Monatsh. Math. 138, 1–29 (2003) DOI 10.1007/s00605-002-0528-6
The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions By 1
D. Alpay , T. Ya. Azizov , A. Dijksma3 , and H. Langer4 1
2
Ben-Gurion University of the Negev, Beer Sheva, Israel 2 Voronezh State University, Voronezh, Russia 3 University of Groningen, The Netherlands 4 Vienna University of Technology, Austria
Received October 31, 2001; in revised form August 21, 2002 Published online November 15, 2002 # Springer-Verlag 2002 Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday Abstract. In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function sðzÞ, which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal sðzÞ1 of the characteristic function sðzÞ and general factorization results for characteristic functions. 2000 Mathematics Subject Classification: 46C20, 47A48, 30D99 Key words: Schur algorithm, generalized Schur function, operator colligation, Schur determinant, Pontryagin space
1. Introduction By S0 we denote the class of all Schur functions: these are the functions s defined and holomorphic on the open unit disc D and such that jsðzÞj 4 1, z 2 D. If s is not a constant of modulus 1 then the Schur transformation (see [14], [15]) 1 sðzÞ sð0Þ s1 ðzÞ ¼ ð1:1Þ z 1 sðzÞsð0Þ is defined and belongs again to the class S0 . The larger class of generalized Schur functions was introduced in [13], [9], see also [5], [4]: It consists of the functions s The research for this paper was supported by grants from the Netherlands Organization for Scientific Research NWO 047-008-008 and NWO 61-453 and from the Russian Foundation for Basic Research RFBR 99-01-00391. A. Dijksma and H. Langer acknowledge support through Harry T. Dozor fellowships at the Ben-Gurion University of the Negev, Beer-Sheva, Israel, in the years 1999 and 2000, respectively. We thank the referee for his=her remarks.
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which are meromorphic in D and such that the kernel 1 sðzÞsðÞ Ss ðz; Þ :¼ ; 1 z which is defined for z; 2 DðsÞ, the domain of holomorphy of s, has a finite number of negative squares; if this number is the function s belongs by definition to the class S . It turns out (see [5]) that s 2 S if and only if s is meromorphic in D with poles, counted according to their multiplicities, and such that lim sup jsðzÞj 4 1: jzj"1 0
Further, by A we denote the set of functions which are defined and holomorphic in 0 a neighbourhood of z ¼ 0, and we set S0 :¼ S \ A0 , S0 :¼ [1 ¼0 S . Evidently, 0 S0 ¼ S0 . For a function s 2 A0 the Schur transformation has been generalized in [5] and [10] as follows: If jsð0Þj < 1 the Schur transformation s1 is still defined by (1.1), if jsð0Þj > 1 it is defined by the formula 1 sðzÞsð0Þ sk ðzÞ ¼ zk ; ð1:2Þ sðzÞ sð0Þ where k ð > 0Þ is the order of the pole at zero of the fraction on the right hand side of (1.2), and if jsð0Þj ¼ 1 then the Schur transformation is defined by s2kþq ðzÞ ¼ zq
ðQðzÞ zk ÞsðzÞ sð0ÞQðzÞ ; sð0Þ QðzÞsðzÞ ðQðzÞ þ zk Þ
ð1:3Þ
here k is given by the Taylor expansion of sðzÞ at z ¼ 0 as the index of the first non-vanishing coefficient k 6¼ 0, k 5 1, that is, sðzÞ ¼ 0 þ k zk þ kþ1 zkþ1 þ ; and QðzÞ ¼ c0 þ þ ck1 zk1 ðck1 zkþ1 þ þ c0 z2k Þ; where the coefficients cj , j ¼ 0; 1; . . . ; ck1 , are defined by the relation ðsðzÞ 0 Þðc0 þ c1 z þ þ cn zn þ Þ 0 zk ; finally, q ð 5 0Þ is the order of the pole at zero of the fraction on the right hand side of (1.3). In (1.2) and (1.3), respectively, the index k or 2k þ q is used because of the fact that, for example, for a rational function s the degree of the transformed function is the degree of s minus k or minus 2k þ q, respectively, whereas for the formula (1.1) the reduction of the degree in passing from s to s1 is just one. To a function s 2 A0 there corresponds a minimal coisometric colligation V in a space H C such that s is the characteristic function of V, see [8], [4] and also Section 2 below. If the coisometric colligation is T u H H V¼ : ! C h; vi C
The Schur Algorithm for Generalized Schur Functions II
3
with elements u; v of the Krein space H, a bounded linear operator T in H and a number 2 C, its characteristic function is by definition sðzÞ ¼ þ zhðI zTÞ1 u; vi: If, in particular, s 2 S0 the Krein space H is in fact a Pontryagin space with negative index . For s 2 A0 with corresponding minimal coisometric colligation V we consider its Schur transformation given by one of the formulas (1.1), (1.2), (1.3) above, and the corresponding minimal coisometric colligation V1 , Vk , or V2kþq, which for a ~ . In [2] the entries of the colligation V ~ were expressed by moment we denote by V ~ is a subspace of H which is the entries of V. It was shown that the Krein space H in case (1.1) of codimension one, in case (1.2) of codimension k, and in case (1.3) of codimension 2k þ q, and that T~ is a compression of a finite dimensional perturbation of T to this subspace. The corresponding formulas, which yield that the ~ , are derived Schur transformation s1 , sk or s2kþq is the characteristic function of V in [2] by straightforward calculations. In the present paper we consider some operators which are related to the colligation V, and their Jordan chains at the eigenvalue zero more closely. The reason for this is that the formulas (1.1), (1.2), (1.3) can all be reduced to fractional linear transformations and multiplications by z1 or a power of z: for (1.1) and (1.2) this is evident, for (1.3) this follows if we write s2kþq ðzÞ as ! zk q : s2kþq ðzÞ ¼ 0 z 1 þ zk 0 sðzÞ0 QðzÞ On the other hand, for an operator colligation V with characteristic function s, a fractional linear transformation ^s of s is in general the characteristic function of a ^ where the main operator T^ is a one-dimensional perturbation of the colligation V main operator T of V, and if zero is an eigenvalue of T^ with a Jordan chain of length k then ^s has a zero at z ¼ 0 of order k. We use these facts in the present paper in order to derive the colligations corresponding to the Schur transformation in (1.1), (1.2), and (1.3). Of course, the formulas are the same as in [2], however, in our opinion, the proofs given here lead to a better insight into the structure of the colligations. Section 2 contains some simple facts about one-dimensional perturbations of an operator S, in particular, the colligation of the inverse of a characteristic function is described. Together with a knowledge of the Jordan chains at the eigenvalue zero of a one-dimensional perturbation of T, the colligations corresponding to the formulas (1.1) and (1.2) can easily be derived, see Section 3. The case of formula (1.3) is more complicated and is split into several steps in Section 4. For the fact that the obtained colligation is coisometric and minimal we refer to the calculations of [2]. Now we start from a function s ¼ s0 2 S0 which is not a constant of modulus one, and consider the Schur transformation s1 , sk or s2kþq according to one of the formulas (1.1), (1.2), or (1.3). If this new function is not a constant of modulus one, its Schur transformation can be considered, and this procedure can be
