Periodic GMP Matrices

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2016
Автор:	Eichinger, B.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2016
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147763
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Periodic GMP Matrices / B. Eichinger // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-147763
record_format	dspace
spelling	irk-123456789-1477632019-02-16T01:26:11Z Periodic GMP Matrices Eichinger, B. We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable ''magic formula'' for this new class. 2016 Article Periodic GMP Matrices / B. Eichinger // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E05; 30F15; 47B36; 42C05; 58J53 DOI:10.3842/SIGMA.2016.066 http://dspace.nbuv.gov.ua/handle/123456789/147763 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable ''magic formula'' for this new class.
format	Article
author	Eichinger, B.
spellingShingle	Eichinger, B. Periodic GMP Matrices Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Eichinger, B.
author_sort	Eichinger, B.
title	Periodic GMP Matrices
title_short	Periodic GMP Matrices
title_full	Periodic GMP Matrices
title_fullStr	Periodic GMP Matrices
title_full_unstemmed	Periodic GMP Matrices
title_sort	periodic gmp matrices
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/147763
citation_txt	Periodic GMP Matrices / B. Eichinger // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT eichingerb periodicgmpmatrices
first_indexed	2025-07-11T02:47:31Z
last_indexed	2025-07-11T02:47:31Z
_version_	1837317032994930688
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 066, 19 pages Periodic GMP Matrices? Benjamin EICHINGER Institute for Analysis, Johannes Kepler University, Linz, Austria E-mail: benjamin.eichinger@jku.at Received January 28, 2016, in final form June 29, 2016; Published online July 07, 2016 http://dx.doi.org/10.3842/SIGMA.2016.066 Abstract. We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP matrices. They form a certain Generalization of matrices related to the strong Moment Problem. This class allows us to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover, due to their structural similarity we can carry over numerous results from the direct and inverse spectral theory of periodic Jacobi matrices to the class of periodic GMP matrices. In particular, we prove an analogue of the remarkable “magic formula” for this new class. Key words: spectral theory; periodic Jacobi matrices; bases of rational functions; functional models 2010 Mathematics Subject Classification: 30E05; 30F15; 47B36; 42C05; 58J53 1 Introduction We start by recalling some known facts from the spectral theory of Jacobi matrices; see [19, Chapter 5]. Let dσ+ be a real scalar compactly supported measure and {Pn(x)}n≥0 the corre- sponding orthonormal polynomials, which we obtain by orthonormalizing the monomials 1, x, x2, . . . . It is easy to see that they obey xPn(x) = anPn−1(x) + bnPn(x) + an+1Pn+1(x), an > 0, that is, the multiplication by the independent variable in the basis {Pn(x)}n≥0 has the matrix J+ =  b0 a1 0 a1 b1 a2 0 . . . . . . . . . . . . . . .  , where \|an\|, \|bn\| ≤ C for C such that dσ+ has support [−C,C]. Matrices of this sort are called one-sided Jacobi matrices. In general, we call an operator one-sided if it is an operator on `2+ = `2(Z≥0) and correspondingly two-sided if it acts on `2 = `2(Z). Moreover, let {en}n∈Z denote the standard basis of `2 and `2− = `2 `2+ with the classical embedding of `2+ into `2. By 〈 (J+ − z)−1e0, e0 〉 = ∫ dσ+(x) x− z , ?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:benjamin.eichinger@jku.at http://dx.doi.org/10.3842/SIGMA.2016.066 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 B. Eichinger one can associate to every one-sided Jacobi matrix a measure dσ+ and in fact, this describes a one-to-one correspondence between real scalar compactly supported measures and one-sided (bounded) Jacobi matrices. To deal with periodic Jacobi matrices (i.e., Jacobi matrices with periodic coefficient sequences) it appears to be useful to extend them naturally to two-sided Jacobi matrices and in the following we will derive another basis, which turned out to be more suitable in their spectral theory than polynomials. This technique applied to reflectionless Jacobi matrices with homogeneous spectra was suggested by Sodin and Yuditskii [20]. Let J+ be a periodic Jacobi matrix, J its two-sided extension and J− = P−JP ∗ −, where P− denotes the orthogonal projection onto `2−. One can show that there exists a polynomial, Tp, of degree p such that the spectrum, E, of J is given by 1) E = T−1 p ([−2, 2]), (1.1) 2) all critical points of Tp (i.e., zeros of T ′p) are real and 3) \|Tp(c)\| ≥ 2 for all critical points c; cf. [19, Theorem 5.5.25]. We define the resolvent functions by rJ−(z) = 〈 (J− − z)−1e−1, e−1 〉 , rJ+(z) = 〈 (J+ − z)−1e0, e0 〉 , z ∈ C+. In the periodic case they can be given explicitly in terms of the orthogonal polynomials and they satisfy 1 rJ+(x+ i0) = a2 0r J −(x+ i0), for almost all x ∈ E. (1.2) If the resolvent functions of a two-sided Jacobi matrix satisfies (1.2) on a set A, we call it reflectionless on A. This property is characteristic in the following sense. Let us define for the given set E = T−1 p ([−2, 2]) the isospectral torus (finite gap class) of Jacobi matrices, J(E), by J(E) = {J : σ(J) = E and J is reflectionless on E}. (1.3) Then J(E) consists of all periodic Jacobi matrices, whose spectrum is the set E. The following well-known parametrization justifies the name torus: J(E) = { J(α) : α ∈ Rg/Zg } , (1.4) where the map α 7→ J(α) is one-to-one; cf. [1, 2, 14, 21]. We recall a proof of this fact in Section 3 based on the previously mentioned second basis. The idea in this construction is the following: Let Γ∗ be the group of all characters of the fundamental group of the domain C \E. Note that Γ∗ ∼= Rg/Zg. Due to the properties of Tp, one can define a function Φ(z) by Tp(z) = Φ(z) + 1 Φ(z) , z ∈ C \ E. (1.5) This is possible since Tp(z) ∈ C \ [−2, 2] for z ∈ C \E, which is the image of the Joukowski map ζ 7→ ζ + 1 ζ . Fix α ∈ Γ∗ and let H2(α) be the Hardy space of character automorphic functions on C \E with character α and L2(α) the corresponding space of character automorphic square integrable functions. The decomposition H2(α) = KΦ(α)⊕ ΦH2(α) defines a natural basis {eαn}∞n=0 of H2(α) and respectively {eαn}∞n=−∞ a basis of L2(α) with the properties that eαn+p = Φeαn and KΦ(α) = {eα0 , . . . , eαp−1}. The multiplication by z in the basis {eαn}∞n=−∞ is a Jacobi matrix J(α). Moreover, if α runs through Γ∗ we obtain J(E), i.e., this construction