by B x = c 4, B x = c 2 for an element x = {f h k) ; F H K) } S, where c 2 and c 4 are as in 3.3) 3.9). In particular, for ν =, i.e., is not a generalized pole not of positive type of q, relations 3.8) and 3.9) become l ν i l σij h ij + c 2 c σ = and k i + c 3 c 4 σ =. i= j= Note that the above notation is justified since from the proof we will see that the relation S is indeed the adjoint of S. Remark 3.2. This realization need not be minimal. For more details see the end of this section. i= i=.

11 Scalar generalized Nevanlinna functions: realizations with operator matrices Remark 3.3. Note that the self-adjoint relation A is independent from the numbers σ ij, and hence it does only depend on the generalized poles α j but not on the generalized zeros β j. In order to prove the theorem we will show that A, S, ϕ ) is unitarily equivalent to a triple Ã, S, ϕ ) in a Pontryagin space K which was introduced in [DLLuSh2] and shown to be a realization for q. For the convenience of the reader we recall these notations here. The unitary operator that yields the equivalence is then defined in 3.4) below. Let the matrix functions M Nκ 2 2 and Q Nκ 3 3 be defined as Mz) := ) r # z), Qz) := rz) q z) r # z) rz) Then the corresponding reproducing kernel Pontryagin space LQ) decomposes as LQ) = Lq ) LM). The self-adjoint relation à was defined as à := { { f; g} LQ)) 2 c C 3 : gζ) ζ fζ) I + Qζ)B)c }, where I is the 3 3 identity matrix and B :=. By ϕ, z) denote the functions where ϕζ, z) := Qζ) Qz) vz), ζ z vζ) := rζ). rζ)q ζ) In [DLLuSh2] it was proved that Ã, ϕ) is a not necessarily minimal realization for q. In the following lemma we identify the symmetry S and its adjoint S such that ϕ, z) LQ) is a corresponding defect function, i.e., ϕ, z) ker S z) for all z ϱã). Define S := { { f; g} à v# ζ) gζ) ζ fζ)) }. Lemma 3.4. The triple Ã, S, ϕ) is a realization of the function q in the space LQ). Moreover, the symmetry S can be written as S = { { f; g} LQ)) 2 c, c 3 C : gζ) ζ fζ) c Qζ) c 3 } 3.) c 3. c

12 2 Matthias Langer and Annemarie Luger and its adjoint is given by S = { { f; g} c,..., c 4 C : gζ) ζ fζ) c c 2 Qζ) c 3 c 4 }. 3.) c 3 c Boundary mappings that are compatible with this realization are given by for elements as in 3.). B es, { f; g} = c 4, B es, { f; g} = c 2 Proof. We first show 3.). By definition the pair { f; g} LQ)) 2 belongs to S if and only if there exists a vector c = ) c c 2 c 3 such that gζ) ζ fζ) c + Qζ)Bc with = v # ζ) gζ) ζ fζ)). The latter expression equals v # ζ) gζ) ζ fζ) ) = r # ζ) r # ζ)q ζ) ) c c 3 q ζ) c 2 c r # ζ) = c 2, which shows 3.). Note that hence S is the restriction of SQ the symmetry S Q corresponds to the canonical model for the matrix function Q, cf. [DLLuSh2, Theorem 2. iii)]), S Q := { { f; g} LQ)) 2 c, d C : gζ) ζ fζ) c Qζ)d) }, 3.2) to elements { f; g} for which c = ) c c 3 and d = Bc. In the following we use the boundary mappings for S Q, which are given by B SQ,{ f; g} = d and B SQ,{ f; g} = c, see, e.g., [DLLuSh2, Theorem 2.4]. Let { f; g} SQ as in 3.2) and { f ; g } S with B SQ,{ f ; g } = d and B SQ,{ f ; g } = c. Then g, f LQ) f, g LQ) = c, d ) C 3 d, c ) C 3 = c 3 d )c + c d 3 )c 3. For fixed { f; g} the latter expression vanishes for all { f ; g } S if and only if d = c 3 and d 3 = c, which shows 3.). It is easy to see that B es,, B es, are possible boundary mappings for S. Since rz)q z) z ϕζ, z) ζ ϕζ, z) = r # z)q z)rz) Qζ) rz) rz) rz)q z) it follows that ϕ, z) ker S z) and that B es, { ϕ, z); z ϕ, z)} =, B es, { ϕ, z); z ϕ, z)} = r # z)q z)rz) = qz), c 3 which shows that B es,, B es, are compatible with the realization Ã, S, ϕ). We define now a mapping Φ : LQ) K [+]C κ +C κ ) that will give the unitary equivalence of Ã, S, ϕ) and A, S, ϕ). Note that since it was assumed that the given realization of q is minimal, it is unitarily equivalent to the canonical