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continued as long as the outcome is not a constant of modulus one. Thus we obtain a finite or infinite sequence of functions sjn , j0 ¼ 0, jn 2 f1; 2; . . .g for n ¼ 1; 2; . . . ; n0 or n ¼ 1; 2; . . . ; such that sjn belongs to a class S0n , where 0 ¼ . The sequence of the n is nonincreasing and it turns out that, beginning from a certain index, it consists only of zeros, that is, the functions sjn are finally Schur functions. In Section 5 we show that for the function s 2 S0 the index is determined by the number of sign changes in the sequence of the Schur determinants of s, and also relations between this sequence of Schur determinants and the sequence ðjn Þ are established. The characteristic function of an operator colligation is, by definition, always holomorphic at zero. For this reason in this note the function s and its Schur transformation are restricted to the class S0 , and in the formula (1.2) the number k may be > 1, and in (1.3) the number q may be > 0. In fact, they have to be chosen as the smallest integers such that the Schur transformation is holomorphic at zero. In another paper we shall study the Schur transformation using reproducing kernel Pontryagin spaces of the functions of class S . Then the function s and also the transformed functions can have a pole at zero. Finally we mention that in the case ¼ 0 according to [11] the reduction process of the coisometric colligation as explained in [2] and in the present paper, which is implied by consecutive application of the Schur transformation, is equivalent to Andersson’s algorithm for the inverse problem for certain Sturm–Liouville equations, see [1]. This paper is dedicated to Professor Edmund Hlawka. Born at the time when Schur discovered his famous transformation, he has influenced mathematics and generations of mathematicians during the last century in a very essential way. 2. Coisometric Colligations Let ðH; h; iÞ be a Krein space. With elements u, v 2 H, a bounded linear operator T in H and a number 2 C we consider the following operator V in H C: T u : ð2:1Þ V¼ h; vi Often V is called a colligation and T its main operator. The colligation V is coisometric if VV ¼ I, where I stands for the identity operator on any space, in this case for the identity operator in H C. Evidently, the coisometry of the colligation (2.1) is equivalent to the relations TT þ h; uiu ¼ I; Tv þ u ¼ 0; hv; vi þ j j2 ¼ 1: ð2:2Þ If follows from the first relation that the adjoint T of the main operator T of the coisometric colligation V is a contraction in H: hT x; T xi ¼ hx; xi jhx; uij2 4 hx; xi; x 2 H: The characteristic function sV ðzÞ of the colligation V is sV ðzÞ :¼ þ zhðI zTÞ1 u; vi:
ð2:3Þ
The Schur Algorithm for Generalized Schur Functions II
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It is defined and holomorphic at least in a neighbourhood of z ¼ 0, and if we write 1 X j zj ð2:4Þ sV ðzÞ ¼ j¼0 j1
then we have 0 ¼ and j ¼ hT u; vi, j ¼ 1; 2; . . . . The relations (2.2) imply for i; j ¼ 0; 1; . . . i ¼ j; 1 j0 j2 j1 j2 ji j2 ; j i hT v; T vi ¼ ð2:5Þ ð0 ij þ 1 ijþ1 þ þ j i Þ; i > j: For the simple proof of this formula we refer to [2, (4.7)]. The space H, if equipped with the inner product hx; yi :¼ hx; yi, x; y 2 H, is denoted by H. We need the following well-known lemma. Its proof is straightforward, see also [6], [2]. Lemma 2.1. If
U¼
S f h; gi
ð2:6Þ
is a coisometric colligation in H C with 6¼ 0, and sU ðzÞ denotes its characteristic function, then the colligation 0 1 h; gif f S B C C b :¼ B U @ h; gi 1 A is coisometric in H C and its characteristic function is given by sU^ ðzÞ¼ 1 . sU ðzÞ b. Moreover, if the coisometric colligation U is minimal, then so is U Remark 2.2. The system theoretic interpretation of Lemma 2.1 is that the input
and the output for the realization U of sðzÞ ¼ sU ðzÞ, h k ¼ ; h; k 2 H;
; 2 C; ð2:7Þ U
b of 1 : the relation (2.7) is equivalent to change roles in the realization U sðzÞ k h b : ¼ U
We do not know of a similar interpretation for the realizations that appear elsewhere in the paper, for example in Lemma 4.1. Lemma 2.3. If S is a bounded linear operator in an inner product space ðH; h; iÞ, f , g 2 H, the integer q 2 f1; 2; . . .g is defined by the relations hf ; gi ¼ hSf ; gi ¼ ¼ hSq2 f ; gi ¼ 0;
hSq1 f ; gi 6¼ 0;
ð2:8Þ
and ~ S :¼ S h; gif , then 1
hðI zSÞ1 f ; gi
zq1 1 þ zhðI zSÞ1 f ; gi
q ðq1Þ gi þ zhðI z~SÞ1 f ; ~S gi: ¼ hf ; ~S
ð2:9Þ
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Proof. From the definition we obtain hðI z~SÞ1 f ; gi ¼
hðI zSÞ1 f ; gi
: 1 þ zhðI zSÞ1 f ; gi The assumptions (2.8) imply the same for ~S: hf ; gi ¼ h~Sf ; gi ¼ ¼ h~Sq2 f ; gi ¼ 0; h~Sq1 f ; gi 6¼ 0:
ð2:10Þ
These relations allow to write 1 1 X X z j h~S j f ; gi ¼ z j h~S j f ; gi hðI z~ SÞ1 f ; gi ¼ j¼0 q1
¼z
j¼q1 1 X ~q
j ~ ðzSÞ f ; g
~Sq1 þ zS
j¼0 q1
¼z
ðhf ; ~Sðq1Þ gi þ zhðI z~SÞ1 f ; ~Sq giÞ:
Combining this formula with formula (2.10) we obtain formula (2.9).
&
Remark 2.4. With the (not necessarily coisometric) colligation U from (2.6) and its characteristic function sU the statement of the lemma can be formulated as follows: The function ~sðzÞ :¼
1 sU ðzÞ zq 1 þ ðsU ðzÞ Þ
is the characteristic function of the colligation ~S f ~ :¼ : U h; ~Sq gi hf ; ~Sðq1Þ gi 3. The Case j0 j 6¼ 1 3.1. Suppose now that j j 6¼ 1. With the colligation (2.1) we associate the operator T^ :¼ T þ h; viu ð3:1Þ 1 j j2 and the characteristic function sV ðzÞ as in (2.3), (2.4). Further, let k be the integer which is determined by the relation k :¼ minfl : l 5 1; l 6¼ 0g;
ð3:2Þ
that is, k 6¼ 0 and, if k > 1, then 1 ¼ ¼ k1 ¼ 0: ð3:3Þ Lemma 3.1. If the colligation V in (2.1) is coisometric and has the property j j 6¼ 1, then T^ v ¼ 0. If, additionally, for its characteristic function sV the integer k in (3.2) is > 1, then the elements v, T v, T 2 v; . . . ; T ðk1Þ v form a Jordan chain of T^ at the eigenvalue zero: T^ v ¼ 0; T^ T v ¼ v; . . . ; T^ T ðk1Þ v ¼ T ðk2Þ v:
