85
Generalized poles and zeros of generalized
Nevanlinna functions - a survey
Annemarie Luger
Abstract
Luger, Annemarie (2005). Generalized poles and zeros of generalized Nevanlinna functions - a survey. In Proceedings of the Algorithmic Information Theory Conference,
Vaasa 2005. Proceedings of the University of Vaasa, Reports 124, 85–94. Eds S. Hassi,
V. Keränen, C.-G. Källman, M. Laaksonen, and M. Linna.
Generalized poles and zeros of a generalized Nevanlinna function Q are defined as
eigenvalues of certain relations that are connected with the function Q. In this note
we give a survey on characterizations of these points, including their order and degree
of non-positivity, in terms of the analytic behaviour of Q.
Annemarie Luger, Department of Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria, E-mail:
[email protected]
Keywords: Generalized Nevanlinna function, generalized pole, generalized zero.
Mathematics Subject Classification (2000): 47B50, 30E99.
1. Introduction
Recall that a function q that maps the upper halfplane C+ holomorphically into
C+ ∪ R is called a Nevanlinna function, q ∈ N0 . These functions, which appear
e.g. as Titchmarsh-Weyl coefficients in Sturm-Liouville problems, are very well studied objects. In particular, such a function admits an integral representation
Z ∞³
1
t ´
(1)
q(z) = a + bz +
−
dσ(t),
1 + t2
−∞ t − z
R∞ 1
where a ∈ R, b ≥ 0, and σ is a real positive measure with −∞ 1+t
2 dσ(t) < ∞. More
abstractly, for every such function there exists a Hilbert space K, a self-adjoint linear
relation (that is, a multi-valued operator) A in K and an element v ∈ K such that with
z0 ∈ %(A) the function q can be represented as
(2)
¢ ¤
£¡
q(z) = q(z0 ) + (z − z0 ) I + (z − z0 )(A − z)−1 v, v .
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Here the representation can always be chosen minimal (see below) and this determines
it uniquely up to unitary equivalence.
In particular, for b = 0 one can choose K = L2σ , where σ is the measure in the integral representation (1), then A is the operator of multiplication by the independent
variable. If b > 0 then A is not an operator, but a relation with one-dimensional
multi-valued part (see e.g. and Dijksma and de Snoo (1987) for the theory of linear
relations, and for the details of this representation e.g. H. Langer and Luger (2000)).
The limit behaviour of q at the real line can be deduced directly from the above representations. In particular, lim (z − α)q(z), where lim denotes the non-tangential limit
z →α
ˆ
z →α
ˆ
to α ∈ R, always exists and is zero or positive. Here the second case is equivalent to
α being an eigenvalue of the minimal representing relation A in (2), or, equivalently,
that {α} is not a zero set for the measure σ.
In several examples, however, e.g. in connection with singular Sturm-Liouville problems (see Dijksma and Shondin (2000), or H. Langer, M. Langer, and Sasvári (2004))
there appear functions that are not as described above, but belong to a so-called
generalized Nevanlinna class. Such functions may have non-real poles and the limit
lim (z − α)q(z) for α ∈ R does not necessarily exist. However, this exceptional behav-
z →α
ˆ
iour can appear at finitely many points only. It is connected to the fact that these
functions admit operator representations of the form (2) but in Pontryagin spaces.
Here self-adjoint linear operators can have nonreal eigenvalues and eigenvalues, for
which there exists a Jordan chain (of finite length).
In this note we present a short survey on what has been done (and we also indicate
some work that is still in progress) on the connection between the limit behaviour of
a scalar, matrix-, and even operator-valued generalized Nevanlinna function and the
structure of the eigenspace of the corresponding representing relation.
2. Definitions
¡
¢
Let H, ( · , · ) be a Hilbert space. An operator function Q with values in L(H)
belongs to the generalized Nevanlinna class Nκ (H), if it is meromorphic in C \ R,
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symmetric with respect to the real axis (that is Q(z)∗ = Q(z) for z ∈ D, the domain
of holomorphy of the function Q), and if the so-called Nevanlinna kernel
NQ (z, ζ) =
Q(z) − Q(ζ)∗
z−ζ
z, ζ ∈ D ∩ C +
has κ negative squares. This means that for arbitrary N ∈ N, z1 , . . . , zN ∈ D ∩ C +
and ξ~1 , . . . , ξ~N ∈ H the Hermitian matrix
³
´N
(NQ (zi , zj )ξ~i , ξ~j )
i,j=1
has at most κ negative eigenvalues, and κ is minimal with this property. For scalar
functions we simply write Nκ instead of Nκ (C). These classes were introduced by
Krein and Langer (1977) for scalar functions and for matrix functions in (1981).