13 Scalar generalized Nevanlinna functions: realizations with operator matrices3 realization in the reproducing kernel space Lq ). Let this unitary equivalence be given by the mapping Φ : Lq ) K. According to [DLLuSh2, Theorem 3.4] an element in LQ) is of the form ν l ν i f = f, hj ζ j + j= i= j= ν hij ζ α i ) j, j= k j ζ j + l ν i i= j= k ) ij ζ α i )j 3.3) with f Lq ) and h ij, k ij C for i =,,..., l and j =,..., ν i. Then define Φ f ) Φ f) := G h i i ) l i=, 3.4) k i ) l i= here we again used the notation h i := hi )... hiνi and accordingly for the vector k i. Obviously Φ is bijective and since the inner product in LQ) is given by f F h, H = f, F l [ Lq) + Gi ki, H i ) C ν i + G h i i, K ] i ) C ν i, k K i= LQ) it follows that Φ is unitary. In the following lemma the isomorphism is applied to kernel elements. Recall that the kernel of a matrix function was defined in 2.4). Lemma 3.5. With the notation in 3.2) the following relations hold, Φ K M, z = ei z) ) ) ) l i= and Φ K M, z ) Proof. For i =,..., l we have r i ζ) r i z) ζ z ν i = j= ν i = = j= ν i k= = ) σ ij ζ α i )j z α i )j ζ z)ζ α i )j z α i )j j σ ij k= ζ α i )k z α i )k ζ α i )j k z α i )j ζ α i )j ν i k+ j= G i e i z)) l i= νi σ i,k+j z α = i )j k=. ) ζ α G i )k i e i z) k

14 4 Matthias Langer and Annemarie Luger and likewise r ζ) r z) ζ z ν = k= ν k+ ζ k j= ν σ,k+j z j = k= ζ k G e z) ) k. Hence we find Φ K M, z ) = ) G i e i z)) l i= and similarly Φ K M, z = G ) i G ) i e i z) ) l i= = ei z) ) l i=. Proof of Theorem 3.. Let again the element f LQ) be given in the form 3.3) with f Lq ) and h ij, k ij C, and similarly an element F LQ) with F Lq ) and H ij, Kij C for i =,,..., l and j =,..., ν i. Moreover, we write Φ f) =: f, h i ) l i=, k i) l i=) and Φ F ) =: F, H i ) l i=, K i) l i=). Inserting f and F in the description of S in 3.) yields that { f; F } S if and only if there exist complex numbers c, c 2, c 3 and c 4 such that ν j= ν j= H j ζ j + K j ζ j + l ν i i= j= H ij ζ α i ) j ν ζ hj ζ j ζ j= l ν i i= j= K ij ζ α i )j ν ζ k j ζ j ζ j= l ν i i= j= l ν i i= j= F ζ) ζ f ζ) = c c 3 q ζ), 3.5) hij ζ α i ) j = c 2 c r # ζ), 3.6) k ij ζ α i )j = c 3 c 4 rζ). 3.7)

15 Scalar generalized Nevanlinna functions: realizations with operator matrices5 According to 2.5) and 2.6), equation 3.5) can be written as 3.3). Comparing coefficients in the partial fractional decomposition 3.6) yields α i σ i, H i = σ i,2. h i + c for i =,..., l,... αi σi,ν i σ, h = σ,2 l. H + c... and H hi = c 2 c σ. σ,ν i= i Observing that for i =,,..., l, α α G i = G i and... α G i σ iν i σ i σi2.. =.. α for any α C and using the relations between h i, H i and h i, H i, we obtain 3.4), 3.5) and 3.8). In the same way one finds that 3.7) can be written as 3.6), 3.7) and 3.9). Hence we have found that { f; F } S if and only if {Φ f); Φ F )} S. Since B i {Φ f); Φ F )} = B es,i { f; F } for i =, cf. Lemma 3.4), B, B are possible boundary mappings and { f; F } Ã {Φ f); Φ F )} A, { f; F } S {Φ f); Φ F )} S. The defect function ϕ, z) for S is given by ϕζ, z) = Qζ) Qz) vz) = ζ z rz) qζ) qz) ζ z rz)q z) r# ζ) r # z) ζ z, rζ) rz) ζ z

