Department of Mathematics and Statistics, 10
Asymptotic expansions of generalized Nevanlinna functions and their
spectral properties Vladimir Derkach, Seppo Hassi,
and Henk de Snoo Preprint, November 2005
University of Vaasa
Department of Mathematics and Statistics P.O. Box 700, FIN-65101 Vaasa, Finland
Preprints are available at: http://www.uwasa.fi/julkaisu/sis.html
of generalized Nevanlinna functions and their spectral properties
Vladimir Derkach, Seppo Hassi and Henk de Snoo
Abstract. Asymptotic expansions of generalized Nevanlinna functionsQare investigated by means of a factorization model involving a part of the gener- alized zeros and poles of nonpositive type of the functionQ. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(SF) of the so-called generalized Friedrichs exten- sion. A classification of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymp- totic expansions. This approach results in various analytic characterizations of the spectral properties of selfadjoint relations in a Pontryagin space and, conversely, translates spectral theoretical properties into analytic properties of the associated Weyl functions.
Mathematics Subject Classification (2000).Primary 46C20, 47A06, 47B50; Sec- ondary 47A10, 47A11, 47B25.
Keywords.Generalized Nevanlinna function, asymptotic expansion, Pontrya- gin space, symmetric operator, selfadjoint extension, operator model, factor- ization, generalized Friedrichs extension.
1. Introduction
LetNκbe the class of generalized Nevanlinna functions, i.e. meromorphic functions onC\RwithQ(¯z) =Q(z) and such that the kernel
NQ(z, λ) = Q(z)−Q(λ)
z−λ¯ , z, λ∈ρ(Q), z6= ¯λ,
The research was supported by the Academy of Finland (project 212150) and the Research Institute for Technology at the University of Vaasa.
has κnegative squares on the domain of holomorphy ρ(Q) of Q, see [20]. If the functionQ∈Nκ belongs to the subclassNκ,−2n,n∈N, (see [6]) then it admits the following asymptotic expansion
Q(z) =γ−
2n+1X
j=1
sj−1
zj +o µ 1
z2n+1
¶
, zc→∞, (1.1)
where γ, sj ∈ R and zc→∞ means that z tends to ∞ nontangentially (0 < ε <
argz < π−ε < 0). Asymptotic expansions for Q ∈ Nκ of the form (1.1) (with γ= 0) were introduced in [21]. They naturally appear, for instance, in the indefi- nite moment problem considered in [22]. The expansion (1.1) is equivalent to the following operator representation of the functionQ∈Nκ,−2n:
Q(z) =γ+ [(A−z)−1ω, ω], (1.2) where ω ∈domAn and A is a selfadjoint operator in a Pontryagin space H; see [21, Satz 1.10] and Corollary 3.4 below. The representation 1.2 can be taken to be minimal in the sense thatω is a cyclic vector forA, i.e.,
H= span{(A−z)−1ω: z∈ρ(A)},
in which case the negative index sq−(H) of H is equal to κ. The representation (1.2) shows that∞is a generalized zero of the functionQ(z)−γ, or equivalently, that∞is a generalized pole of the functionQ∞(z) =−1/(Q(z)−γ). This means that the underlying symmetric operatorS is nondensely defined inHwith
domS={f ∈domA: [f, ω] = 0} (1.3) and that
SF =S+ ({0} ×b span{ω}) (1.4)
is a selfadjoint extensions of S in H with ∞ ∈ σp(SF). Here + stands for theb componentwise sum in the Cartesian productH×H. In other words, the extension SF is multivalued and, in fact, can be interpreted as the generalized Friedrichs extension of S, see [5] and the references therein. It follows from (1.1) and (1.2) that
s0= [ω, ω]∈R.
Ifκ > 0 then it is possible thats0 ≤0, in which case∞ is a generalized pole of nonpositive type (GPNT) of the function Q∞, cf. [23]. More precisely, if ∞ is a GPNT ofQ∞with multiplicityκ∞:=κ∞(Q∞) (see (2.2) below for the definition), then in (1.1) one automatically has
s0=· · ·=sj = 0, for every j <2κ∞−2.
Furthermore, if m is the first nonnegative index in (1.1) such that sm 6= 0 (if exists), then, equivalently, the function Q∞ admits an asymptotic expansion of the form
Q∞(z) =pm+1zm+1+· · ·+p2`+1z2`+1+o¡ z2`+1¢
, zc→∞, (1.5)
wherepm+1= 1/sm,pi ∈R,i=m+ 1, . . . ,2`+ 1, and the integersm,n, and`are connected by`=m−nwith m≥2`; see Theorem 5.4 below for further details.
It turns out that (1.5) holds for some `≤0 if and only if∞ is a regular critical point ofSF, or equivalently, if and only if the corresponding root subspace
R∞(SF) ={h∈H: {0, h} ∈SFk for somek∈N}
of the generalized Friedrichs extensionsSF in (1.4) is nondegenerate. In this case the GPNT ∞ of Q∞ as well as the corresponding root subspace R∞(SF) are shortly called regular. On the other hand, if ∞is a singular critical point of SF, then in (1.5)` >0 and, moreover, the minimal integer` such that the expansion (1.5) exists coincides with the dimensionκ0∞ of theisotropic subspace of the root subspace R∞(SF), see Theorem 5.6. In this case the GPNT ∞ of Q∞ and the corresponding root subspaceR∞(SF) are shortly calledsingular withthe index of singularityκ0∞.