It was shown in Krein and Langer (1977) (see also Hassi, deSnoo, and Woracek (1998)
for arbitrary H) that an operator function belongs to the class Nκ (H) if and only if it
admits a minimal representation of the form
³
´
(3)
Q(z) = Q∗0 + (z − z0 )Γ+ I + (z − z0 )(A − z)−1 Γ
¡
¢
with some Pontryagin space K, [ · , · ] with negative index κ. Here Γ ∈ L(H, K), the
linear relation A in K is self-adjoint with non-empty resolvent set, z0 ∈ %(A) ∩ C + , and
Q0 ∈ L(H) with
Q0 − Q∗0 = (z0 − z0 )Γ+ Γ.
The representation (3) is called minimal if
n¡
o
¢ ¯¯
−1
K = span I + (z − z0 )(A − z) Γf~ ¯ z ∈ %(A), f~ ∈ H .
Note that for n = 1 and κ = 0 representation (3) reduces to (2). Also the integral representation (1) has been generalized for these classes. We refer to Krein et al. (1977)
in the scalar case and Daho and Langer (1985) for matrix functions.
Let now Q ∈ Nκ (H) be given. Then the points, which are in the focus of our interest
in this note, are given as follows. The point α ∈ C ∪ {∞} is called generalized pole of Q
if α is an eigenvalue of the relation A in the minimal representation (3). Note that α
is an (ordinary) pole of the function Q, if and only if α is an isolated eigenvalue of A.
The generalized pole α is called to be of positive type if the eigenspace of A at α is
positive. An important role play those generalized poles that are not of positive type,
that is there exists at least one eigenvector that is negative or neutral.
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Remark 1. In the literature one finds also the notation of a generalized pole of
negative type (Borogovac and Langer (1988)) or of non-positive type (Dijksma, Langer,
Luger and Shondin (2000)). However, we follow here Luger (2002), since this notation
of the type of a general pole fits the type of the corresponding eigenspace. In particular,
in case of dim H > 1 for a generalized pole that is not of positive type the corresponding
eigenspace of the representing relation need not be non-positive, in fact there can also
exist positive eigenvectors.
With a generalized pole α there are connected two characteristic numbers. The first
is the dimension of the corresponding algebraic eigenspace, it is called the order of α.
If α is an (ordinary) pole of Q then the order coincides with its polar multiplicity
(in the sense of Gohberg and Sigal (1971)). The second number is connected with
the indefinite inner product. The degree of non-positivity of a generalized pole α is
by definition the index of non-positivity of the algebraic eigenspace. Recall that the
index of non-positivity of a subspace M of a Pontryagin space is the dimension of
a maximal non-positive subspace M0 ⊆ M. Hence for Q ∈ Nκ (H) there are only
finitely many generalized poles, which are not of positive type and their degrees of
non-positivity sum up to κ. Note that in the early papers the degree of non-positivity
was also called (non-positive, or negative) multiplicity.
b
Recall that if Q ∈ Nκ (H), then also Q(z)
:= −Q(z)−1 belongs to the class Nκ (H).
b are called the generalized zeros of Q and the type and
The generalized poles of Q
degree of non positivity are defined accordingly.
Given a function Q ∈ Nκ (H) there arises the question how to read off from the function directly – without using the operator representation – which points are generalized
poles (or zeros) and, in particular, to characterize analytically their order and degree
of non-positivity. Note that it is essentially the same to deal with generalized poles or
zeros, since the results can be translated directly.
3. Characterizations
As a first step Langer (1986) characterized the degree of non-positivity of a generalized
zero of a scalar generalized Nevanlinna function in terms of nontangential limits.
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Theorem 2. The point β ∈ R is a generalized zero with degree of non-positivity κβ of
the function q ∈ Nκ if and only if
q(z)
lim
≤0
z →β
ˆ (z − β)2κβ −1
and
lim
z →β
ˆ
q(z)
> 0 or ∞.