16 6 Matthias Langer and Annemarie Luger and hence rz)ϕ z) Φ ϕ, z)) = rz)q z)e i z) ) l i= G i e i z)) l i= because of Lemma 3.5, where ϕ z) := Φ Kq, z ) ) is the defect function corresponding to S in K. It follows from Lemma 3.4 that A, S, ϕ) is a realization of the function q and that B, B are boundary mappings that are compatible with this realization. The compression of the resolvent of A to the Hilbert space K is given in the next proposition. Proposition 3.6. For z ρa) the following relation holds, P K A z) K = A z), where P K denotes the orthogonal projection onto the subspace K. Proof. We calculate the element F from F h = A z) f, that is k F h ; f + zf zh A. k zk From Theorem 3. it follows in particular that {F ; f + zf } S, B S,{F ; f + zf }) = c 3 3.8) and α i zk i =... k i for i =,..., l. α i Since z α i, this implies k i =, and if ν >, we also find k =. In both cases relation 3.9) gives c 3 =. But then 3.8) implies {F ; f + zf } A and hence F = A z) f. Minimality. In [DLLuSh2] it is discussed that the realization given there and hence also that given in Theorem 3. need not be minimal. This happens exactly if a zero pole) of r is a generalized pole zero) of q. But in this case there exists a positive subspace K red K such that A K [ ], ϕ ) red forms a minimal realization of q in K [ ] red. Here K[ ] red denotes the orthogonal complement of K red in K with respect to [, ]. Note that in [DLLuSh2] it was shown

17 Scalar generalized Nevanlinna functions: realizations with operator matrices7 that K [ ] red is an invariant subspace for the resolvent of A and hence the restriction should be understood in this sense, moreover ϕz) K [ ] red for z ϱa). In order to describe the space K red we need some more notations. Put α :=, β :=. In the sets I +, J + we collect those indices i j, respectively) such that the generalized pole α i zero β j, respectively) of q is also a generalized zero pole) of q : I + := { ) } { } i l α i σ p Abq, J + := j k β j σ p A q ), where again q z) = /q z). Note that the α i s and the β j s are real numbers. Moreover, define the elements ŷ αi and y βj in K such that for i I + \ {} : {ŷ αi ; α i ŷ αi } ker B S, with B S,{ŷ αi ; α i ŷ αi } =, if I + : {; ŷ α } ker B S, with B S,{; ŷ α } =, for j J + \ {} : {y βj ; β j y βj } ker B S, with B S,{y βj ; β j y βj } =, if J + : {; y β } ker B S, with B S,{; y β } =. Note that y βj and ŷ αi are eigenvectors of the self-adjoint relations A = ker B S, and ker B S,, respectively, where ker B S, is unitarly equivalent to A bq. Since the eigenspaces of ker B S, and ker B S, are one-dimensional, the elements ŷ αi and y βj are uniquely determined by the above characterization. Define the following elements in K: ŷ αi ŷ αi = δik e ) for i I +, l y βj k= y βj = ek β j ) ) l y β = y βj e) l k= k= for j J + \ {}, if J +. We can now give an explicit description of the space K red. Proposition 3.7. With the notations as above and as in the beginning of this section the Hilbert space K red such that A [ ] K, ϕ) is a minimal realization of q) is given red by K red = span{ŷ αi, y βj i I +, j J + }.

18 8 Matthias Langer and Annemarie Luger Proof. In [DLLuSh2] a space K red was constructed such that à K e[ ], ϕ) is a minimal realization of q in K red [ ] red. Here K red is spanned by the vectors where and It remains to show that x α i, i I +, q x αi and x β j r # x βj, j J +, q ζ) for i I + \ {}, x αi ζ) := ζ α i q ζ) for i = I + for j J + \ {}, x βj ζ) := ζ β j. for j = J + ŷ αi = Φ x αi q x αi ) ) and y βj = Φ x βj r # x βj ) ). 3.9) It follows from the facts that x βj is an eigenvector of A q and B S,m{y βj ; β j y βj } = B es,m {x β j ; β j x βj } for m =, that Φ x βj ) = y βj since the eigenspaces are one-dimensional and the boundary mappings yield the correct scaling. For j > Lemma 3.5 implies that r Φ # ζ) = Φ = ek β j ) ) l. 3.2) k= ζ β j r # ζ) r # β j) ζ β j If J +, then is a zero of r and hence r =. It follows from the definition of Φ that Φ r # ) = G k σ l = e) l k k= k=. 3.2) The relations Φ x βj ) = y βj and 3.2), 3.2) show the validity of the second equality in 3.9). The first equality in 3.9) is similar but even easier to show. Remark 3.8. In [DLLuSh2] it was shown that those generalized poles and zeros of the function q that contribute to the space K red can also be characterized analytically. It holds: and qz) β σ p A bq ) σ p A q ) κ β N : lim z ˆ β z β) 2κ β <, α σ p A q ) σ p A bq ) ν α N : lim z ˆ α z α) 2να qz) >.