The above mentioned results reflect the close connections between the as- ymptotic expansions (1.1), (1.5), and the root subspaceR∞(SF) ofSF. The given assertions are examples of the results in the present paper which have been derived by means of the factorization model of the functionQ∞ recently constructed by the authors in [9]. This model is based on the following “proper” factorization of the functionQ∞∈Nκ:
Q∞(z) =q(z)q](z)Q0(z), (1.6) whereqis a (monic) polynomial, q](z) =q(¯z), andQ0∈Nκ0 such that
κ∞(Q0) = 0 and κ0=κ−degq,
see Lemma 4.3 below. Such a factorization forQ∞is in general not unique, but the factorization model based on such a factorization carries the complete information about the root subspaceR∞(SF) ofSF.
A major part of the results presented in this paper is associated with the structure of the root subspace R∞(SF) of SF in a model space and the various connections to the asymptotic expansions (1.1) and (1.5). By using the factoriza- tion model based on a proper factorization (1.6) ofQ∞ maximal Jordan chains in R∞(SF) are constructed in explicit terms. Their construction leads to three dif- ferent types of maximal Jordan chains in R∞(SF). Each of these three types of maximal Jordan chains admits its own characteristic features, reflecting various properties of the root subspace R∞(SF). The construction shows explicitly, for instance, when the root subspaceR∞(SF) is regular and when it is singular. The length of the maximal Jordan chain as well as the signature of the root subspace R∞(SF) can be easily read off from their construction. In the case that the root subspaceR∞(SF) is regular, the three types of maximal Jordan chain can be char- acterized by their length. The first type of maximal Jordan chain is of length 2k+1, where k = degq = κ∞(Q∞), and the second and third type of maximal Jordan chains are of length 2kand 2k−1, respectively. The classification of these maximal Jordan chains remains the same in the case when the root subspace R∞(SF) is singular. In that case the index of singularity κ0∞ as introduced above enters to
the formulas, while the differenceκ−(R∞(SF))−κ+(R∞(SF)) of the negative and the positive index of R∞(SF) remains unaltered, see Theorem 4.12. All of these facts can be translated into the analytical properties of the functions Q∞ and Q=γ−1/Q∞ via the asymptotic expansions (1.1) and (1.5), and conversely.
The classification of maximal Jordan chains inR∞(SF) motivates an analo- gousclassification of generalized zeros and poles of nonpositive typeof the function Q∈ Nκ, which turns out to be connected with the characterization of the mul- tiplicities of GZNT and GPNT of the function Q due to H. Langer in [24]; see Subsection 3.2 for the definitions of generalized zeros and poles of types (T1)–
(T3). This induces a classification for the asymptotic expansions for the functions QandQ∞; see Theorems 5.3 and 5.4. Some further characterizations of the three different types of generalized zeros and poles are obtained by means of the fac- torized integral representations of the functions Q and Q∞, which are based of their canonical factorizations, see [11]; for definitions, see Subsection 2.1, cf. also [5]. In particular, Theorem 6.1 and Theorem 6.3 extend some earlier results by the authors in [6] (whereκ= 1) and in [8], from the regular case to the singular case in an explicit manner involving the index of singularityκ0∞, which is characterized in Theorem 5.6 below.
The construction of the maximal Jordan chains in R∞(SF) using the fac- torization model for Q∞ in (1.6) is carried out in Section 4. The most careful treatment of the model is required in the construction of maximal Jordan chains which are of the third type (T3). The reason is that the factorization ofQ∞ does not produce a minimal model for the function Q∞ directly. In the minimal fac- torization model the maximal Jordan chains of type (T3) are roughly speaking the shortest ones, cf. (4.21), (4.24), (4.28); see also Theorem 5.3 and Theorem 6.3.
The results in Lemma 4.11 and part (iii) of Theorem 4.12 characterize maximal Jordan chains of type (T3). In this case the underlying symmetric relationS(Q) is multivalued (before the auxiliary part of the space is factored out). This statement is true more generally: for an arbitraryNκ-functionQthe occurrence of general- ized zeros and poles of type (T3) inR∪ {∞}is an indication that point spectrum σp(S(Q)) ofS(Q) is nonempty, see Lemma 6.4 below, which by part (i) of Theo- rem 4.6 is equivalent toS(Q) being not simple. In fact, the existence of maximal Jordan chains of type (T3) or, equivalently, the existence of GZNT and GPNT of type (T3) can be used to give criteria for minimality of various factorization models forNκ-functions, see Propositions 6.6 and 6.7 below.
The topics considered in this paper have connections to some other recent studies involving asymptotic expansions ofNκ-functions, see in particular [6], [8], [9], [12], [13], [15], and their canonical factorization, see e.g. [3], [5], [7], [11], [14].
For instance, in [13] the authors investigate the subclass ofNκ-functions withκ= κ∞(Q) and extend some results e.g. from [6], [8]. General operator models based on the canonical factorization ofNκ-functions have been introduced in [3]; for another model not using the canonical factorization of Q, see [18]. The construction of a minimal canonical factorization model by using reproducing kernel Pontryagin space methods has been recently worked out in [12], cf. also [3, Theorem 4.1].
Some of the results in the present paper can be naturally augmented by the results which can be found from [15], where characteristic properties of the generalized zeros and poles ofNκ-functions have been studied with the aid of their operator representations.
The present paper forms a continuation of the paper [9], where the details concerning the construction of the announced factorization model can be found.