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If V is minimal then for each l 2 f1; 2; . . . ; kg, ker T^ ¼ spanfv; T v; . . . ; T ðl1Þ vg: l
Proof. The relations (2.2) imply hv; viu ¼ Tv þ u ¼ 0; T^ v ¼ Tv þ 1 j j2 and, similarly, for 1 4 j 4 k 1, T^ T j v ¼ TT j v þ
1 j j
2
hv; T j viu
¼ T ðj1Þ v hT ðj1Þ v; uiu ¼ T ðj1Þ v
1 1 j j2
j j2 1 j j2
hv; T j1 uiu
j u ¼ T ðj1Þ v:
We prove the last equality of the lemma. The first part of the lemma implies l spanfv; T v; . . . ; T ðl1Þ vg ker T^ :
ð3:4Þ
To prove that equality prevails when V is minimal we use a dimension argument. Note that by (2.5) and (3.2) the space on the left has dimension l: The Gram matrix 2 ðhT j v; T i viÞl1 i;j¼0 is equal tol ð1 j j Þ times thell l identity matrix. The minimality implies that x 2 ker T^ if and only if hx; T^ T j vi ¼ 0, j ¼ 0; 1; 2; . . . . By l computing T^ T j v for j ¼ 0; 1; 2; . . . , we find that l
ker T^ ¼ M? ;
ð3:5Þ
where M ¼ fT ðlþjÞ v þ pj ðT Þv j j ¼ 0; 1; 2; . . .g, and each pj ðzÞ is some polynomial in z of degree at most l 1. Evidently, span M þ spanfv; T v; . . . ; T ðl1Þ vg ¼ H: l This decomposition implies dim M? 4 l and the inclusion (3.4) yields ker T^ 5 l. Because of (3.5) these inequalities can be replaced by equalities. Consequently, equality prevails in the inclusion (3.4). &
Now we consider a coisometric colligation with the operator T^ from (3.1) as the main operator. Here we have to distinguish between the cases j j < 1 and j j > 1. Set u v ^u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; v^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 j1 j j j j1 j j2 j Lemma 3.2. If j j < 1 then the colligation ^u T^ ^ V :¼ h; ^vi 0
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is coisometric in the space H C; if j j > 1 then the colligation ^ T^ u ^ V :¼ h; v^i 0 is coisometric in the space H C. In both cases the characteristic function of ^ or V ^ is the new coisometric colligation V sV ðzÞ : 1 sV ðzÞ ^ if j j < 1 or V ^ if j j > 1 is also minimal. If V is minimal, then V ^V ^ ¼ I is equivalent to Proof. Suppose that j j < 1. The coisometry relation V the following identities; here we use the relations (2.2): þ T^ T^ þ h; ^ui^u ¼ T þ h; viu T h; uiv 1 j j2 1 j j2 1 þ h; uiu; 1 j j2 h; Tviu þ h; uiTv ¼ TT þ 2 1 j j 1 j j2 þ
j j2
h^ v; v^i ¼
hv; vi 1 j j2
hv; vih; uiu þ
1
h; uiu ð1 j j Þ 1 j j2 ¼ TT þ h; uiu ¼ I; 1 ^ hv; viu ¼ 0; T v^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tv þ 1 j j2 1 j j2 2 2
¼ 1:
In order to find the characteristic function sV^ ðzÞ we consider the element ^x ¼ ðI zT^ Þ1 ^ u. It follows that z ^u ¼ ðI zT^ Þ^x ¼ ^x zT^x h^x; viu; 1 j j2 ðI zTÞ1 u z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ^x h^x; viðI zTÞ1 u; 2 2 1 j j 1 j j hðI zTÞ1 u; vi z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ h^x; vi h^x; vihðI zTÞ1 u; vi; 2 2 1 j j 1 j j hðI zTÞ1 u; vi ; h^x; vi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 z 1 j j 1 1j j2 hðI zTÞ u; vi
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and we obtain h^x; vi sV^ ðzÞ ¼ zhðI zT^ Þ1 ^u; v^i ¼ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j j2 ¼
zhðI zTÞ1 u; vi 2 1 z ð1 j j Þ 1 1j j2 hðI zTÞ u; vi
¼
sV ðzÞ : 1 sV ðzÞ
The equality spanf^ v; T^ v^; . . .g ¼ spanfv; T v; . . .g: ^ is minimal also. In the case j j > 1 the proof is implies that if V is minimal, then V similar. & P j 3.2. Now we consider a function s 2 S0 , sðzÞ ¼ 1 j¼0 j z , with j0 j 6¼ 1. By kð 5 1Þ we denote the integer given by (3.2) and by T u ð3:6Þ V¼ h; vi a minimal coisometric colligation in the space H C having s as its characteristic function, H being a Pontryagin space with negative squares, and ¼ 0 . The assumption j0 j 6¼ 1 and (2.5) imply that for l 2 f1; 2; . . . ; kg the subspace spanfv; T v; . . . ; T ðl1Þ vg is nondegenerate, in fact it is positive if j0 j < 1 and negative if j0 j > 1. (We used this fact already in the proof of Lemma 3.1.) In the second case the inequality j0 j > 1 implies that k 4 . For l 2 f1; 2; . . . ; kg, denote by Hl the subspace H :¼ fv; T v; . . . ; T ðl1Þ vg? l
of H and by Pl the orthogonal projection onto Hl . Theorem 3.3. Given the function s 2 S0 with j0 j < 1 and a corresponding minimal coisometric colligation V in H C as in (3.6) with ¼ 0 . Let k be given by (3.2), let l 2 f1; 2; . . . ; kg, and define Pl u Pl T l v l ul ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vl ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Tl ¼ Pl TPl ; l ¼ : 1 j j2 1 j j2 1 j j2 Then the function sl , sl ðzÞ :¼ zl
sðzÞ 0 ; 1 0 sðzÞ
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belongs to the same class S0 and it is the characteristic function of the minimal coisometric colligation Tl ul Vl ¼ h; vl i l in the space Hl C. ^ from Lemma 3.2, and observe that for Proof. We consider the colligation V j j ^ j 2 f0; 1; . . . ; k 1g we have T u ¼ T u and ^j ¼ j =ð1 j j2 Þ ¼ 0. For the characteristic function sV^ we find l þ ^lþ1 z þ Þ ¼ zhðI zT^ Þ1 ^u; v^i sV^ ðzÞ ¼ zl ð^ l1 l lþ1 ¼ zl ðhT^ ^u; v^i þ zhT^ ^u; v^i þ z2 hT^ ^u; v^i þ Þ
l l ¼ zl ð l þ zh^u; T^ v^i þ z2 hT^ ^u; T^ v^i þ Þ l ¼ zl ð l þ zhðI zT^ Þ1 ^u; T^ v^iÞ ¼ zl ð l þ zhðI zT^ l Þ1 ^ul ; v^l iÞ with ^ ul :¼ ^u;
l v^l :¼ T^ v^;
T^ l :¼ T^ ;
l :¼ ^l ¼
l 1 j j2
:
? ? (with H? The Lemma 3.1 implies T^ l H? l ¼ Hl1 Hl 0 :¼ f0g) and 1 ? ? ? ker T^ l ¼ Hl . It follows that ðI zT^ Þ Hl ¼ Hl , T^ Hl Hl and ran T^ l ¼ l Hl and, in particular, Pl v^l ¼ Pl T^ v^ ¼ v^l . So we finally get
hðI zT^ l Þ1 ^ul ; v^l i ¼ hðI zT^ l Þ1 ^ul ; Pl v^l i ¼ hðI zT^ Þ1 ðPl þ ðI Pl ÞÞ^ul ; Pl v^l i ¼ hðI zT^ Þ1 Pl ^ul ; Pl v^l i ¼ hðI zPl T^ Pl Þ1 Pl ^ul ; Pl v^l i ¼ hðI zTl Þ1 ul ; vl i: For the last equality we used that ^ h; viu Pl ¼ Pl TPl ¼ Tl ; Pl T Pl ¼ Pl T þ 1 j j2 1 Pl ^ul ¼ Pl ^u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pl u ¼ ul ; 1 j j2
The Schur Algorithm for Generalized Schur Functions II
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and 1 Pl v^l ¼ Pl T^ l v^ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pl T^ l v 1 j j2 l 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pl T þ h; uiv v ¼ 1 j j2 1 j j2 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pl T l v ¼ vl : 1 j j2 The coisometry of V is proved in the following three steps in which we show the analogs of the three equations in (2.2). In each step we use that, on account of ^ is coisometric. Lemma 3.2, V (1) From the inclusion T^ Hl Hl we see that Pl T^ Pl ¼ T^ Pl and hence Tl Tl þ h; ul iul ¼ Pl T^ Pl T^ Pl þ h; Pl ^uiPl ^u ¼ Pl ðT^ T^ þ h; ^ui^uÞjH ¼ I: l