proves (1.4). Note that eαn+p = Φeαn means that Φ is the symbol of Sp, where S Periodic GMP Matrices 3 is the right shift on `2, i.e., Sen = en+1. Since zΦ(z) = Φ(z)z implies J(α)Sp = SpJ(α), the existence of the function Φ shows the periodicity of the Jacobi matrices J(α). Finally, we would like to point out that the relation (1.5) is the so-called magic formula in terms of the functional model, which is a surprising characterization of the isospectral torus of periodic Jacobi matrices. It states that J ∈ J(E) ⇐⇒ Tp(J) = Sp + S−p. We have seen that spectra of periodic Jacobi matrices are a finite union of intervals of a very special structure. Namely, there has to be a polynomial Tp with the properties 1), 2) and 3). In fact, it is not hard to show that this condition is also sufficient. Nevertheless, the isospectral torus (1.3) can be defined for more general sets, in particular for so-called finite gap sets, E, of the form E = [b0,a0] \ g⋃ j=1 (aj ,bj), g ∈ N, aj < bj < aj+1. (1.6) The corresponding bases {eαn} can be defined as well, which leads exactly in the same way to a parametrization of J(E) by Γ∗. Moreover, it gives explicit formulas for the coefficients of J(α) by means of continuous functions A, B on Γ∗, i.e., an(α) = A(α− nµ), bn(α) = B(α− nµ), (1.7) where µ is a fixed character defined only by the spectrum. This in particular implies that all elements of J(E), which may not be periodic, are for sure almost periodic. Note that (1.7) coincides with the formulae given in [21, Theorem 9.4]. In fact, this holds even for Jacobi mat- rices with infinite gap, homogeneous spectra. Hardy classes on finitely or infinitely connected domains are discussed, e.g., in [9, 10, 24]. Since it is based on the existence of a polynomial, Tp, with the properties 1), 2) and 3), the characterization of the isospectral torus in terms of the magic formula is only possible for spectra of periodic Jacobi matrices, which turned out to be a powerful tool in the past. It was crucial in proving the first generalization of the remarkable Killip–Simon theorem; cf. [5, 13]. More specifically, Damanik, Killip and Simon [5] were able to generalize the Killip–Simon theorem for the case that E is the spectrum of periodic Jacobi matrices and \|Tp(c)\| > 2 for all critical points c, which is a strong restriction on E. The idea of GMP matrices is to substitute the polynomial Tp by a rational function. In [6], we carried out this idea for the simplest case, namely if E is the arbitrary union of two distinct intervals. Note that the Damanik, Killip and Simon theorem only covers two intervals of equal length. Nevertheless, we can always find a rational function, ∆E , of the form ∆E(z) = λ0z + c0 + λ1 c1 − z , λ0, λ1 > 0, (1.8) such that ∆−1 E ([−2, 2]) = E. By a linear change of variable we may assume that c1 = 0. This suggests to consider matrices obtained by orthonormalizing the family of functions 1, −1 x , x, (−1)2 x2 , . . . , (1.9) for a given real compactly supported measure dσ+. Denoting this basis by ϕn, we call the matrix of multiplication by the independent variable w.r.t. this basis a one-sided SMP matrix; see also [12]. Let us mention that they are also called Jacobi-Laurent matrices; cf. [11]. 