Some basic definitions and concepts which will be used throughout the paper are given in Section 2. In Section 3 some additions concerning the subclasses Nκ,−` as introduced in [6] are given, including a proof for [6, Proposition 6.2]
as announced in that paper, cf. Theorem 3.3 below; see also Theorem 5.4 for an extension of these results. Asymptotic expansions are introduced in Section 3 and a classification of generalized zeros and poles is given. In Section 4 the main ingredients concerning the factorization model are given and the construction of maximal Jordan chains in R∞(SF) is carried out. The connection between the properties of the root subspaceR∞(SF) and the asymptotic expansions of the form (1.1) and (1.5) is investigated in Section 5. Finally, in Section 6 the classification of GZNT and GPNT is connected with factorized integral representations of the functions Q and Q∞(z). In this section also the generalized zeros and poles of nonpositive type of Nκ-functions which belong toRare briefly treated and some consequences as announced above are established.
2. Preliminaries
2.1. Canonical factorization ofQ∈Nκ
The notions of generalized poles and generalized zeros of nonpositive type were introduced in [23]. The following definitions are based on [24]. A point α∈Ris called ageneralized poleof nonpositive type (GPNT) of the functionQ∈Nκwith multiplicityκα(Q) if
−∞< lim
zc→α(z−α)2κα+1Q(z)≤0, 0< lim
zc→α(z−α)2κα−1Q(z)≤ ∞. (2.1) Similarly, the point∞is called a generalized pole of nonpositive type (GPNT) of Qwith multiplicityκ∞(Q) if
0≤ lim
zc→∞
Q(z)
z2κ∞+1 <∞, −∞ ≤ lim
zc→∞
Q(z)
z2κ∞−1 <0. (2.2) A point β ∈ R is called a generalized zero of nonpositive type (GZNT) of the functionQ∈Nκ ifβ is a generalized pole of nonpositive type of the function
−1/Q. The multiplicity πβ(Q) of the GZNT β of Q can be characterized by the inequalities:
0< lim
zc→β
Q(z)
(z−β)2πβ+1 ≤ ∞, −∞< lim
zc→β
Q(z)
(z−β)2πβ−1 ≤0. (2.3)
Similarly, the point∞is called a generalized zero of nonpositive type (GZNT) of Qwith multiplicityπ∞(Q) if
−∞ ≤ lim
zc→∞z2π∞+1Q(z)<0, 0≤ lim
zc→∞z2π∞−1Q(z)<∞. (2.4) It was shown in [23] that forQ∈Nκ the total number (counting multiplic- ities) of poles (zeros) in C+ and generalized poles (zeros) of nonpositive type in R∪ {∞} is equal to κ. Let α1, . . . , αl (β1, . . . , βm) be all the generalized poles (zeros) of nonpositive type in R and the poles (zeros) in C+ with multiplicities κ1, . . . , κl (π1, . . . , πm). Then the function Q admits a canonical factorization of the form
Q(z) =r(z)r](z)Q00(z), Q00∈N0, r= pe e
q, (2.5)
where p(z) =e Qm
j=1(z−βj)πj and eq(z) = Ql
j=1(z−αj)κj are relatively prime polynomials of degreeκ−π∞(Q) andκ−κ∞(Q), respectively; see [11], [5]. It follows from (2.5) that the functionQadmits the (factorized) integral representation
Q(z) =r(z)r](z) µ
a+bz+ Z
R
µ 1
t−z− t 1 +t2
¶ dρ(t)
¶
, r= ep e
q, (2.6) wherea∈R,b≥0, andρ(t) is a nondecreasing function satisfying the integrability
condition Z
R
dρ(t)
t2+ 1 <∞. (2.7)
2.2. The subclassesNκ,1 andNκ,0
A functionQ∈Nκ is said to belong to the subclassNκ,1, if
zclim→∞
Q(z)
z = 0 and Z ∞
η
|ImQ(iy)|
y dy <∞,
withη >0 large enough. SimilarlyQ∈Nκ is said to belong to the subclassNκ,0, if
zclim→∞
Q(z)
z = 0 and lim sup
zc→∞
|zImQ(z)|<∞,
see [5]. In the following theorems the subclassesNκ,1 andNκ,0 are characterized both in terms of the integral representation (2.6) and in terms of operator repre- sentations of the form (1.2). LetEtbe a spectral function of a selfadjoint operator A in a Pontryagin space H, see [1]. Denote byH` :=H`(A), `∈N, the set of all elements h∈Hsuch thatR
∆|t|`d[Eth, h]<∞for some neighborhood ∆ of ±∞.
Moreover, letH−`(A),`∈N, be the corresponding dual spaces. Here, for instance, H−1(A) can be identified as the set of all generalized elements obtained by com- pletingHwith respect to the inner productR
∆(1 +|t|)−1d[Eth, h]<∞with some neighbourhood ∆ of±∞. The operator A admits a natural continuationAefrom HintoH−1, see [5] for further details. The classesNκ,1andNκ,0are characterized in the following two theorems, see [5].
Theorem 2.1. ([5])ForQ∈Nκ the following statements are equivalent:
(i) Qbelongs toNκ,1;
(ii) Q(z) = γ+ [(Ae−z)−1ω, ω], z∈ρ(A), for some selfadjoint operatorA in a Pontryagin space H, a cyclic vector ω∈H−1, andγ∈R;
(iii) Q has the integral representation (2.6) with degqe−degep=π∞(Q)>0, or withdegep= degqe(π∞(Q) = 0),b= 0, and
Z
R
(1 +|t|)−1dρ(t)<∞. (2.8) Theorem 2.2. ([5])ForQ∈Nκ the following statements are equivalent:
(i) Qbelongs toNκ,0;
(ii) Q(z) =γ+O(1/z),zc→∞;
(iii) Q(z) = γ+ [(A−z)−1ω, ω], z∈ρ(A), for some selfadjoint operatorA in a Pontryagin space H, a cyclic vector ω∈H, andγ∈R;
(iv) Q has the integral representation (2.6)with degqe−degpe=π∞(Q) >0, or withdegep= degqe(π∞(Q) = 0),b= 0, and
Z
R
dρ(t)<∞. (2.9)
Remark 2.3. If Q ∈ Nκ,0 then the operator representation of Q in part (iii) of Theorem 2.2 implies that
zclim→∞−z(Q(z)−γ) = [ω, ω].