19 Scalar generalized Nevanlinna functions: realizations with operator matrices9 4. Block operator representations of A In this section we consider some cases, where the realization A, ϕ) and in particular the description of the relation A simplify. In the first two subsections we consider the case that A is an operator, which is the case if neither q nor r have a generalized pole at infinity. There are two cases to consider, whether S the corresponding symmetric operator to q ) is densely defined or not. Finally, we give an example where A is a relation. 4.. The case that S is not densely defined We consider the case that S is not densely defined and A is an operator, that is the function q possesses a minimal u-resolvent representation q z) = s + [ A z) ] u, u 4.) with s R and u K. Moreover, we assume that rz) has no pole at infinity, i.e., ν =. Theorem 4.. Assume that q has the representation 4.) and that ν =. Then the operator A in the realization of q has the following block operator matrix representation A, e)u, e)u [, u ] e Jα ) s, e)e s, e)e A = [, u ] e Jα l ) s, e)e s, e)e, 4.2) Jα )... Jα l ) where empty blocks are zero. Moreover, the function q has the u-resolvent representation where qz) = s σ 2 + [ A z) u, u ], u := σ u s σ e s σ e σ σ l ). 4.3) Proof. According to Section 2.4 we have S = { {f ; F } DA ) H c C : F = A f cu }, 4.4) and B S,{f ; F } = c, B S,{f ; F } = s c + [f, u ],

20 2 Matthias Langer and Annemarie Luger where c is the constant appearing in 4.4), are boundary mappings which are compatible with the realization of q. Now let f h ; F H A. k K Relations 3.3) and 3.9) yield note that c 4 = ) l F = A f B S,{f ; F }u = A f c 3 u = A f + k i u = A f + and 3.5) gives l k i, e)u, i= i= H i = Jα i )h i + c e = Jα i )h i + s c 3 + [f, u ] l = Jα i )h i s k i, e) + [f, u ]. i= Relation 3.7) reduces to K i = Jα i ) k i and 3.8) gives no constraint since c 2 is arbitrary, which proves 4.2). Since ϕz) is in the domain of A, the function qz) admits again a u-resolvent representation with u = A z)ϕz). The constant s σ 2 is obtained by taking the limit z. So it remains to prove 4.3). Let u = A z)ϕz) = F H K). Then F = rz)a z)ϕ z) + = rz)u + l ν i i= j= l i= G i e i z), e) u σ ij z α i )j u = σ u, H i = rz) [ ϕ z), u ] e + rz)q z)jα i )e i z) s = rz) q z) s ) e + rz)q z)e s l k= j= = s rz)e s rz) + σ ) e = s σ e, ν k l G e k z), e) e k= k σ kj z α ) j e K i = Jα i ) G i e i z) = G i Jα i )e i z) = G i e = σ i, which finishes the proof.

21 Scalar generalized Nevanlinna functions: realizations with operator matrices2 To make the representation even more explicit, one can choose A to be the multiplication operator by the independent variable in the space L 2 σ and u =, where σ is the measure in the integral representation of q The case that S is densely defined Now we consider the case that S is a densely defined operator, that is q has the integral representation q z) = a + t z t ) + t 2 dσt), 4.5) where a R and σ is a measure with dσt) < and + t2 R R R dσt) =. As explained in Section 2.4 the operator A can be chosen to be the multiplication operator by the independent variable in L 2 σ with maximal domain, S is given by DS ) = {f L 2 σ c f : tft) c f L 2 σ} S f)t) = tft) c f and t ) B S,f := c f, B S,f := c f + ft) c f R + t 2 dσt) are boundary mappings that are compatible with the realization. Theorem 4.2. Suppose that q has the integral representation 4.5) with an infinite measure σ. Moreover, assume that ν =. Then the operator A has the following block operator representation t, e), e) B S, )e Jα ).... A = B S, )e Jα l ) 4.6) Jα )... with domain { DA) = f h k) K tf t) + i= Jα l ) l } k i, e) L 2 σ. Remark 4.3. Note that the components of vectors in the domain of A are coupled by an extra condition.