(2) Since ran T^ l ¼ Hl we have Pl T^ l ¼ T^ l and hence u Tl vl þ l ul ¼ Pl T^ Pl T^ l v^ þ Pl ^ l ¼ P1 ðT^ T^ v^ þ l ^uÞ ðl1Þ ðl1Þ ¼ Pl ðT^ v^ hT^ v^; ^ui^u þ l ^uÞ ðl1Þ ^ui þ ÞP ^u ¼ 0: ¼ ðh^ v; T^ l
l
(3) We distinguish between 1 4 l 4 k 1 and l ¼ k. In the first case l ¼ 0 and l l l l hvl ; vl i þ j l j2 ¼ hPl T^ v^; Pl T^ v^i ¼ hT^ v^; T^ v^i ðl1Þ ðl1Þ ðl1Þ ðl1Þ l ^ui^u; T^ i v^i ¼ hT^ v^ h^ v; T^ ¼ hT^ T^ v^; T^ ðl1Þ ðl1Þ ¼ hT^ v; T^ vi ¼ ¼ h^ v; v^i ¼ 1: In the second case similar calculations give ðk1Þ ^uij2 þ j k j2 ¼ 1 j k j2 þ j k j2 ¼ 1: hvk ; vk i þ j k j2 ¼ h^ v; v^i jh^ v; T^ ^ which is Finally, the minimality of Vl can be derived from the minimality of V guaranteed by Lemma 3.2. We omit the details since the minimality of Vl was shown in [2, Corollary 5.2]. &
3.3. Now we suppose that j0 j > 1. Then the steps in the proof of Theorem 3.3 can be carried out leading to a characteristic function sl , which we now denote by s l , and a colligation Vl . The main operator of this colligation is a contraction in 1 the space ðHl Þ . In an extra step, for l ¼ k we take the inverse function ðs k ðzÞÞ having in its colligation a main operator which is a contraction in Hk , and this is
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the desired Schur transformation. Observe that the last step cannot be carried out 1 for l < k since in this case the inverse function ðs has a pole at z ¼ 0 of l ðzÞÞ order k l. Theorem 3.4. Given the function s 2 S0 with j0 j > 1 and a corresponding minimal coisometric colligation V in H C as in (3.6) with ¼ 0 . Let k be given by (3.2). Then the function sk ðzÞ, 1 0 sðzÞ sk ðzÞ ¼ zk sðzÞ 0 belongs to the class S0k , and it is the characteristic function of the minimal coisometric colligation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 2 k viP u j j 1 h; P T k k B Pk TPk Pk u C B C k k C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q Vk ¼ B B C 2 @ A j j2 1 1 j j k h; Pk T vi k k in the space Hk C. Proof. By the same reasoning as in the proof of Theorem 3.3 we find sðzÞ 0 ¼ sV^ ðzÞ ¼ zl sVl ðzÞ; 1 0 sðzÞ
ð3:7Þ
where l 2 f1; 2; . . . ; kg and Vl is the coisometric colligation 0 1 Pl u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q TP P l l B C 2 B C j j 1 B C Vl ¼ B C l l B h; Pl T vi C @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A 1 j j j j2 1 in the space ðHl Þ C. The relation (3.7) yields for l ¼ k 1 1 0 sðzÞ zk ¼ sVk ðzÞ ; sðzÞ 0 and according to Lemma 2.1 the function on the right hand side is the characteristic function of the colligation Vk in the space Hk C. For the proof of the minimality and the coisometry of the colligation Vk we refer to [2, Theorem 6.1]. & 4. The Case j0 j ¼ 1 4.1. In this section we suppose that in the expansion (2.4) we have j0 j ¼ 1. As before, the integer kð 5 1Þ is determined by the relation (3.2), that is, k 6¼ 0 and, if k > 1, then ð4:1Þ 1 ¼ ¼ k1 ¼ 0: With the functions ^sðzÞ :¼
0 zk ; sðzÞ 0
^s1 ðzÞ :¼
^sðzÞ QðzÞ ; zk
~s1 ðzÞ :¼
1 ^s1 ðzÞ ; zq 1 þ ^s1 ðzÞ
ð4:2Þ
The Schur Algorithm for Generalized Schur Functions II
13
the Schur transformation (1.3) can be written in the form s2kþq ðzÞ ¼ 0 z 1 þ
zk
q
0 zk sðzÞ0
1 ¼ 0 z 1 þ ^sðzÞQðzÞ q
QðzÞ 1 ¼ 0 zq 1 þ ¼ 0~s1 ðzÞ1 : ^s1 ðzÞ
zk
We shall find, step by step according to this formula, colligations with the characteristic functions ^s, ^s1 , ~s1 1 (only if q > 0) and, finally, s2kþq . Define ^u T^ ^ :¼ ð4:3Þ V h; v^i ^ with the entries h; T k viu u T^ :¼ T k1 ; ^u :¼ k1 ; hT u; vi hT u; vi ð4:4Þ T k v : ; ^ :¼ k1 v^ :¼ hT u; vi hT k1 u; vi Note that the definition of these entries here differs from the definition in Section 3 according to the different values of . Lemma 4.1. Let s be the characteristic function of the coisometric colligation (2.1) and let k be defined by (3.2). Then the function 0 zk ^sðzÞ ¼ sðzÞ 0 is the characteristic function of the colligation (4.3). Proof. We have ^sðzÞ ¼ ¼
0 zk sðzÞ 0 0 zk
zhðI zTÞ1 u; vi 0 zk ¼ P1 jþ1 hT j u; vi j¼k1 z 0 ¼ P1 j k1 j T u; vi j¼0 z hT P 0 j j1 ¼ k1 hT u; vi þ 1 u; T k vi j¼1 z hT 0 ¼ : k1 hT u; vi þ zhðI zTÞ1 u; T k vi
It remains to apply Lemma 2.1.
&
14
D. Alpay et al.
In the sequel we assume that V in (2.1) is coisometric and use the notation of Section 2. We often need the following relations for m 2 f0; 1; . . . ; k 1g and n 2 f0; 1; . . .g: hT m v;T n vi ¼ hT ðm1Þ v;T ðn1Þ vihT ðm1Þ v;uihu;T ðn1Þ vi 8 ðnmÞ vi ¼ hTv;T ðnm1Þ vi ¼ ; m 0 nm < hv;T ð4:5Þ ¼ hv;vi ¼ 0; m ¼ n; > : 0 mn ¼ 0; n : T ðl1Þ v þ 0 lk u if l ¼ 2k; 2k þ 1; . . . : k In particular, the elements v; T v; . . . ; T ð2k1Þ v form a Jordan chain (of length 2k) of the operator T^ at zero: T^ v ¼ 0; T^ T v ¼ v; . . . ; T^ T ð2k1Þ v ¼ T ð2k2Þ v: 4. The elements T ðk1Þ v; T ðk2Þ v; . . . ; v form a Jordan chain of length k of the operator T^ at zero, that is, T^ T ðk1Þ v ¼ 0;
T^ T ðk2Þ v ¼ T ðk1Þ v; . . . ; T^ v ¼ T v:
Proof. The first two claims follow from (4.5) and (4.1). In order to prove 3. we first observe that hv; T k viu h0 u; T ðk1Þ viu ¼ Tv ¼ Tv þ 0 u ¼ 0: T^ v ¼ Tv k k Further, hT l v; T k vi u T^ T l v ¼ T T l v k ¼ T ðl1Þ v hT ðl1Þ v; uiu hT ðl1Þ v; T ðk1Þ vi hT ðl1Þ v; uihu; T ðk1Þ vi u k hT ðl1Þ v; T ðk1Þ vi u; ¼ T ðl1Þ v k
The Schur Algorithm for Generalized Schur Functions II
15
and it remains to use (4.5). Finally, 4. follows from hT ðk1Þ v; ui k T v ¼ 0; T^ T ðk1Þ v ¼ T k hv; T k1 ui and, for 0 4 j 4 k 2, hT j v; ui k ðjþ1Þ v jþ1 T k v ¼ T ðjþ1Þ v: T v ¼ T T^ T j v ¼ T ðjþ1Þ v hv; T k1 ui hv; T k1 ui & ^ 1 having ^s1 ðzÞ as its characteristic function. Since Next we find a colligation V ^sðzÞ ¼ ^ þ zh^u; v^i þ z2 hT^ ^u; v^i þ and by Lemma 4.1, the function QðzÞ in the Schur algorithm (1.3) becomes k2 QðzÞ ¼ ^ þ zh^u; v^i þ z2 hT^ ^u; v^i þ þ zk1 hT^ ^u; v^i k2 k3 zkþ1 hT^ ^u; v^i zkþ2 hT^ ^u; v^i z2k ^ :
It follows that k k1 k2 ^sðzÞ QðzÞ ¼ zk fhT^ ^u; v^i þ zðh^u; T^ v^i þ hT^ ^u; v^i Þ k k3 þ z2 ðhT^ ^u; T^ v^i þ hT^ ^u; v^i Þ þ þ k k k1 k þ zk ðhT^ ^u; T^ v^i þ ^ Þ þ zkþ1 hT^ ^u; T^ v^i þ g:
ð4:6Þ
We introduce the new element k v^1 :¼ T^ v^
k1 X
j T j v;
ð4:7Þ
j¼0
and show that the coefficients j can be chosen such that ( ) 1 X k1 j k jþ1 ^sðzÞ QðzÞ ¼ z hT^ ^u; v^i þ z hT^ ^u; v^1 i j¼0
¼ z fhT^ k
k1
^u; v^i þ zhðI zT^ Þ1 ^u; v^1 ig:
ð4:8Þ
To this end we first observe that Lemma 4.2 implies for j ¼ 0; 1; . . . ; k 1 the relations k j ðklÞ j 0 if j 5 l; T^ T v ¼ 0; T^ ð4:9Þ T v¼ T ðklþjÞ v if j < l: Now it follows that hT^
kþl
k kþl ^u; T^ v^i ¼ hT^ ^u; v^1 i; 2kþ1
hence the terms containing z to choose the j ’s such that hT^
k1
2kþ2
;z
l ¼ 0; 1; . . . ;