Hence, the statement (ii) in Theorem 2.2 can be strengthened in the sense that for every functionQ∈Nκ,0 there are real numbersγ ands0, such that
Q(z) =γ−s0
z +o µ1
z
¶
, zc→∞. (2.10)
3. Asymptotic expansions of generalized Nevanlinna functions
Asymptotic expansions of generalized Nevanlinna functions (as in (2.10)) can be used for studying operator and spectral theoretical properties of selfadjoint exten- sions of symmetric operators in Pontryagin and Hilbert spaces, see [20], [16]. In this section a subdivision of the class Nκ of generalized Nevanlinna functions is given along the lines of [16], [6]. Moreover, a classification for generalized zeros of nonpositive type is introduced and interpreted via asymptotic expansions.
3.1. The subclassesNκ,−` of generalized Nevanlinna functions
Definition 3.1. A functionQ∈Nκis said to belong to the subclassNκ,−2n,n∈N, if there are real numbersγ ands0, . . . , s2n−1 such that the function
Q(z) =e z2n
Q(z)−γ+ X2n
j=1
sj−1
zj
(3.1)
isO(1/z) aszc→∞. Moreover,Q∈Nκis said to belong to the subclassNκ,−2n+1
if the functionQe in (3.1) belongs toNκ0,1 for someκ0 ∈N.
The next lemma clarifies the above definition of the subclassesNκ,−`,`∈N.
Lemma 3.2. If the functionQbelongs to the subclassNκ,−2n (Nκ,−2n+1) for some n∈N, then the functionQe in(3.1)belongs to the subclassNκ0,0 (resp.Nκ0,1) with κ0≤κ. Moreover, the following inclusions are satisfied
· · · ⊂Nκ,−2n−1⊂Nκ,−2n⊂Nκ,−2n+1⊂ · · · ⊂Nκ,0⊂Nκ,1. (3.2) Proof. Rewrite the expression for the function Qe in (3.1) in the form Q(z) =e z2nQ(z). Thenb Q(z) as a sum of two generalized Nevanlinna functions is also ab generalized Nevanlinna function, and therefore, in view of (2.5),Qeis a generalized Nevanlinna function, too. Next it is shown that the inequality κ(Q)e ≤ κ(Q) is satisfied. First observe that the conditionQ(z) =e O(1) and hence, in particular, the conditionQ(z) =e O(1/z) as zc→∞implies that
κ∞(Q) = 0,e (3.3)
cf. (2.2), (2.4). Clearly, κα(Q) =e κα(Q) for everyα6= 0,∞, while for α= 0 one derives from (2.1) the estimate
κ0(Q)e ≤κ0(Q). (3.4)
Therefore, one can conclude from (3.3) and (3.4) that κ(Q)e ≤ κ(Q). Now by Theorem 2.2 the condition Q(z) =e O(1/z), zc→∞, is equivalent to Qe ∈ Nκ0,0
with κ0 ≤ κ, which proves the first statement for the subclasses Nκ,−2n. If Q ∈ Nκ,−2n+1, thenQ(z) =e O(1) and sinceκ(Q)e ≤κ(Q), one actually hasQe∈Nκ0,1
forκ0 ≤κ.
Since Nκ,0 ⊂Nκ,1 the inclusions Nκ,−2n ⊂Nκ,−2n+1, n ∈ N, follow from the first part of the lemma. Now letQ∈Nκ,−2n−1. Then by definition
z2Q(z) +e zs2n+s2n+1∈Nκ0,1, (3.5) where Qe is as in (3.1) and κ0 ≤κ. It is clear from (3.5) (see Theorem 2.1) that Q(z) =e O(1/z) as zc→∞. Hence, Q ∈ Nκ,−2n and this proves the remaining
inclusions in (3.2). ¤
The subclassesNκ,−`,`∈N, are now characterized by means of the operator and the integral representation ofQin (1.2) and (2.6), respectively.