4 B. Eichinger The connection to CMV matrices should not go unmentioned. CMV matrices are the Jacobi matrix analogue for measures supported on the unit circle. Already Szegő discussed orthogonal polynomials, ψn, w.r.t. a measure, dµ, supported on the unit circle and showed that there are constants {αn}∞n=0 in D, called Verblunsky coefficients, so that√ 1− \|αn\|2ψn+1(z) = zψn(z)− αnznψn(1/z). Due to Verblunsky [22], who defined them in another context, the map dµ 7→ {αn} is one-to-one and onto all of D∞. Recent developments are due to Cantero, Moral and Velázques [4]. They considered bases obtained by orthonormalizing families of the sort (1.9) and showed that the matrix of the multiplication operator is a special structured five-diagonal matrix. For a given measure, dµ, the entries can be given in terms of the Verblunsky coefficients. Recognizing this characteristic structure they could use it to give a constructive definition of CMV matrices, which uniquely defined them, in the sense that there is a one-to-one correspondence between measures and CMV matrices. For a review on CMV matrices see [18]. In [6], we were also able to identify this characteristic structure for SMP matrices and to give a constructive definition of them. Again, it was then more convenient for us to define them as two-sided matrices. Roughly speaking, a SMP matrix A and its shifted inverse −S−1A−1S (note that we assumed that 0 is not in the spectrum) are five-diagonal matrices such that all even entries on the most outer diagonal vanish and the odd ones are positive. This structure perfectly fits to the following “generalized magic formula”: Proposition 1.1 ([6]). Let E be an arbitrary union of two intervals around zero and ∆E the corresponding rational function of (1.8). Moreover, let A(E) be the set of all two-periodic SMP matrices with its spectrum on E. Then A ∈ A(E) ⇔ ∆E(A) = S2 + S−2. To deal with arbitrary finite gap sets of the form (1.6), one first has to generalize (1.8), which is done by the following lemma. The Ahlfors function, Ψ, of the domain C \ E is the function that maximizes the value Capa(E) = ∣∣ lim z→∞ zΨ(z) ∣∣ (the so-called analytic capacity), among all functions, which vanish at infinity and are bounded by one in modulus. Lemma 1.2. The function ∆E(z) := Ψ(z) + 1 Ψ(z) (1.10) is a rational function of the form ∆E(z) = λ0z + c0 + g∑ j=1 λj cj − z , (1.11) with λj > 0, j ≥ 0, cj ∈ (aj ,bj), j ≥ 1 and E = [b0,a0] \ g⋃ j=1 (aj ,bj) = ∆−1 E ([−2, 2]). (1.12) In fact, if we demand that Im ∆E(z) > 0 for Im z > 0 and lim z→∞ ∆E(z) = ∞, ∆E is the unique rational function with the property (1.12). Periodic GMP Matrices 5 Proof. Due to [16], we have 1−Ψ(z) 1 + Ψ(z) = √√√√ g∏ j=0 z − aj z − bj =: G(z). Therefore, ∆E(z) := Ψ(z) + 1 Ψ(z) = 2 1 +G2(z) 1−G2(z) , is of the form Pg+1(z)/Qm(z), where m ≤ g. Like in [15, Chapter VII], we see that ImG2(z) > 0 for Im z > 0, which then clearly also holds for ∆E . Therefore, G2 is increasing on the interval (aj ,bj) and has a zero at aj and a pole at bj . Hence, there is exactly one pole of ∆E in each gap. Since G2(∞) = 1, there is also a pole at infinity. To prove uniqueness let ∆(z) be a rational function with the claimed properties. First, we notice that, due to the argument principle, it is not possible that ∆ has a pole in the upper half plane. This and lim z→∞ ∆(z) =∞ already implies that ∆ is of the form (1.11). Since ∆′(x) > 0 on