(z − β)2κβ +1
The situation for matrix functions (and even operator valued) functions is more complicated. There can exist linearly independent eigenvectors and Jordan chains. Moreover,
in general, the limit of Q(z)~x depends on ~x ∈ H. So, even if Q has a generalized zero,
limits of the function may exist only in ”certain directions”, in particular, a point can
be both a generalized zero and a generalized pole of Q. As an appropriate tool, the
notion of so-called pole-cancellation functions has been adjusted to this situation.
Let Uα be a neighbourhood of α ∈ C. A holomorphic function ~η : Uα ∩ C+ → H is
called a pole cancellation function of Q at α if:
P1 lim ~η (z) = ~0,
z →α
ˆ
P2 there exists an operator function H, that is holomorphic at α such that the
function Q(z)~η (z) − H(z)~η (z) can be continued holomorphically into α and
lim Q(z)~η (z) 6= ~0,
z →α
ˆ
µ
¶
Q(z) − Q(w)
P3 ∃ lim
~η (z), ~η (w) .
z,w→α
ˆ
z−w
Note that if Q has an isolated pole at α, then P1 and P2 mean that the functions ~η
and Q~η can be continued holomorphically into α, and α is a zero of ~η , but not of Q~η .
The condition P3 is then no further restriction, and the definition reduces to the one
already used in Gohberg et al. (1971)
In Borogovac et al. (1988) the position of a generalized zero was characterized by the
b The proof is using
existence of a pole cancellation function for the inverse function Q.
b More detailed results were obtained in special cases,
the integral representation of Q.
see Borogovac (1991) for meromorphic functions and Dijksma, Langer, and de Snoo
(1993) for non-degenerate eigenspaces (which excludes Jordan chains of length > 1).
In the scalar case the above characterization of the degree of non-positivity became
more transparent with the following so-called basic factorization: let the points αi ,
i = 1, 2, . . . , `, (βj , j = 1, 2, . . . , k, respectively) be the generalized poles (zeros, respectively) of q in C+ ∪ R that are not of positive type, denote by νi (κj , respectively)
90
the degree of non-positivity of αi (βj , respectively), and define
(z − β1 )κ1 . . . (z − βk )κk
.
(z − α1∗ )ν1 . . . (z − α`∗ )ν`
Then the following factorization result holds.
r(z) :=
Theorem 3. Let q ∈ Nκ be given. Then there exists a function q0 ∈ N0 such that
q(z) = r# (z) q0 (z) r(z),
where r# (z) := r(z ∗ )∗ .
Both proofs of this result (see Derkach, Hassi, and de Snoo (1999) and Dijksma et
al. (2000)), made use of Theorem 2. In the matrix case it was done the other way
round. First the factorization result was extended to arbitrary generalized Nevanlinna
b does not exist). Then
functions (see Luger (2002), and (2003) for the case that Q
using this factorization the result by Borogovac and Langer could be extended. We
formulate it for a generalized pole. Let us first recall the following definition.
The order of a pole-cancellation function ~η , denoted by ord(~η ), is defined as the maximal number l0 ∈ N ∪ {∞} such that for 0 ≤ j < l0 it holds
• lim ~η (j) (z) = ~0 and
z →α
ˆ
¶
µ
d2j
Q(z) − Q(w)
• ∃ lim
~η (z), ~η (w) .
z,w→α
ˆ dz j dw j
z−w
Theorem 4. Let Q ∈ Nκ (H) be given with the operator representation (3) and let ~η (z)
be a pole cancellation function of Q at α. Then it holds:
(i) The point α is a generalized pole of Q.
(ii) If ord(~η ) ≥ l, then there exist elements x0 , . . . , xl−1 which form a Jordan chain
of the representing relation A at the eigenvalue α.