Theorem 3.3. ForQ∈Nκ the following statements are equivalent:
(i) Q∈Nκ,−`,`∈N;
(ii) Q(z) = γ+ [(A−z)−1ω, ω], z∈ρ(A), for some selfadjoint operatorA in a Pontryagin space H, a cyclic vector ω∈H`, andγ∈R;
(iii) Qhas an integral representation (2.6)withπ∞(Q) = degqe−degpe≥0 (and b= 0if π∞(Q) = 0), such that
Z
R
(1 +|t|)`−2π∞dρ(t)<∞. (3.6) Proof. (i)⇒(iii) LetQ∈Nκ,−`, where`is either 2nor 2n−1,n∈N. In view of (3.2) and Theorems 2.1, 2.2 one hasπ∞(Q) = degqe−degpe≥0, and ifπ∞(Q) = 0 then b = 0 and (2.8) or (2.9) is satisfied. By Lemma 3.2 the functionQe in (3.1) belongs toNκ0,2n−`withκ0≤κ. Hence,Qe admits the factorization
Qe=er(z)er](z) µ
e a+ebz+
Z
R
µ 1
t−z − t 1 +t2
¶¶
deρ(t), er=pe2
e
q2, (3.7) where pe2 and eq2 are the polynomials associated to Q, cf. (2.5). Moreover, thee inequality π∞(Q) = dege qe2−degpe2 ≥0 holds by Theorems 2.1 and 2.2. On the other hand, it follows from (2.6) and (3.1) thatQe admits also the representation
Q(z) =e z2nr(z)r](z) µ
a+bz+ Z
R
µ 1
t−z− t 1 +t2
¶¶
dρ(t) +p1(z), (3.8) where p1 is a polynomial with degp1 ≤ 2n. An application of the generalized Stieltjes inversion formula (see [19]) shows that the measures deρ(t) in (3.7) and dρ(t) in (3.8) are connected by
|er(t)|2deρ(t) =t2n|r(t)|2dρ(t). (3.9) Therefore, ifQe∈Nκ0,1\Nκ0,0 so that`= 2n−1, then degpe2= degqe2 anddeρ(t) satisfies the condition (2.8) in Theorem 2.1. The condition (3.6) follows now from (3.9). IfQe∈Nκ0,0 so that`= 2n, then either degpe2= degeq2 in which casedeρ(t) satisfies the condition (2.9) in Theorem 2.2, or π∞(Q) = dege qe2−degpe2 > 0 in which casedeρ(t) satisfies the condition (2.7). In both cases
Z
|t|>M
|er(t)|2deρ(t)<∞, forM >0 large enough.
Hence, again the condition (3.6) follows from (3.9).
(ii)⇔(iii) LetEt be the spectral function of a selfadjoint operatorAin the minimal representation (1.2) ofQ. It follows from (1.2), (2.6), and the generalized Stieltjes inversion formula that
d[Etω, ω] =|r(t)|2dρ(t), t∈∆, in some neighbourhood ∆ of±∞. This implies that
Z
R
(1 +|t|)`−2π∞dρ(t)<∞ if and only if Z
∆
(1 +|t|)`d[Etω, ω]<∞, i.e.,ω∈H`, which proves the equivalence of (ii) and (iii).
(ii) ⇒ (i) First consider the case ` = 2n. Then ω ∈ H` means that ω ∈ domAn. Define the functionQe in (3.1) by setting
sj = [Ajω, ω], sn+j= [Ajω, Anω], j= 0, . . . , n. (3.10) Then a straightforward calculation shows thatQeadmits the operator representa- tion Q(z) = [(Ae −z)−1ω0, ω0], ω0=Anω∈H. (3.11) Therefore,Q(z) =e O(1/z) andQ∈Nκ,−`.
Now let ` = 2n−1. Then ω ∈ H` means that ω0 := Anω ∈ H−1. Hence s2n−1:= [Aωe 0, ω0] is well defined. Moreover, by definings0, . . . , s2n−2 as in (3.10) it follows that the functionQein (3.1) admits the operator representation
Q(z) = [(e Ae−z)−1ω0, ω0], ω0=Anω∈H−1. (3.12) Hence, by Theorem 2.1 Qe ∈ Nκ0,1 for some κ0 ∈ N and thus Q ∈ Nκ,−`. This
completes the proof. ¤
From Theorem 3.3 one obtains the following result of M.G. Kre˘ın and H.
Langer, see [21, Satz 1.10]
Corollary 3.4. ([21]) The function Q ∈ Nκ admits an operator representation Q(z) = γ+ [(A−z)−1ω, ω] with γ ∈ R and ω ∈ H2n(= domAn) if and only if there are real numbersγ ands0, . . . , s2n, such that
Q(z) =γ−
2n+1X
j=1
sj−1
zj +o µ 1
z2n+1
¶
, zc→∞. (3.13)
In this case the numberss0, . . . , s2n are given by (3.10).
Proof. The proof of Theorem 3.3 shows that the conditionω ∈domAn is equiv- alent to the operator representation (3.11) of the function Q(z) in (3.1). Nowe by applying (2.10) in Remark 2.3 to the functionQ(z) in (3.11) and taking intob account (3.1) the equivalence to the expansion (3.13) follows. ¤ The criterion of M.G. Kre˘ın and H. Langer in Corollary 3.4 does not hold in the case of an odd number`= 2n−1. However, it is clear that ifω∈H2n−1then the analog of the expansion (3.13) exists.
Corollary 3.5. If the function Q∈Nκ admits an operator representationQ(z) = γ+ [(A−z)−1ω, ω] with γ ∈ R and ω ∈ H2n−1(= dom|A|n−1/2) then there are real numbersγ ands0, . . . , s2n, such that
Q(z) =γ− X2n
j=1
sj−1
zj +o µ 1
z2n
¶
, zc→∞. (3.14)
Proof. Sinceω ∈ H2n−1 the operator representation (3.12) in the proof of Theo- rem 3.3 shows thatQ(z) =e o(1). The expansion (3.14) for the functionQfollows
now from the definition ofQein (3.1). ¤
It is emphasized that the existence of the expansion (3.14) does not imply that ω ∈ H2n−1 or, equivalently, that ω0 := Anω belongs to H−1. In this case [Aωe 0, ω0] need not be defined and hence it cannot coincide with the coefficient s2n−1in (3.14).
3.2. A classification of generalized zeros of nonpositive type
For what follows it will be useful to give a classification for generalized zeros and poles of nonpositive type of a functionQ∈Nκ.