R \ {c1, . . . , cg}, ∆−1([−2, 2]) = E implies ∆(aj) = 2 and ∆(bj) = −2 for j ≥ 0. This defines ∆ uniquely. � Let C = {c1, . . . , cg} be a collection of distinct real points and dσ− a measure such that the points ck don’t belong to its support. Like SMP matrices, we define a one-sided GMP matrix, A−, as the matrix of the multiplication operator w.r.t. the basis obtained by orthonor- malizing the family of functions 1, 1 cg − x , 1 cg−1 − x , . . . , 1 c1 − x , x, 1 (cg − x)2 , . . . . This is discussed in detail in the Appendix of [25]; see also [3]. Their characteristic structure, which looks quite complicated at the first glance, will be used in the following definition, but first we would like to point out another property of multiplication operators w.r.t. rational functions. Namely, since ck does not belong to the support of the measure, we can also consider multiplication by 1 ck−x . Hence, if the first block of A− corresponds to the basis[ 1, 1 c1 − x , . . . , 1 cg − x ] (1.13) then the linear change of variable y = 1 ck−x leads to the basis related to[ −1 y , 1 y(c1)− y , . . . , 1, . . . , 1 y(cg)− y ] , (1.14) which says that the shifted resolvents should be of the same shape. In fact, in the construction the spaces (1.13) and (1.14) serve as cyclic subspaces for ∆(x) and ∆̃(y) = ∆(x), respectively. See also proof of Theorem 1.5. The structure of A− and this certain invariance property of the resolvents is now used as a definition for two-sided GMP matrices. By T ∗ we denote the conjugated operator to an operator T , or the conjugated matrix if T is a matrix. In particular, for a column vector ~p ∈ Cg+1, (~p)∗ is a (g + 1)-dimensional row vector. The notation T− denotes the upper triangular part of a matrix T (excluding the main diagonal), and T+ = T − T− is its lower triangular part (including the main diagonal). First of all, the GMP class depends on an ordered collection of distinct points C={c1, . . ., cg}. 6 B. Eichinger Definition 1.3. We say that A is of the class A if it is a (g + 1)-block Jacobi matrix A =  . . . . . . . . . A∗(~p−1) B(~p−1) A(~p0) A∗(~p0) B(~p0) A(~p1) . . . . . . . . .  such that ~p = (~p, ~q ) ∈ R2g+2, A(~p) = δg~p ∗, B(~p) = (~q~p ∗)− + (~p~q ∗)+ + C̃, and C̃ =  c1 . . . cg 0  , ~pj = p (j) 0 ... p (j) g  , ~qj = q (j) 0 ... q (j) g  , p(j) g > 0. We call {~pj}j∈Z the generating coefficient sequences (for the given A). Definition 1.4. A matrix A ∈ A belongs to the GMP class if the matrices {ck − A}gk=1, for 1 ≤ k ≤ g, are invertible, and moreover S−k(ck −A)−1Sk are also of the class A. To abbreviate we write A ∈ GMP(C). We call a GMP matrix one-block periodic or simply periodic if ~pj = ~p for all j ∈ Z. Thus, we pay a quite high price in giving up the simple structure of Jacobi matrices, but in return we get (1.10), which will in particular allow us to parametrize the finite gap class of almost periodic Jacobi matrices by periodic GMP matrices. In the same way as for Jacobi matrices, the decomposition H2(α) = KΨ(α)⊕ΨH2(α), leads to a new basis {fαn }∞n=−∞ such that fαn+p = Ψfαn , KΨ(α) = {fα0 , . . . , fαg } and the multipli- cation by z is a GMP matrix. Note that the property fαn+p = Ψfαn shows that the corresponding matrix is periodic. Defining the isospectral torus of GMP matrices by A(E,C) = {A ∈ GMP(C) : σ(A) = E and A is