22 22 Matthias Langer and Annemarie Luger Proof. Let {f h k) ; F H K) } A. Since S is an operator, f determines F, c, and c 3 by 3.3), which in particular implies F t) = tft) c 3. Relation 3.7) reduces to K i = Jα i ) k i, 3.8) is void, and 3.9) reduces to l i= k i, e) + c 3 =, which is a constraint for elements in the domain of A An example In this subsection we consider the example q µ z) = π 2 sin πµ z)µ for µ >, µ / Z. This function appears as a Titchmarsh Weyl coefficient in connection with the Bessel operator on the half line, see [DSh]. We choose the branch such that for z C + it holds z) µ = ρ µ e iµφ π) where z = ρe iφ. Write µ = µ + 2κ with < µ < and κ Z. Then the basic factorization of q µ is given by q µ z) = r # z)q µ z)rz) with rz) = z κ. It is easy to see that q µ is an N function which has the following integral representation: t z π q µ z) = + 4 sin πµ 2 2 which for µ < reduces to q µ z) = 2 t z tµ dt. t ) + t 2 t µ dt, Since the measure σ is infinite, the operator S is given by Sf = tft) c f, where c f is such that this expression is in L 2 σ. Corresponding boundary mappings cf. Section 2.4) are given by B S,f = c f where c f is as above), π B S,f = c 4 sin πµ f t ) ft) c f + t 2 t µ dt. In the case µ < we just have B S,f = 2 ft)t µ dt. With the notation of Section 3 we have l =, ν = κ, σ = = σ,κ =, and σ κ =. Therefore there exists a realization of q µ in the space K = K [+]C κ +C κ ), where K = L 2 σ with dσ = 2 dt. The self-adjoint relation tµ A in this realization is described in the following theorem. Theorem 4.4. Let the notation be as above. With respect to the decomposition K = K + C κ + C κ the operator part A op of A has the representation A op = S J) B S, )e J) with domain DA op ) = {f h k ) k κ =, h = B S,f }. The multivalued part of A is spanned by the vector ;... ;... ).

23 Scalar generalized Nevanlinna functions: realizations with operator matrices23 Proof. Since S is an operator, the constants c and c 3 in Theorem 3. are determined by f. Equations 3.4) and 3.8) yield H = J) h + c 2... ) with c 2 C arbitrary and h = B S,f. Equations 3.6) and 3.9) give K = J)k + B S,f )e and k κ =. References [De] V. A. Derkach, On Weyl function and generalized resolvents of a Hermitian operator in a Krein space, Integral Equations Operator Theory, ), [DeH] V. A. Derkach and S. Hassi, A reproducing kernel space model for N κ-functions, Proc. Amer. Math. Soc. 3 23), no. 2, [DeHS] V. A. Derkach, S. Hassi, and H. de Snoo, Operator models associated with Kac subclasses of generalized Nevanlinna functions, Methods Funct. Anal. Topology 5 999), no., [DeM] V. A. Derkach and M. M. Malamud, The extension theory of hermitian operators and the moment problem, J. Math. Sciences, ), [DLLuSh] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, A factorization result for generalized Nevanlinna functions of the class N κ, Integral Equations Operator Theory 36 2), [DLLuSh2] A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization, to appear in: Oper. Theory Adv. Appl. 54, 69-9, Birkhäuser, Basel, 24 [DSh] A. Dijksma and Yu. Shondin, Singular Point-like Perturbations of the Bessel Operator in a Pontryagin Space, J. Differential Equations 64 2), [KL] [LLu] M. G. Krein and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr ), H. Langer and A. Luger, A class of 2 2-matrix functions. Dedicated to the memory of Branko Najman, Glas. Mat. Ser. III 3555) 2), no., Matthias Langer Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8/ 4 Wien, Austria mlanger@mail.zserv.tuwien.ac.at Annemarie Luger Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8/ 4 Wien, Austria aluger@mail.zserv.tuwien.ac.at