Let∞be a GZNT ofQwith multiplicity π∞>0. It follows from (2.4) that precisely one of the following three cases can occur:
(T1) −s2π∞:= limzc→∞z2π∞+1Q(z)<0, limzc→∞z2π∞−1Q(z) = 0;
(T2) limzc→∞z2π∞+1Q(z) =∞, limzc→∞z2π∞−1Q(z) = 0;
(T3) limzc→∞z2π∞+1Q(z) =∞, −s2π∞−2:= limzc→∞z2π∞−1Q(z)>0.
In these cases∞is said to be a generalized zero of type (T1), (T2), or (T3), respec- tively; the shorter notations GZNT1, GZNT2, and GZNT3 are used accordingly.
The corresponding classification for a finite generalized zeroβ∈RofQis defined analogously:
(T1) limzc→β Q(z)
(z−β)2πβ+1 >0, limzc→β Q(z)
(z−β)2πβ−1 = 0;
(T2) limzc→β Q(z)
(z−β)2πβ+1 =∞, limzc→β Q(z)
(z−β)2πβ−1 = 0;
(T3) limzc→β Q(z)
(z−β)2πβ+1 =∞, limzc→β Q(z)
(z−β)2πβ−1 <0.
A generalized pole of nonpositive type β ∈ R∪ {∞} of Q is said to be of type (T1), (T2) or (T3), ifβ is a generalized zero of nonpositive type of the function
−1/Qwhich is of type (T1), (T2) or (T3), respectively.
To give some immediate implications of the above classification consider the generalized zero∞ of Q. If it is of the first type, then it follows from (T1) that Q∈Nκ,−2π∞. Moreover,Qhas the following asymptotic expansion:
Q(z) =− s2π∞
z2π∞+1 +o µ 1
z2π∞+1
¶
, zc→∞, s2π∞ >0. (3.15) If the generalized zero ∞of Qis of type (T3), then Q∈ Nκ,−2π∞−2 and Q has the following asymptotic expansion
Q(z) =−s2π∞−2
z2π∞−1 +o µ 1
z2π∞−1
¶
, zc→∞, s2π∞−2<0. (3.16) In the case that the generalized zero∞is of type (T2) there are two possibilities:
eitherQbelongs toNκ,−2π∞, in which case both of the momentss2π∞−1ands2π∞
are finite andQhas the asymptotic expansion Q(z) =−s2π∞−1
z2π∞ − s2π∞
z2π∞+1 +o µ 1
z2π∞+1
¶
, zc→∞, s2π∞−16= 0, (3.17)
orQbelongs toNκ,−2(π∞−1)\Nκ,−2π∞ and it has the asymptotic expansion Q(z) =−s2π∞−1
z2π∞ +o µ 1
z2π∞
¶
, zc→∞, (3.18)
or
Q(z) =o µ 1
z2π∞−1
¶
, zc→∞. (3.19)
Observe, that the expansions (3.17) and (3.18) are also special cases of the ex- pansion (3.19). Hence, if ∞ is a generalized zero of type (T2), then Q has an expansion of the form (3.16), but now withs2π∞−2= 0; however,Qdoes not have an expansion of the form (3.15). Similar observations remain true for generalized zerosβ∈Rand polesα∈R∪ {∞}. For instance, to get the analogous expansions for a generalized zeroβ ∈Rapply the transform−Q(1/(z−β)) to the expansions in (3.15)–(3.19); cf. also [15]. The role of the above classification for generalized zeros and poles of nonpositive type will be described in detail in Sections 4–6.
4. An operator model for the generalized Friedrichs extension
4.1. Boundary triplets and Weyl functions
The construction of the model uses the notion of a boundary triplet in a Pontryagin space setting. LetHbe a Pontryagin space with negative indexκ, letSbe a closed symmetric relation inHwith defect numbers (n, n), and letS∗be the adjoint ofS.
A triplet Π ={Cn,Γ0,Γ1}is said to be a boundary triplet forS∗, if the following two conditions are satisfied:
(i) the mapping Γ : fb→ {Γ0f ,bΓ1fb}fromS∗ to Cn⊕Cn is surjective;
(ii) the abstract Green’s identity
[f0, g]−[f, g0] = (Γ1f ,bΓ0bg)−(Γ0f ,bΓ1bg) (4.1) holds for allfb={f, f0}, bg={g, g0} ∈S∗,
see e.g. [2], [10]. It is easily seen thatA0= ker Γ0 andA1= ker Γ1 are selfadjoint extensions ofS. Associated to every boundary triplet there is the Weyl function Qdefined by
Q(z)Γ0fbz= Γ1fbz, z∈ρ(A0),
wherefbz:={fz, zfz} ∈Nbz, andNz= ker (S∗−z) denotes the defect subspace of S atz∈C. It follows from (4.1) that the Weyl functionQis also aQ-function of the pair{S, A0}in the sense of Kre˘ın and Langer, see [20]. IfS issimple, so that
H= span{Nz: z∈ρ(A0)},
then the Weyl functionQbelongs to the classNκ, otherwiseQ∈Nκ0 withκ0≤κ.
Moreover, ifS is simple andH is a selfadjoint extension ofS inH, then the point spectrum of H is also simple, that is, every eigenspace ofH is one-dimensional, and ifα∈R∪ {∞}, then the root subspace atαis at most 2κ+ 1-dimensional.
In the case whereS is given by (1.3) one can define a boundary triplet for S∗ as follows.
