CHARACTERIZATIONS OF GENERALIZED POLES BY POLE CANCELLATION FUNCTIONS OF HIGHER ORDER MUHAMED BOROGOVAC AND ANNEMARIE LUGER http://arxiv.org/abs/1309.3677 Abstract In this paper, a new, more general definition of pole cancelation functions at a generalized pole of a generalized Nevanlinna function is introduced and it is proven that for each such pole cancelation function of a given order there exists a corresponding Jordan chain of the representing linear relation of length . The converse statement is first proven under the condition that the general pole is not a zero of in the same time. For a given Jordan chain of at eigenvalue it is proven that the function satisfies all requirements of our definition of pole cancelation functions of order .
If pole is also a zero of , then pole cancelation function of that form need not to exist. An example is provided that shows this. Moreover, it is shown how the methodology of characterization of generalized poles by our pole cancelation functions can be applied even in the case when is both pole and zero of . Generalized Nevanlinna class Definition 2.1 A function : belongs to generalized Nevanlinna class if , , , has negative squares i.e. for arbitrary and the Hermitian matrix has at most negative squares, and is minimal with this property. The following representation is the main tool in the research of the generalized Nevanlinna functions. Linear relation representation of
Proposition 2.2 An -valued function is generalized Nevanlinna function iff , z (2.1) where is a self-adjoint linear relation in a Pontryagin space , , is bounded operator. Moreover, this representation can be chosen minimal, that is, . Then iff negative index of Pontryagin space equals . Definition 2.4 A point is called generalized pole of if is an eigenvalue of the is called generalized pole of if is an eigenvalue of the representing relation A in a minimal realization of the form (2.1). History of the problem The problem of analytic characterization of the algebraic eigen-space
structure of the representing operator A in terms of Q has been around from time when generalized Nevanlinna functions have been introduced, around 40 years ago, by M. G. Krein and H. Langer. For scalar generalized Nevanlinna functions, analytic characterization of the degree of non-positivity was proven in 1986 by H. Langer. Analytic characterization of the whole chain was proven in 2006 by S. Hassi and A. Luger . Pole cancelation function (PCF) For operator generalized Nevanlinna functions the appropriate tool are socalled pole cancellation functions (PCFs).
Solving the problem of characterization of generalized poles requires finding appropriate definition of PCF. It is a very big part of the solution. Several partial results have been proven by H. Langer, M. Borogovac, A. Dijksma, H. de Snoo, A. Luger. For example, characterization of eigen-vectors of embedded poles and characterization of algebraic eigen-space structure of isolated singularities were proven in 1988 and 1991 by H. Langer and M. Borogovac . Definition PCF
Definition 3.1 A holomorphic function :is called a pole cancelation function of at if: , , is bounded when . This definition was basically used in [BLa] 1988 for matrix generalized Nevanlinna function Q (where weak limits above coincide with strong limits). Definition PCF of higher order
Definiton 3.2 The order of pole cancelation function is defined as the maximal number such that for 0 it holds: , is bounded when . Note that Def. 3.1 follows from Def. 3.2 for 0. We keep separate Def. 3.1 from Def. 3.2 because of the historical reasons and clarity. Def. 3.2 of the PCF of higher order is new. It is the just right definition that enables characterization of the structure of algebraic eigen-space. Strong PCF
We say that is a strong PCF if it satisfies (A), (B) and Strong PCF is of order if it satisfies (D), (E) and Note that we need not to assume that PCF is strong in order to obtain a Jordan Chain at a generalized pole of Q, see Theorem 3.3. Conversely, for every Jordan chain, we have a strong PCF of given by (3.4), see Theorem 3.5 and Corollary 3.6. Jordan chains by means of PCFs
Theorem 3.3 Let have minimal operator or relation representation and . Let . Then the following holds: (1) If satisfies also , and then is a generalized pole of , precisely , when , where is an eigenvector of . (2) If satisfies , and , for some then for it holds , and , form a Jordan chain of A at . Remarks about Theorem 3.3
Statement (2), the construction of Jordan chain by means of PCF is the necessary part of the characterization of the algebraic eigen-space at , while the converse statements, i.e. a construction a PCF by means of a given Jordan chain will be proven in the Theorem 3.5. The description of the structure of the algebraic eigen-space will be completed in the Corollary 3.6. Statement (1) is a special case of (2). We separated them because of historical reasons. Characterization of generalized poles by means of Def. 3.1, i.e. statement (1) and its converse, was proven in [BLa] for matrix generalized Nevanlinna functions. The converse statement there was proven in a different way, by means of irreducible representation of Q, see [DLa]. Basic formulae used in the proofs
For and we have: , , =, . (2.2) (2.3) (2.4)
Hint of the proof of Theorem 3.3 The proof is based on the Lemma 1 [BLa]: The sequence converges weakly if and only if the sequence is bounded and is a Cauchy sequence for all elements of some total subset U of . From (D), (E) and (F) it follows: is bounded and , is Cauchy, when According to above lemma . Then we verify that , is a Jordan chain of A, i.e. we verify . Remark about Strong PCF Let have minimal linear relation representation and . Let . Then the following holds: () If satisfies (A), (B) and then , when .