(iii) Let for i = 1, 2 the functions ~ηi be pole cancellation functions of Q ∈ Nκ (H)
at α of order li and denote by xi,k for i = 1, 2 and k = 0, 1, . . . , li − 1 the
corresponding Jordan chains. Then
´
1 dk+l ³ Q(z) − Q(w)
~
η
(z),
~
η
(w)
.
i
j
z →α
ˆ w→α
ˆ k!l! dz k dw l
z−w
[xi,k , xj,l ] = lim lim
Note that (iii) shows how to obtain the inner product on the algebraic eigenspace in
terms of the function Q. Conversely, in order to describe the degree of non-positivity
one has construct pole-cancellation functions explicitly and then consider maximal
91
systems of such functions. This is done by using the analogue of Theorem 3 for operator valued functions (for details see Luger (2005)). However, this approach gives only
information on the non-positive part of the algebraic eigenspace of the representing
relation.
In order to characterize analytically also the order, that is the dimension of the algebraic eigenspace, a direct study of the representing relation is necessary. In Hassi and
Luger (2005) this has been done for scalar functions. For the formulation of the result,
which gives the connection between the order of a generalized zero and an asymptotic
expansion of q ∈ Nκ at this point, the following notation is needed.
Let the generalized Nevanlinna function q ∈ Nκ be given. Then for β ∈ C define
q(z)
s0 (β) := lim
z →β
ˆ z−β
and then recursively
£
¤
1
n
sn (β) := lim
q(z)
−
(z
−
β)s
(β)
−
.
.
.
−
(z
−
β)
s
(β)
,
0
n−1
z →β
ˆ (z − β)n+1
and for β = ∞
s0 (∞) := lim −z q(z)
z →∞
ˆ
and
£
s0 (∞)
sn−1 (∞) ¤
sn (∞) := lim −z n+1 q(z) +
+ ... +
,
z →∞
ˆ
z
zn
whenever these limits exist and are real if β ∈ R ∪ {∞}. These numbers will be
referred to as the moments of q at β. A justification for this notation is that for a
(usual) Nevanlinna function q0 ∈ N0 they coincide with the moments of the measure
σ in the integral representation with respect to β.
Theorem 5. Let q ∈ Nκ with q(z) 6≡ 0 be given. Then β ∈ C ∪ {∞} is a generalized
zero of q of order dβ ≥ 1 if and only if dβ is the maximal integer with the following
properties:
∃ s0 (β), s1 (β), . . . , s2dβ −2 (β) and s0 (β) = s1 (β) = . . . = sdβ −2 (β) = 0.
In the proof of this theorem a Jordan chain x0 , . . . , xdβ −1 of the representing relation
b is constructed and then it turns out that for the inner product on the algebraic
of Q
eigenspace it holds
[xi , xj ] = si+j (β)
for i, j = 0, . . . , dβ − 1.
92
This gives another proof for Theorem 2.
In the case of a matrix valued generalized Nevanlinna function Q ∈ Nκ (Cn ) the same
idea can be applied. It will be presented in detail elsewhere.
In this situation a certain subspace H0 ⊆ Cn plays an important role. If Q has a limit
at β ∈ R, then H0 = ker Q(β). In any case H0 can be obtained as the set of all possible
b η (z), where ~η (z) are pole-cancellation functions of Q
b at β. Then
pole vectors lim Q(z)~
z →β
ˆ
the following holds:
The point β is a generalized zero of Q and there exists a corresponding Jordan chain
b of length l (not necessarily maximal) if and
of the minimal representing relation of Q
only if there exist vectors ~h1 , . . . , ~hl−1 ∈ H0 such that
¢
¡
1
~h1 + (z − β)~h2 + . . . + (z − β)l−2~hl−1 ), ~h = 0 for all ~h ∈ H2
(i) lim
Q(z)(
z →β
ˆ (z − β)l−1
(ii) and for the scalar function
¢
¡
Q(z)(~h1 +(z−β)~h2 +. . .+(z−β)l−2~hl−1 , ~h1 +(z− β)~h2 +. . .+(z −β)l−2~hl−1
there exist moments s0 (β), . . . , s2l−2 (β).
Again the moments of the function in (ii) are related to the inner product on the root
subspace. This statement gives the possibility to describe the root subspace and hence
the order (and degree of non-positivity) in terms of limits.
We finally want to remark that the structure of a generalized pole of Q ∈ Nκ (H)
which is of positive type is comparably simple. Locally at such a point the function Q
behaves like a usual Nevanlinna function. In particular, in this case there do not exist
Jordan chains (since this implied that the corresponding eigenvector would be neutral).
93
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