Proposition 4.1. (cf. [5]) Let A be a selfadjoint operator in a Pontryagin space H and let the restrictionS of Abe defined by (1.3)withω∈H. Then the adjointS∗ of S inHis of the form
S∗={{f, Af+cω}: f ∈domA, c∈C} and a boundary tripletΠ∞={C,Γ∞0 ,Γ∞1 } forS∗ is determined by
Γ∞0 fb= [f, ω], Γ∞1 fb=c, fb={f, Af+cω} ∈S∗. The corresponding Weyl functionQ∞ is given by
Q∞(z) =− 1
[(A−z)−1ω, ω], z∈ρ(A). (4.2) 4.2. The model operatorS(Q∞)corresponding to a proper factorization
Operator models for generalized Nevanlinna functions whose only generalized pole of nonpositive type is at∞have been constructed in [6] and [14]. Such functions admit a canonical factorization of the form
Q∞(z) =q(z)q](z)Q0(z), (4.3) where Q0 ∈ N0, q(z) = zk +qk−1zk−1+· · ·+q0 is a polynomial, and q](z) = q(¯z). In general, models which are based on the canonical factorization of Q ∈ Nκ are not necessarily minimal, i.e., the underlying model operatorS(Q∞) need not be simple and it can even be a symmetric relation (multivalued operator).
However, with the canonical factorization the nonsimple part of S(Q∞) can be easily identified and factored out to produce a simple symmetric operator from S(Q∞), cf. [3]. The model constructed forS(Q∞) in [6] uses an orthogonal coupling of two symmetric operators. In [9] this model was adapted to the case where the functionQ0 is allowed to be a generalized Nevanlinna function, too. In this case the situation becomes more involved and, in general, one cannot representS(Q∞) as an orthogonal sum of a simple symmetric operator and a selfadjoint relation.
However, such a simple orthogonal decomposition forS(Q∞) can still be obtained if the factorization (4.3) ofQ∞is proper. This concept is defined as follows.
Definition 4.2. ([9]) The factorizationQ∞(z) =q(z)q](z)Q0(z) is said to beproper if q is a divisor of degree κ∞(Q∞) > 0 of the polynomial eq in the canonical factorization of the functionQ∞, cf. (2.5) .
Clearly, proper factorizations of Q∞ ∈ Nκ always exist, but they are not unique ifqehas more than one zero andκ∞(Q∞)< κ. Proper factorizationsQ∞= qq]Q0can be characterized also without using the canonical factorization ofQ∞. Lemma 4.3. Let Q∞∈Nκ have a factorization of the form
Q∞(z) =q(z)q](z)Q0(z), degq=k≥1, (4.4)
whereq(z)is a monic polynomial, and let α∈σ(q)be a zero ofqwith multiplicity kα. Then the following statements are equivalent:
(i) the factorization (4.4)of Q∞ is proper;
(ii) the multiplicities κ∞(Q∞) andπα(Q∞)satisfy the following relations:
κ∞(Q∞) = degq andπα(Q∞)≥kαfor all α∈σ(q); (4.5) (iii) κ∞(Q0)andκ(Q∞) =κsatisfy the following identities
κ∞(Q0) = 0andκ(Q∞) = degq+κ(Q0). (4.6) Proof. (i)⇔(ii) In a proper factorization (4.4) κ∞(Q∞) = degqand clearly the inequalities in (4.5) just mean that q divides the polynomial eq in the canonical factorization ofQ∞.
(i) ⇒ (iii) If the factorization (4.4) is proper, then in the canonical factor- ization of the function Q0 the numerator eq0 and denominator pe0(= p) of thee corresponding rational factor r0 are of the same degree κ(Q0), and this implies (4.6).
(iii) ⇒(i) It follows from the second equality in (4.6) that q and the poly- nomial pe0 in the canonical factorization of the function Q0 are relatively prime and, therefore, q is a factor of the polynomial eq in the canonical factorization of Q∞. Moreover, π∞(Q0) = 0. Now the assumption κ∞(Q0) = 0 implies that
κ∞(Q∞) = degq. ¤
The construction of factorization models is now briefly described. Letqbe a polynomial as in (4.4) of degreek= degq. Define thek×kmatricesBq andCq by
Bq =
q1 . . . qk−1 1 ... . .. 1 0 qk−1 . ..
. .. ... 1 0 . . . 0
, Cq =
0 1 . . . 0
... . .. ... 0
0 0 . . . 1
−q0 −q1 . . . −qk−1
,
so that σ(Cq) = σ(q). Moreover, let Hq be a 2k-dimensional Pontryagin space defined by
(Ck⊕Ck,hB·,·i), B=
µ 0 Bq
Bq] 0
¶ .
A general factorization model for functionsQ∞of the form (4.4) was constructed in [9] and can be applied, in particular, for proper factorizations ofQ∞.
Theorem 4.4. (cf. [9])Let Q∞∈Nκ be a generalized Nevanlinna function and let Q∞(λ) =q(λ)q](λ)Q0(λ), (4.7) be a proper factorizationQ∞, whereqis a monic polynomial of degreek= degq≥ 1. LetS0be a closed symmetric relation in a Pontryagin spaceH0with the boundary tripletΠ0={H,Γ00,Γ01} whose Weyl function isQ0. Then:
(i) the function Q0 in (4.7)belongs to the class Nκ−k;
(ii) the linear relation S(Q∞) =
f0
f fe
,
f00 Cq]f Cqfe+ Γ00fb0ek
:
fb0={f0, f00} ∈S0∗, f1= Γ01fb0,
fe1= 0
(4.8) is closed and symmetric in H:=H0⊕Hq and has defect numbers (1,1);
(iii) the adjoint S(Q∞)∗ of S(Q∞)is given by S(Q∞)∗=
f0
f fe
,
f00 Cq]f+ϕee k
Cqfe+ Γ00fb0ek
:
fb0={f0, f00} ∈S0∗, f1= Γ01fb0,
e ϕ∈C
; (iv) a boundary triplet Π ={H,Γ0,Γ1}forS(Q∞)∗ is determined by
Γ0(fb0⊕Fb) =fe1, Γ1(fb0⊕Fb) =ϕ,e fb0⊕Fb∈S(Q∞)∗; (4.9) (v) the corresponding Weyl function coincides with Q∞.