periodic}, we obtain the following analogue of (1.4): Theorem 1.5. Let E be a finite gap set and C be a fixed ordering of the zeros of the corre- sponding Ahlfors function. Then A(E,C) = { A(α,C) : α ∈ Rg/Zg } , where A(α,C) is the multiplication by the independent variable w.r.t. the basis {fαn }. The map α 7→ A(α,C) is one-to-one up to the identification (pj , qj) 7→ (−pj ,−qj) for j = 0, . . . , g − 1. Moreover, (1.10) is the magic formula for GMP matrices in terms of our functional model. Theorem 1.6. Let A ∈ GMP(C). Then A ∈ A(E,C) ⇐⇒ ∆E(A) = Sg+1 + S−(g+1). Periodic GMP Matrices 7 This was one of the main observations which allowed Yuditskii in [25] to generalize the Killip– Simon theorem to arbitrary systems of intervals. Finally, we would like to point out that the direct spectral theory of periodic GMP matrices has numerous similarities to the one of periodic Jacobi matrices. It is based on the fact that like Jacobi matrices GMP matrices can be written as a two-dimensional perturbation of a block diagonal matrix. Let a0 = ‖~p‖, ẽ0 = 1 a0 P+Ae−1 and A± = P±AP ∗ ±, P+ = I − P−. Then A = [ A− 0 0 A+ ] + a0(〈·, e−1〉ẽ0 + 〈·, ẽ0〉e−1). Definition 1.7. Let A ∈ GMP(C) be a periodic GMP matrix with coefficients ~p and ~q. Let p = [ p q ] ∈ R2. We introduce the matrix functions a(z; p) = a(z,∞; p, q) = [ 0 −p 1 p z−pq p ] , (1.15) and a(z, c; p) = I − 1 c− z [ p q ] [ p q ] j, j = [ 0 −1 1 0 ] . Then the product A(z) = a(z, c1; p0)a(z, c2; p1) · · · a(z, cg; pg−1)a(z; pg) is called the transfer matrix associated with the given A. Moreover, we define its discriminant by ∆A(z) = trA(z). (1.16) Theorem 1.8. Let A ∈ GMP(C) be a periodic GMP matrix with coefficients ~p and ~q. Then A has purely absolutely continuous spectrum, which is given by σ(A) = σac(A) = { z ∈ C : ∆A(z) ∈ [−2, 2] } . If for a given set E A ∈ A(E,C), then we show in Lemma 3.9 that indeed ∆E = ∆A. This allows us to explain an alternative definition of GMP matrices. We define for a periodic GMP matrix with generating coefficients ~p and ~q Λk(~p) = −tr { k−2∏ m=0 a(ck, cm+1; pm)pk−1p ∗ k−1j g−1∏ m=k a(ck, cm+1; pm)a(ck; pg) } , (1.17) where by definition Λk(~p) = −Res cktrA(z). Let us consider the last non-zero diagonal of ∆E(A), i.e., ∆E(A)j,g+1+j for 0 ≤ j ≤ g. By definition of GMP matrices, (ck − A)−1 k−1,g+k > 0 for 1 ≤ k ≤ g, whereas (cl − A)−1 k−1,g+k = Ak−1,g+k = 0 for l 6= k. Moreover, (ck − A)−1 g,2g+1 = 0 for 1 ≤ k ≤ g and Ag,2g+1 > 0. Thus, on the last non-zero diagonal of ∆E(A) only one of the summands is non-zero. Note that the relation ∆E(z) = ∆A(z) implies λk = Λk(~p). The previous consideration, the magic formula and this identity yield Λk(~p)(ck −A)−1 k−1,g+k = 1 for 1 ≤ k ≤ g. It is important to mention that all this just served as explanation, but is not necessary to prove the following alternative definition of GMP matrices. 