ð4:10Þ
; . . . in (4.6) and (4.8) coincide. It remains
k k1 ^u; T^ v^i þ ^ ¼ hT^ ^u; v^1 i
ð4:11Þ
16
D. Alpay et al.
and hT^
kl
k l2 kl ^ u; T^ v^i þ h^ v; T^ ^ui ¼ hT^ ^u; v^1 i;
that is, such that k1 k1 X ^ ¼ T^ ^u;
j T j v ;
hT^
l2
l ¼ 2; . . . ; k;
ð4:12Þ
k1 kl X ^u; v^i ¼ T^ ^u;
j T j v :
j¼0
j¼0
By (4.9), these relations imply
0 ¼ and hT^
l2
20 ; k
k1 kl X ^ u; v^i ¼ T^ ^u;
j T j v j¼0
¼
l1 X
kl
j hT^ ^u; T j vi
j¼0
¼
l2 X
h^u; T ðk1Þ vi;
j h^u; T ðklþjÞ vi l1
j¼0
h^u; T ðk1Þ vi; ¼ l1 therefore
j ¼
20 ^ j1 hT u; v^i; k
j ¼ 1; 2; . . . ; k 1:
Thus, finally, ^s1 ðzÞ ¼
^sðzÞ QðzÞ k1 ¼ hT^ ^u; v^i þ zhðI zT^ Þ1 ^u; v^1 i; k z
^ 1 becomes and the colligation V ^1 ¼ V
^u T^ : k1 u; v^i h; v^1 i hT^ ^
Now suppose first that ^s1 ð0Þ ¼ hT^
k1
^u; v^i 6¼ 0. Then the function s2k is given by 0 : s2k ðzÞ ¼ 0 þ ^s1 ðzÞ An application of Lemma 2.1 yields s2k ðzÞ ¼ sV2k0 ðzÞ with
0
0 V2k
h; v^1 i^u ^ B T ^ k1 hT ^u; v^i B ¼B h; v^1 i @ k1 0 hT^ ^u; v^i
1
^u hT^ 0 þ
k1
hT^
^u; v^i 0
k1
^u; v^i
C C C A
ð4:13Þ
The Schur Algorithm for Generalized Schur Functions II
17
Thus, we have found a colligation having s2k as its characteristic function. This colligation, however, need not be minimal. It will be reduced to a minimal one in Subsection 4.3. Here we first find an analogous nonminimal colligation in the case k1 that ^s1 ð0Þ ¼ hT^ ^u; v^i ¼ 0. We introduce for q ¼ 1; 2; . . . the following condition: ðCq Þ
hT^
k1
^ u; v^i ¼ 0;
h^u; v^1 i ¼ ¼ hT^
q2
^u; v^1 i ¼ 0;
hT^
q1
^u; v^1 i 6¼ 0;
k1 where ðC1 Þ means that hT^ ^u; v^i ¼ 0, h^u; v^1 i 6¼ 0:
If ðCq Þ is satisfied we consider the function ~s1 ðzÞ :¼
1 ^s1 ðzÞ ; zq 1 þ ^s1 ðzÞ
which is up to the factor 0 the inverse of the function s2kþq . According to Lemma ~ 1: 2.3, it is the characteristic function of the colligation V ~1 ¼ V
T^ 1 q h; T^ 1 v^1 i
^u ; q1 hT~ 1 ^u; v^1 i
where h; T k viu h; v^1 i^u: T~ 1 :¼ T^ h; v^1 i^u ¼ T k
ð4:14Þ
An application of Lemma 2.1 yields that s2kþq ðzÞ ¼ 0~s1 ðzÞ1 is the characteristic 0 function of the colligation V2kþq : 1 0 ~ 1 q v^1 i^u ^ u h; T C B T~ 1 q1 q1 B hT~ 1 ^u; v^1 i hT~ 1 ^u; v^1 i C 0 C: ð4:15Þ V2kþq ¼ B q C B h; T~ 1 v^1 i0 0 A @ q1 q1 hT~ 1 ^u; v^1 i hT~ 1 ^u; v^1 i k1 0 Hence, if hT^ ^ u; v^i ¼ 0, then the characteristic function of the colligation V2kþq in (4.15) is the Schur transformation s2kþq from (1.3) of the given function s. However, also this colligation need not be minimal. In order to get a minimal colligation, we have to reduce the space H.
4.2. First we prove some more relations between T and T^ . Lemma 4.3. The following relation holds: r1 T^ u ¼ T r1 u
r1 X kþrj j¼1
k
j1 T^ u;
r ¼ 1; 2; . . . :
ð4:16Þ
18
D. Alpay et al.
Proof. We prove (4.16) by induction. For r ¼ 1 it is trivially true. If it is true for r ¼ l we get l1 hT^ u; T k viu l l1 T^ u ¼ T T^ u k l1 X kþlj ^ j1 l1 ¼T T u T u k j¼1 l1 X 1 kþlj ^ j1 l1 k T u T u; T v u k k j¼1 j1 l1 X kþlj ^ ^ j1 hT^ u; T k vi l TT u þ u ¼T u k k j¼1 l1 X 1 kþlj ^ j1 l1 k k hT u; T viu hT u; T viu k k j¼1
¼ T lu
l1 X kþlj j¼1
¼ T lu
k
kþl j u T^ u k
l X kþljþ1
k
j¼1
j1 T^ u:
& The relation (4.16) can also be written in the form r X kþrj ^ j1 T r1 u T u ¼ 0; r ¼ 1; 2; . . . ; k j¼1
ð4:17Þ
which implies that kþr þ
r X j1 hT^ u; v^ikþrj ¼ 0:
ð4:18Þ
j¼1
Lemma 4.4. The element v^1 from (4.7) has the following properties: h^ v1 ; T l vi ¼ 0; l ¼ 0; 1; . . . ; 2k 1:
ð4:19Þ
For s ¼ 0; 1; . . . ; it holds minðk1;sÞ X ðkj1Þ h^ v1 ; T ð2kþsÞ vi ¼ kþs þ 0 s 0 þ kþsj hT^ v^; ^ui k j¼0
þ
s X
kþj1 ^u; v^i; kþsj hT^
j¼0
where 0 0 ¼ 0 and s 0 ¼ s if s > 0.
ð4:20Þ
The Schur Algorithm for Generalized Schur Functions II
19
Proof. If l ¼ 0; 1; . . . ; k, the relation (4.19) follows from (4.5) and Lemma 4.2. If l ¼ k þ r, r ¼ 1; . . . ; k 1, then by (4.5), Lemma 4.2 and (4.18) we get k h^ v1 ; T ðkþrÞ vi ¼ hT^ v^; T ðkþrÞ vi
k1 X
j hT j v; T ðkþrÞ vi
j¼0
hT k v; T r vi 0 ¼ kþr k k r 2 X 0 ^ j1 hT u; v^i ð0 kþrj Þ ¼ 0: k j¼1 It remains to prove (4.20). Using hT j v; T ð2kþsÞ vi ¼ 0
2kþsj
if
j < k;
and the relation (4.16), we obtain h^ v ; T ð2kþsÞ vi 1
k1 2 2 X k j1 ¼ hT^ v^; T ð2kþsÞ vi þ 0 hv; T ð2kþsÞ vi þ 0 hT^ u; v^ihT j v; T ð2kþsÞ vi k k j¼1
X 2 0 kþs1 j1 k 2kþsj hT^ u; v^i ¼ hT^ v^; T ð2kþsÞ vi þ 0 hv; T ð2kþsÞ vi k k j¼1 þ
X 0 kþs1 j1 2kþsj hT^ u; v^i k j¼k 2
k kþs1 ¼ hT^ v^; T ð2kþsÞ vi þ 0 hv; T ð2kþsÞ vi 0 hT kþs1 u T^ u; v^i k X 0 kþs1 j1 2kþsj hT^ u; v^i þ k j¼k 0 k ¼ hT^ v^; T ð2kþsÞ vi þ k 0 k ¼ hT^ v^; T ð2kþsÞ vi þ k
kþs X j¼k s X
j1 2kþsj hT^ u; v^i
kþsj hT^
jþk1
u; v^i:
j¼0
The first summand in the last expression becomes with Lemma 4.2, 3.: k hT^ v^; T ð2kþsÞ vi ðk1Þ ðk1Þ ¼ hT^ v^; T ð2kþs1Þ vi þ 0 kþs hT^ v^; ui k