Proof. Since the factorization (4.7) is proper the statement (i) is immediate from the equality (4.6) in Lemma 4.3. All the other statements are contained in [9]. ¤ In fact, the statement (iv) in Theorem 4.4 can be obtained directly also from Proposition 4.1, sinceS(Q∞) in (4.8) is a restriction of the selfadjoint relation
A(Q∞) =
f0
f fe
,
f00 Cq]f Cqfe+ Γ00fb0ek
: fb0={f0, f00} ∈S0∗, f1= Γ01fb0
(4.10) to the subspaceHªω0, whereω0= col(0, ek,0); compare (1.3). The generalized Friedrichs extension ofS(Q∞) is given by
SF(Q∞) =
f0
f fe
,
f00 Cq]f+ϕee k
Cqfe+ Γ00fb0ek
:
fb0={f0, f00} ∈S0∗, f1= Γ01fb0, fe1= 0,ϕe∈C
. (4.11) According to (4.2) in Proposition 4.1 the Weyl function Q∞(z) corresponding to the boundary triplet (4.9) is of the form
Q∞(z) =− 1
[(A(Q∞)−z)−1ω0, ω0]. Thus the functionQ=−1/Q∞ has the representation
Q(z) = [(A(Q∞)−z)−1ω, ω], (4.12) which is, however, not necessarily minimal, since mulS and ker (S−α),α∈σ(q), can be nontrivial. The following lemma describes these subspaces.
Lemma 4.5. ([9]) Under the assumptions of Theorem4.4letS0 be a simple closed symmetric operator in the Pontryagin space H0, let A0i = ker Γ0i(⊃S0), i= 0,1, and letkα be the multiplicity ofα∈Cas a zero of the polynomialq. Then:
(i) mulS(Q∞)is nontrivial if and only if mulA01 is nontrivial and in this case mulS(Q∞) ={(g,0,Γ00bgek)> : gb={0, g} ∈A01}; (4.13) (ii) if mulS(Q∞)is nontrivial, then it is spanned by a positive vector;
(iii) if mulA00 is nontrivial, then it is spanned by a positive vector;
(iv) σp(S(Q∞)) =σp(A00)∩σ(q]) and forα∈σp(A00)∩σ(q])one has
ker (S(Q∞)−α) ={(g0,Γ01bg0Λ|λ=α,0)>: g0∈ker (A00−α)}; (4.14) whereΛ = (1, λ, . . . , λk−1),λ∈C;
(v) if ker (S(Q∞)−α) or, equivalently, ker (A00 −α) is nontrivial, then it is spanned by a positive vector.
It follows from (ii) and (iv) that the linear relationS(Q∞) can be decomposed into a direct sum of an operatorS0with an empty point spectrum and a selfadjoint part in a Hilbert space which is the sum of mulS(Q∞) and ker (S(Q∞)−α), α ∈ σp(A00)∩σ(q]). The next theorem shows that the reduced operator S0 is simple.
Theorem 4.6. Let the assumptions of Theorem 4.4 be satisfied and let S(Q∞), A(Q∞), andSF(Q∞)be given by (4.8),(4.10), and (4.11), respectively. Then:
(i) S(Q∞) is simple if and only ifσp(S(Q∞)) =∅. In this case the linear rela- tionsS=S(Q∞),A=A(Q∞), andSF =SF(Q∞)satisfy the equalities (1.3) and (1.4)withω=ω0and the operator representation (4.12)ofQ=−1/Q∞
is minimal.
(ii) If S(Q∞)is not simple, then the subspace
H00= span{mulS(Q∞),ker (S(Q∞)−α) : α∈σp(A00)∩σ(q)} (4.15) is positive and reducing forS(Q∞). The simple part ofS(Q∞)coincides with the restrictionS0 of S(Q∞) toH0:=HªH00. The compressions S0,A0, and SF0 of S(Q∞),A(Q∞), and SF(Q∞)to the subspace H0 satisfy the equalities (1.3)and (1.4), with ω∈H0 given by
ω=
½ ω0, if k >1, (g,−1/Γ0bg,Γ0bg)>, if k= 1,
and the function Q=−1/Q∞ admits the minimal representation Q(λ) =−1/Q∞(λ) = [(A0−λ)−1ω, ω].
4.3. The root subspace of the generalized Friedrichs extension at∞.
Let the assumptions of Theorem 4.4 be satisfied and letS(Q∞),A(Q∞),SF(Q∞) and S0, A0, SF0 be the same as in Theorem 4.6. Since S0 is a simple symmetric operator in the Pontryagin space H0 its selfadjoint extension SF0 has a simple point spectrum. In particular, the multivalued part mulSF0 ofS0F is at most one- dimensional. The corresponding root subspace
R∞(SF0 ) = span{g∈H0: {0, g} ∈SF0 k, for somek∈N}