() If satisfies , and then for some then for , form a Jordan chain of A in and . The proof of the above statements () and () is based on the characterization of strong convergence in a Pontryagin space similar to above quoted Lemma 1 [BLa] but the condition is bounded is replaced by the requirement is Cauchy sequence. Statement () is a special case of () and it was proven earlier, in [DLaS]. A construction of a Jordan chain by means of was done by A. Luger in 2006 under some stronger assumptions. Pole cancellation functions by means of Jordan chains. Theorem. 3.5 Let the regular generalized Nevanlinna function Q be given with a minimal realization and assume that is not a generalized zero of Q. If is a generalized pole of Q, that is , and , is a Jordan chain of A at , then
(3.4) is a strong pole cancelation function of at of order . Corollary 3.6 Under assmption of Th. 3.5 (strong convergence) . Hint of the proof of Theorem 3.5 For the proof of Th. 3.5 we will need the two technical lemmas below. However, for the proofs of the lemmas, we need the following representation of . , where and =, see [LaLu].
For a given Jordan chain , we frequently use the following formulae . Lemma 3.7 If we denote , then and = Lemma 3.8 Technical proofs of the lemmas are omitted in this presentation. Proof of Theorem 3.5 continuation The proof of Theorem 3.5 we start by rewriting from (3.4), where, in particular, Lemma 3.8 is used in the fourth equality
(3.11) Also note, due to the assumption that is not a generalized zero of Q, it holds as : . Now we can verify (D), (E) and (). Verification of () and Corollary 3.6 Verifications of the conditions (D) and (E) is straightforward. For verification of () we need first to prove
+, where for , . This implies (), and even more, the statement of the Lemma 3.6: Structure of the algebraic eigen-space Note that for the construction of a Jordan chain we needed only PCF, a function that satisfies (D), (E) and (F). We do not need a strong PCF. Conversely, for any Jordan chain formula (3.4) gives a strong PCF, the function that satisfies (D), (E) and , see Th. 3.5, and it satisfies even
stronger condition given in Corollary 3.6. Also note that pole cancelation functions corresponding to different Jordan chains can be employed in the proof of Corollary 3.6. This makes possible to fully define inner product in the algebraic eigen-space in terms of Jordan chains. Examples summarized In the Theorem 3.5 it was shown that given a Jordan chain the corresponding function is of order at least . So the question arises how the order of is related to the maximality of the Jordan chain. We summarize our observations in the following corollary.
Corollary 3.11. Let and be given as in Theorem 3.5. If is a maximal Jordan Chain of then has order . Conversely, if has order , then need not be maximal. If is not maximal, then the order of is or larger then and there are examples for both situations. What if a generalized pole is also a generalized zero? In Theorem 3.5 there are still restrictive assumptions, namely that Q is a regular function and is only a generalized pole, not a generalized zero. These restrictions are removed in the following theorem.
Theorem 3.12. Let be given and assume that is a generalized pole of Q such that the representing relation A has a Jordan chain at of length .Then there exists a strong pole cancellation function of order at least of the form where and is an -valued polynomial of degree . Conclusion The results in this paper give a complete answer to the longstanding problem of an analytic characterization of generalized poles including multiplicities. The concrete form of the pole cancellation functions appears to be much simpler than expected. In the following theorem we summarize the situation.
Theorem 4.1. Let and be given. Then the following statements are equivalent: I The point is a generalized pole of Q and there exists a Jordan chain of the representing relation of length . II There exists a pole cancellation function of Q at of order at least . III There exists a strong pole cancellation function of Q at of order at least . IV There exist and an -valued polynomial of degree such that is a strong pole cancellation function of Q at of order at least . References
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