k1 X ðkj1Þ kþsj hT^ ¼ h^ v; T ðkþsÞ vi þ 0 v^; ui k j¼0
20
D. Alpay et al.
k k1 T v ðkþsÞ 0 X ðkj1Þ ¼ ; T v þ kþsj hT^ v^; ui k j¼0 k k1 1 X ðkj1Þ kþsj hT^ ¼ ðhT ðk1Þ v; T ðkþs1Þ vi k kþs Þ þ 0 v^; ui k j¼0 k k1 X ðkj1Þ kþsj hT^ v^; ui; ¼ kþs þ 0 s 0 þ 0 k k j¼0
&
and (4.20) follows. Corollary 4.5. Under the assumption ðCq Þ and with v^1 in (4.7) we have h^ v1 ; T ð2kþsÞ vi ¼ kþs ;
s ¼ 0; 1; . . . ; q 1:
Proof. Formula (4.20) can be rewritten as k1 X ðkj1Þ ðkþj1Þ ^ui þ hT^ ^u; v^iÞ kþsj ðh^ v; T^ h^ v1 ; T ð2kþsÞ vi ¼ kþs þ 0 s 0 þ k j¼0
þ
s X
kþsj hT^
kþj1
^u; v^i:
j¼k
In the first sum the summand corresponding to j ¼ 0 is zero since hT^ ðk1Þ ^u; v^i ¼ 0. j1 The other summands, because of (4.12), are equal to kþsj hT^ ^u; v^1 i, j ¼ 1; 2; . . . ; k 1. If s 5 k, by (4.10) and (4.11), the second sum equals: s hT^
2k1
^u; v^i þ
s X
kþsj hT^
j¼kþ1 s X
¼ s ^ þ
j1
^u; v^1 i
j1 kþsj hT^ ^u; v^1 i:
j¼k
Thus s X ðj1Þ ^u; v^1 i; kþsj hT^ h^ v1 ; T ð2kþsÞ vi ¼ kþs þ 0 ðs 0 s 00 Þ þ k j¼1
where 00s ¼ 0 if s < k and ¼ s if s 5 k, that is, 00 ¼ 0 . The corollary now easily follows. & Lemma 4.6. Let v^1 and T~ 1 be as in (4.7) and (4.14). Under the assumption ðCq Þ, with v^1 from (4.7) it holds q hT s v; T~ 1 v^1 i ¼ 0;
s ¼ 0; 1; . . . ; 2k þ q 1:
ð4:21Þ
Proof. To prove (4.21) for s ¼ 0; . . . ; 2k 1, we make an induction with respect to q and to k. For q ¼ 0 the relations (4.21) coincide with (4.19). For
The Schur Algorithm for Generalized Schur Functions II
21
1 4 j 4 q we have by the induction assumption ðj1Þ j hv; T~ 1 v^1 i ¼ hT~ 1 v; T~ 1 v^1 i ðj1Þ ¼ hT^ v hv; v^1 i^u; T~ 1 v^1 i ðj1Þ ¼ hv; v^1 ih^u; T~ 1 v^1 i ¼ 0; and for 1 4 i 4 2k 1, j ðj1Þ hT i v; T~ 1 v^1 i ¼ hT~ 1 T i v; T~ 1 v^1 i ðj1Þ ¼ hT^ T i v hT i v; v^1 i^u; T~ 1 v^1 i ðj1Þ ^ 1 ih^u; T~ 1ðj1Þ v^1 i ¼ 0; ¼ hT ði1Þ v; T~ 1 v^1 i hT i v; V where we have used Lemma 4.2, 3., and the last expression vanishes because of the induction assumption. Next, for r ¼ 0; 1; . . . ; q 1, q ðq1Þ hT ð2kþrÞ v; T~ 1 v^1 i ¼ hT~ 1 T ð2kþrÞ v; T~ 1 v^1 i hT ð2kþrÞ v; T k vi ðq1Þ u hT ð2kþrÞ v; v^1 i^u; T~ 1 ¼ TT ð2kþrÞ v v^1 k ðq1Þ ðq1Þ ¼ hT ð2kþr1Þ v; T~ v^ i hT ð2kþr1Þ v; uihu; T~ v^ i 1
1
1
1
hT ð2kþr1Þ v; T ðk1Þ vi hT ð2kþr1Þ v; uihu; T ðk1Þ vi
ðq1Þ hu; T~ 1 v^1 i
k ðq1Þ 0 hu; T~ 1 v^1 i hT ð2kþrÞ v; v^1 i k ð2kþr1Þ ~ ðq1Þ ¼ hT v; T 1 v^1 i 1 ðq1Þ ðhT ð2kþr1Þ v; T ðk1Þ vi þ 0 hT ð2kþrÞ v; v^1 iÞhu; T~ 1 v^1 i: k The second summand vanishes because of hT ð2kþr1Þ v; T ðk1Þ vi ¼ 0 kþr ;
see (4.5), and Corollary 4.5, the first summand vanishes by an induction argument with respect to q. & 0 0 and V2kþq from (4.13) and (4.15) to mini4.3. We reduce the colligations V2k mal ones. To this end we introduce the spaces L :¼ spanfv; T v; . . . ; T ð2k1Þ vg; 2k
L2kþq :¼ spanfv; T v; . . . ; T ð2kþq1Þ vg; and H1 :¼ L? 2k ;
Hq1 :¼ L? 2kþq :
22
D. Alpay et al.
The relations (4.5) imply that for the space L2k the negative and the positive index are equal to k, hence L2k and H1 are nondegenerate, and the negative index of H1 is k. Denote the orthogonal projection onto H1 by P1. Now we suppose that the condition hT^
k1
^u; v^i 6¼ 0
0 from (4.13). Lemma 4.2, 3., (4.19) and (4.4) imply is satisfied and consider V2k j P1 T^ ¼ ðP1 T^ P1 Þj ¼ ðP1 TP1 Þj ;
j ¼ 1; 2; . . . ;
k k T k v v^1 ¼ P1 v^1 ¼ P1 T^ v^ ¼ P1 T^ k h; uiT k v ^ ðk1Þ T k v T ¼ P1 T k k P1 T ð2kÞ v ðk1Þ T k v ¼ P1 T T^ ¼ ¼ ; k
k
and we obtain hðI zT^ Þ1 ^u; v^1 i ¼ hðI zT^ Þ1 ^u; P1 v^1 i ¼
1 X
j
zj hP1 T^ ^u; P1 v^1 i
j¼0
1 ð2kÞ v X j P1 T j u; z ðP1 TP1 Þ ^ ¼ k j¼0 P1 u P1 T ð2kÞ v ; : ¼ 0 ðI zP1 TP1 Þ1 k 8k It remains to apply Lemma 2.1, and the following theorem is proved, except for the last statement. Theorem 4.7. Assume that for s 2 S0 we have j0 j ¼ 1, and that q ¼ 0. If V is a minimal coisometric colligation in the space H C such that sV ðzÞ ¼ sðzÞ then the function ðQðzÞ zk ÞsðzÞ 0 QðzÞ s2k ðzÞ ¼ 0 QðzÞsðzÞ ðQðzÞ þ zk Þ belongs to the class S0k , and it is the characteristic function of the colligation 0 1 0 h; P1 T 2k viP1 u 0 P1 u k1 B P1 TP1 þ ^ k1 C hT ^u; v^ihT k1 u; vi2 hT^ ^u; v^ihT k1 u; vi C B V2k :¼ B C 0 h; P1 T 2k vi 0 @ A k1 0 þ k1 k1 ^ ^ hT ^u; v^i hT ^u; v^ihT u; vi in the space H1 C. This colligation is minimal and coisometric. The theorem coincides with [2, Theorem 7.1]; indeed, the number t2k in that k1 theorem can be written as t2k ¼ k hT^ ^u; v^i=0 . The minimality and coisometry of V2k can be proved as in [2, Theorem 7.1].
The Schur Algorithm for Generalized Schur Functions II
23
Now we suppose that hT^
k1
^u; v^i ¼ 0:
Then for some q 5 1 the condition ðCq Þ is satisfied. For the convenience of the reader we repeat it here: ðCq Þ hT^
k1
^ u;^ vi ¼ 0; h^u;^ v1 i ¼ hT^ ^u;^ v1 i ¼ ¼ hT^
q2
^u;^ v1 i ¼ 0; hT^
q1
^u;^ v1 i6¼ 0;
where v^1 is given by (4.7). It is easy to see that the equalities in the middle are equivalent to h^u; v^1 i ¼ hT^u; v^1 i ¼ ¼ hT q2 ^u; v^1 i ¼ 0: ð4:22Þ We prove some more properties of the operator T~ 1 (given by (4.14)). By Pq1 we denote the orthogonal projection Hq1 Lemma 4.8. It the condition ðCq Þ is satisfied, the following statements hold: 1. The elements v; T v; . . . ; T ð2kþq1Þ v form a Jordan chain of the operator T~ 1 at zero: T~ 1 v ¼ 0; T~ 1 T v ¼ v; . . . ; T~ 1 T ð2kþq1Þ v ¼ T ð2kþq2Þ v: s 2. The elements T~ 1 v^1 , s ¼ 0; 1; . . . ; q 1, are orthogonal to the space L2k and belong to the space L2kþq . 3. Pq T ð2kþqÞ v q Pq1 T~ 1 v^1 ¼ 1 : k Proof. 1. Firstly, by (4.19) and Lemma 4.2, 3., T~ 1 v ¼ T^ v hv; v^1 i^u ¼ 0: If 1 4 j 4 2k þ q 1 we have on account of (4.5) hT j v; T k vi u T^ T j v ¼ TT j v k ¼ T ðj1Þ v hT ðj1Þ v; uiu ¼ T ðj1Þ v þ
0 jk u k
hT j v; T k vi u k
Now, again by (4.19) for 1 4 j 4 2k 1, T~ 1 T j v ¼ T^ T j v hT j v; v^1 i^u ¼ T ðj1Þ v 2 L2kþq1 ; and by (4.20) with s ¼ 0 T~ 1 T ð2kÞ v ¼ T^ T ð2kÞ v hT ð2kÞ v; v^1 i^u hT ð2kÞ v; v^1 i0 u ¼ T ð2k1Þ v þ 0 k u k k ¼ T ð2k1Þ v;
24
D. Alpay et al.
and for 1 4 s 4 q 1 T~ 1 T ð2kþsÞ v ¼ T^ T ð2kþsÞ v hT ð2kþsÞ v; v^1 i^u 0 0 ¼ T ð2kþs1Þ v þ kþs u kþs u k k ð2kþs1Þ v: ¼T 2. The first claim follows from 1. and (4.19), the proof of the second claim is left to the reader. 3. This relation is obtained as follows: h; ui ðq1Þ q v1 T~ 1 Pq1 T~ 1 v^1 ¼ Pq1 T T k v h; ^ui^ v^1 k h; ui k ðq2Þ q ¼ P1 T T T v h; ^ui^ v1 T~ 1 v^1 ¼ k h; ui v1 v^1 ¼ Pq1 T ðq1Þ T T k v h; ^ui^ k k1 k T k v X j v ¼ Pq1 T q v^1 ¼ Pq1 T q T^
T j k
¼
Pq1 T ð2kþqÞ v k
j¼0
: &
The following lemma implies that the space L2kþq has positive index k and negative index k þ q. Lemma 4.9. For r, s ¼ 0; 1; . . . ; q 1 the relations r s hT~ 1 v^1 ; T~ 1 v^1 i ¼ rs hold, where v^1 and T~ 1 are given by (4.7) and (4.14). Proof. For 1 4 j 4 k 1 the relation hT k v; T ðkjÞ v^i ¼ hT j v; v^i ¼
hT j v; T k vi ¼0 k
holds (see (4.5)). We leave it to the reader to show that for 1 4 l 4 q 1 T l v^1 2 spanfL2k ; u; . . . ; T l1 ug:
ð4:23Þ
The space on the right hand side is orthogonal to v^1 because of (4.19) and (4.22). Now, k1 X k k k k k h^ v1 ; v^1 i ¼ h^ v1 ; T^ v^i ¼ T^ v^
j T j v; T^ v^ ¼ hT^ v^; T^ v^i j¼0
ðk1Þ ^ ðk1Þ ðk1Þ ^ ðk1Þ ¼ hTT T^ v^; T v^i ¼ hT^ v^; T v^i
The Schur Algorithm for Generalized Schur Functions II
25
h; uiT k v ^ ðk2Þ ^ ðk1Þ ¼ T T v^; T v^ k 1 ðk2Þ ðk1Þ ðk2Þ ^ ðk1Þ v^; T v^i hT^ v^; uihT k v; T^ v^i ¼ hT T^ k ðk2Þ ^ ðk2Þ v^; T v^i ¼ ¼ h^ v; v^i ¼ hT^ ðk1Þ k k T v T v T v T ðk1Þ v ; ; ¼ 1 ¼ 1: ¼ k
k
k
k
For arbitrary s ¼ 1; 2; . . . ; q 1 we obtain, using again ðCq Þ, s s hT~ 1 v^1 ; T~ 1 v^1 i ¼ hT1s v^1 ; T1s v^1 i ¼ h^ v1 ; v^1 i jh^ v1 ; uij2 ¼ 1: Finally, if 0 < r < q 1 and 0 4 s < r we obtain with l ¼ r s: r l s l hT~ 1 v^1 ; T~ 1 v^1 i ¼ hT~ 1 v^1 ; v^1 i ¼ h^ v1 ; T~ 1 v^1 i h; T k viu ~ l1 T 1 v^1 ¼ v^1 ; T h; v^1 i^u k l2 v1 ; T l v^1 i; ¼ h^ v1 ; T T~ 1 v^1 i ¼ ¼ h^
and this is zero because of the remark following (4.23).
&
The Lemma 4.8 1. implies that T~ 1 L2kþq L2kþq . As above it follows now that Pq1 T~ 1 ¼ Pq1 T~ 1 Pq1 ¼ Pq1 TPq1 ; l
Pq1 T~ 1 ¼ ðPq1 TPq1 Þl ;
l ¼ 2; 3; . . . ;
and we have proved the following theorem, except for the last statement. Theorem 4.10. Assume that for s 2 S0 we have j0 j ¼ 1, and that q > 0. If V is a minimal coisometric colligation in the space H C such that sV ðzÞ ¼ sðzÞ then the function s2kþq ðzÞ ¼
1 ðQðzÞ zk ÞsðzÞ 0 QðzÞ zq 0 QðzÞsðzÞ ðQðzÞ þ zk Þ
belongs to the class S0kq , and it is the characteristic function of the colligation 1 0 h; Pq1 T ð2kþqÞ vi^u Pq1 ^u q q q1 q1 C B P1 TP1 þ hT~ 1 ^u; v^1 i C k hT~ 1 ^u; v^1 i B V2kþq :¼ B C q ð2kþqÞ A @ 0 h; P1 T vi 0 q1 q1 hT~ 1 ^u; v^1 i k hT~ 1 ^u; v^1 i in the space Hq1 C. This colligation is minimal and coisometric. This theorem coincides with [2, Theorem 8.1]: the number there is equal to . The minimality and coisometry of V2kþq then follow from [2, Theorem 8.1].
0 q1 v1 i hT~ 1 ^u;^
26
D. Alpay et al.
5. The Sequence of Schur Determinants Consider again a function s 2 S0 , and let 1 X n zn sðzÞ ¼
ð5:1Þ
n¼0
be its Taylor expansion at x ¼ 0. We consider the matrices 1 0 0 0 0 ... 0 B 1 0 0 ... 0 C C B B 2 1 0 . . . 0 C Aj1 ¼ B C; C B .. .. .. .. .. A @ . . . . . j1 j2 j3 . . . 0 and Gj1 ¼ Ij Aj1 Aj1 ; where Ij stands for the j j identity matrix. The matrix Gj1 is the Gram matrix for the space Lj ¼ spanfv; T v; . . . ; T ðj1Þ vg, and hence it determines the signature of this space. Further, set j1 ðsÞ :¼ det Gj1 ;
j ¼ 1; 2; . . . :
In [5] it is shown that the series in (5.1) defines a function s in the class S0 if and only if there is a natural number n0 such that either n ðsÞ ¼ 0 for all n 5 n0 or n ðsÞ nþ1 ðsÞ > 0 for all n 5 n0 . The case n0 1 ðsÞ 6¼ 0 and n ðsÞ ¼ 0 for all n 5 n0 holds if and only if there exist complex numbers c and 1 ; . . . ; n0 with jcj ¼ 1 and i j 6¼ 1 such that n0 Y z j : sðzÞ ¼ c 1 j z j¼1 In the following we show that the number of negative eigenvalues of the matrix Gn , if it is invertible, is determined through the number of sign changes of the sequence 1; 0 ðsÞ; 1 ðsÞ; . . . ; n ðsÞ; provided the zero entries in this sequence located between two nonzero entries are replaced by nonzero numbers following the simple rules for Toeplitz matrices. Although the matrix Gn is not Toeplitz these rules can be applied here. In order to see this, following [3] we introduce for n ¼ 0; 1; 2; . . . the matrices ð5:2Þ M ¼ B diagð1 G ÞB ; n
where
n
0 B B Bn :¼ B @
n
n
c0 c1 .. .
0 c0 .. .
... ... .. .
cnþ1
cn
. . . c0
0 0 .. .
1 C C C; A
The Schur Algorithm for Generalized Schur Functions II
27
and the ck are determined recursively by the relations c0 ¼ 1;
ck ¼
k1 X
k ¼ 1; 2; . . . ; n þ 1:
cj k1j ;
j¼0
It was shown in [3] that Mn is a hermitian Toeplitz matrix. Since c0 ¼ 1 we have det Mn ¼ n ðsÞ;
n ¼ 0; 1; . . . :
Let "n :¼ sign n ðsÞ
if
n ðsÞ 6¼ 0:
ð5:3Þ
If, for example, h1 ðsÞ 6¼ 0, h ðsÞ ¼ hþ1 ðsÞ ¼ ¼ hþp1 ðsÞ ¼ 0, hþp ðsÞ 6¼ 0, where p > 0, then since Mn is a hermitian Toeplitz matrix the number p is odd (see [12, Lemma 16.1]) and we have pþ1
"hþp ¼ ð1Þ 2 "h1 (see [12, Corollary 2 to Lemma 16.1]). For the indices h; h þ 1; . . . ; h þ p 1 with vanishing determinants we define the numbers (see [12, Formula (16.7)]) jðj1Þ 2
"hþj1 :¼ ð1Þ
"h1 ;
j ¼ 1; . . . ; p:
ð5:4Þ
Now the following theorem is an immediate consequence of [12, Theorem 16.2]. For a matrix B by ðBÞ we denote the number of its negative eigenvalues. Theorem 5.1. If for 0 4 m < n the matrices Gm , Gn are invertible and the numbers "j , j ¼ 0; 1; . . . ; n, are defined according to (5.3), (5.4), then ðGn Þ equals the number of sign changes in the sequence 1, "0 ; "1 ; . . . ; "n , and the difference ðGn Þ ðGm Þ equals the number of sign changes in the sequence "m ; "mþ1 ; . . . ; "n . For the function s 2 S0 we denote by JðsÞ the set of the indices j 2 N for which the Schur transformation sj is defined. The set JðsÞ cannot be characterized through the sequence of the n ðsÞ only. However, we prove the following theorem. Theorem 5.2. Let s 2 S0 and assume that h 2 JðsÞ, h 5 1. Then h1 ðsÞ 6¼ 0, and the following holds: (i) If h h1 > 0, then for sh ðzÞ case (1.1) of the Schur transformation applies; (ii) if h h1 < 0, then for sh ðzÞ case (1.2) applies; (iii) if h ðsÞ ¼ hþ1 ðsÞ ¼ ¼ hþp1 ðsÞ ¼ 0, hþp ðsÞ 6¼ 0, then for sh ðzÞ case (1.3) applies with k ¼ pþ1 2 . Proof. The first claim follows from the fact that in all cases (1.1)–(1.3) of the Schur transformation the subspace Lh is nondegenerate; see [2, Theorem 9.1]. Next we prove (iii), the statements (i) and (ii) follow in the same way. First we observe that the assumptions imply that the space Lh is nondegenerate, the spaces Lhþ1 ; Lhþ2 ; . . . ; Lhþp are degenerate and Lhþpþ1 is nondegenerate.
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D. Alpay et al.
Since h1 6¼ 0 and h ¼ 0, we have h ¼ rank Gh1 ¼ rank Gh . Since hþp 6¼ 0 and hþp1 ¼ 0, we have rank Ghþp ¼ h þ p þ 1 and rank Ghþp1 ¼ h þ p 1, see [12, Corollary to Lemma 6.1]. Now we use that rank Gj is nondecreasing as a function of j and conclude that there is a largest number k0 such that rank Ghþpk0 > h. In view of (5.2) we may treat the matrices Gn as if they were Toeplitz matrices, therefore k0 can be viewed as the characteristic of Ghþp1 (see [12, (14.3)]) and applying [12, Theorem 15.1 and (15.8)] we obtain h þ p 1 ¼ h þ 2k0 . Hence k0 ¼ ðp 1Þ=2, and, on account of [12, Theorem 15.6], rank Gh ¼ rank Ghþ1 ¼ ¼ rank Ghþk1 ¼ h
ð5:5Þ
and rank Ghþkþj1 ¼ h þ 2j;
j ¼ 1; 2; . . . ; k;
0
where we have set k :¼ k þ 1 ¼ ðp þ 1Þ=2. By (5.5), Lhþk contains a k-dimensional isotropic subspace. The elements T h v; T ðhþ1Þ v; . . . ; T ðhþk1Þv span a complement of Lh in Lhþk . Therefore the orthogonal projections (modulo a multiplicative constant) ðk1Þ v^; T^ v^; . . . ; T^ v^ ð5:6Þ of the elements T h v; T ðhþ1Þ v; . . . ; T ðhþk1Þ v onto the orthogonal complement of Lh in Lhþpþ1 ¼ Lhþ2k also span a neutral subspace. On the other hand the elements in (5.6) are the first k basis elements of the state space of the colligation corresponding to sh , see [2, Theorem 9.1] and its proof. Hence jsh ð0Þj ¼ 1 and the coefficient of zk is the first nonzero coefficient with index 51 in the Taylor expansion of sh ðzÞ around z ¼ 0. & We mention that the numbers k in the case of (1.2) and q in the case of (1.3) can, apparently, not be determined from the sequence ð n ðsÞÞ. As a consequence of this theorem the following relations between the set JðsÞ of all integers j such that the Schur transformation sj is defined and the sequence of the numbers j1 ðsÞ, j ¼ 1; 2; . . . , can be formulated. If j 2 N is such that j1 ðsÞ 6¼ 0, j ðsÞ ¼ ¼ jþp1 ðsÞ ¼ 0, jþp ðsÞ 6¼ 0 then j 2 JðsÞ, j þ 1; jþ 2; . . . ; j þ p 2 = JðsÞ and j þ p þ 1 can belong to JðsÞ or not. In any case the function sjþpþ1 can be formed according to [2, Remark 6.2]: If it has a pole at zero then j þ p þ 1 2 = JðsÞ, otherwise, if it is holomorphic at zero, then j þ p þ 1 2 JðsÞ. If jþpþ1 ðsÞ ¼ 0, then the next index j þ p þ 2 does not belong to JðsÞ. If jþpþ1 ðsÞ 6¼ 0 and sign jþp ðsÞ ¼ sign jþpþ1 ðsÞ then j þ p þ 2 2 JðsÞ, if sign jþp ðsÞ 6¼ sign jþpþ1 ðsÞ then in order to see if j þ p þ 2 2 JðsÞ we form the function sjþpþ2 in accordance with [2, Remark 6.2]. If it has a pole at z ¼ 0 of order q then j þ p þ 2, j þ p þ 3; . . . ; j þ p þ q 2 = JðsÞ, but j þ p þ q þ 1 2 JðsÞ. If j 2 N is such that j1 ðsÞ and j ðsÞ are nonzero and have the same sign, then the functions sj and sjþ1 are defined and their Schur kernels have the same number of negative squares. If sign j1 ðsÞ sign j ðsÞ ¼ 1 and sj is defined then sjþ1 can always be defined in accordance with [2, Remark 6.2], but it may not belong to the class S0 .
The Schur Algorithm for Generalized Schur Functions II
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References [1] Andersson LE (1990) Algorithms for solving inverse eigenvalue problems for Sturm–Liouville equations. In: Sabatier PC (ed) Inverse Problems in Action. Berlin Heidelberg New York: Springer [2] Alpay D, Azizov TYa, Dijksma A, Langer H (2001) The Schur algorithm for generalized Schur functions I: Coisometric realizations. Operator Theory: Adv Appl 129: 1–36 [3] Alpay D, Constantinescu T, Dijksma A, Rovnyak J (2002) Notes on interpolation in the generalized Schur class I. Applications of realization theory. Operator Theory: Adv Appl 134: 67–97 [4] Alpay D, Dijksma A, Rovnyak J, de Snoo H (1997) Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces. Operator Theory: Adv Appl 96 [5] Bertin MJ, Decomps-Guilloux A, Grandet-Hugot M, Pathiaux-Delfosse M, Schreiber JP (1992) Pisot and Salem Numbers. Basel: Birkh€auser [6] Bart H, Gohberg I, Kaashoek MA (1979) Minimal Factorization of Matrix and Operator Functions. Operator Theory: Adv Appl 1 [7] Chamfy C (1958) Fonctions meromorphes sur le circle unite et leurs series de Taylor. Ann Inst Fourier 8: 211–251 urgus B, Dijksma A, Langer H, de Snoo HSV (1989) Characteristic functions of unitary [8] C colligations and of bounded operators in Krein spaces. Operator Theory: Adv Appl 41: 125–152 [9] Dufresnoy J (1958) Sur le probleme des coefficients par certaines fonctions dans le cercle unite. Ann Acad Sci Fenn Ser AI 250(9): 1–7 [10] Delsarte P, Genin Y, Kamp Y (1991) Pseudo-Caratheodory functions and hermitian Toeplitz matrices. Philips J Res 41: 1–54 [11] Gladwell GML (1991) The application of Schur’s algorithm to an inverse eigenvalue problem. Inverse Problems 7: 557–565 [12] Iokhvidov IS (1982) Hankel and Toeplitz Matrices and Forms, Algebraic Theory. Boston, Mass: Birkh€auser € ber einige Fortsetzungsprobleme, die eng mit der Theorie [13] Krein MG, Langer H (1977) U hermitescher Operatoren im Raume zusammenh€angen, Teil I: Einige Funktionenklassen und ihre Darstellungen. Math Nachr 77: 187–236 € ber die Potenzreihen, die im Innern des Einheitskreises beschr€ankt sind. J Reine [14] Schur I (1917) U Angew Math 147: 205–232; English translation in: I. Schur Methods in Operator Theory and Signal Processing. Operator Theory: Adv Appl 18: 31–59 € ber die Potenzreihen, die im Innern des Einheitskreises beschr€ankt sind; [15] Schur I (1918) U Fortsetzung. J Reine Angew Math 148: 122–145; English translation in: I. Schur Methods in Operator Theory and Signal Processing. Operator Theory: Adv Appl 18: 31–59 Authors’ addresses: D. Alpay, Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel, e-mail:
[email protected]; T. Ya. Azizov, Department of Mathematics, Voronezh State University, 394693 Voronezh, Russia, e-mail:
[email protected]; A. Dijksma, Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, e-mail:
[email protected]; H. Langer, Department of Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria, e-mail: hlanger@ mail.zserv.tuwien.ac.at