8 B. Eichinger Theorem 1.9. Let A ∈ A be periodic with generating coefficients ~p. Then A ∈ GMP(C) if and only if Λk(~p) > 0 for 1 ≤ k ≤ g. Proof. The proof is based on the idea that one can find the entries of the inverse matrices (ck −A)−1 explicitly; cf. [25, Lemma 3.2 and Theorem 3.3] � The relation ∆E(z) = ∆A(z) finally leads to an algebraic description for A(E,C). Theorem 1.10. Let A ∈ GMP(C) be periodic with generating coefficients ~p. Then A ∈ A(E,C) if and only if pg = 1 λ0 , qg = −c0 − λ0 g−1∑ j=1 pjqj , Λk(~p) = λk for k = 1, . . . , g, (1.18) where Λk(~p) is defined as in (1.17). The organization of the paper is as follows. In Section 2 we deal with the direct spectral theory of periodic GMP matrices. That is, we prove Theorem 1.8. In Section 3 we show that periodic GMP matrices arise as the multiplication by the independent variable w.r.t. {fαn } and prove Theorems 1.6 and 1.10. In both sections we first recall the known theory for Jacobi matrices and then adapt this construction to the GMP case. The results of the paper were first announced in [7]. 2 Direct spectral theory of periodic GMP matrices Let J be a p-periodic two-sided Jacobi matrix with generating coefficients {aj , bj}gj=0. Its transfer matrix is defined by AJ(z) = a(z, a1, b0/a1)a(z, a2, b1/a2) · · · a(z, ap, bp−1/ap), where a(z, aj , bj−1/aj) = [ 0 −aj 1 aj z−bj−1 aj ] , is defined as in (1.15). Moreover, the discriminant is given by Tp(z) = trAJ(z). This is not a notational conflict, but trAJ(z) is indeed the polynomial in (1.1). The spectrum of J is purely absolutely continuous and σ(J) = σac(J) = T−1 p ([−2, 2]), (2.1) cf. [19, Chapter 5]. One way of proving this is to write J as J = [ J− 0 0 J+ ] + a0(〈·, e−1〉e0 + 〈·, e0〉e−1), (2.2) where {e0, e−1} spans a cyclic subspace for J . From this representation it is easy to deduce that the matrix resolvent function RJ(z), defined by RJ(z) = [ 〈(J − z)−1e−1, e−1〉〈(J − z)−1e0, e−1〉〈(J − z)−1e−1, e0〉〈(J − z)−1e0, e0〉 ] , Periodic GMP Matrices 9 admits the representation RJ(z) = [ rJ−(z)−1 a0 a0 rJ+(z)−1 ]−1 , where rJ±(z) = 〈(J± − z)−1e−1±1 2 , e−1±1 2 〉. Using what is called coefficient stripping in [19, Theorem 3.2.4] one can then find a representation for RJ(z) that implies (2.1). In this chapter we will follow this strategy for GMP matrices. Let A be a one-block periodic GMP matrix. Due to [25, Proposition 5.5], {e−1, ẽ0} span a cyclic subspace for A. Like in (2.2), we can represent A as A = [ A− 0 0 A+ ] + a0(〈·, e−1〉ẽ0 + 〈·, ẽ0〉e−1), where a0 = ‖~p ‖. Hence, defining R(z) = [ 〈(A− z)−1e−1, e−1〉〈(A− z)−1ẽ0, e−1〉〈(A− z)−1e−1, ẽ0〉〈(A− z)−1ẽ0, ẽ0〉 ] , we obtain R(z) = [ r−(z)−1 a0 a0 r+(z)−1 ]−1 , (2.3) where r−(z) = 〈(A− − z)−1e−1, e−1〉, r+(z) = 〈(A+ − z)−1ẽ0, ẽ0〉. The following theorem is an analogue of [5, Theorem 3.2.4] for periodic GMP matrices. First, we introduce some notations, which will be used in the proof. For ~x ∈ Rg+1, we define sk~x =  x0 ... xg−k  . Moreover, let the Mk’s be upper triangular matrices such that B(~p, ~q )− ~p(~q )∗ = M(~p, ~q ) := M0 = [ M1 0 0 0 ] + (−~p qg + ~q pg)δ ∗ g and Mk = [ Mk+1 0 0 cg+1−k ] + (−sk~p qg−k + sk~q pg−k)(skδg−k) ∗ for 1 ≤ k ≤ g − 1. Theorem 2.1. Let[ R(z, p, p) R(z, g, p) R(z, p, g) R(z, g, g) ] = [ 〈(B(~p, ~q )− z)−1~p, ~p 〉〈(B(~p, ~q )− z)−1~δg, ~p 〉〈(B(~p, ~q )− z)−1~p, ~δg〉〈(B(~p, ~q )− z)−1~δg, ~δg〉 ] . Let A be a periodic GMP matrix, r+(z) the resolvent function of A+, i.e., r+(z) = 〈 (A+ − z)−1ẽ0, ẽ0 〉 . 10 B. Eichinger Then we have a2 0r+(z) = a2 0r+(z)Ã11(z) + Ã12(z) a2 0r+(z)Ã21(z) + Ã22(z) , (2.4) where a2 0 = ‖~p‖2 and Ã(z) = [ Ã11(z) Ã12(z) Ã21(z) Ã22(z) ] = 1 R(z, p, g) [ R(z, p, p)R(z, g, g)−R(z, p, g)2 −R(z, p, p) R(z, g, g) −1 ] . Proof. We write A+ = [ B(~p, ~q ) 0 0 A+ ] + a0(〈·, eg〉ẽ1 + 〈·, ẽ1〉eg), where ẽ1 = Sg+1ẽ0 and apply the Sherman–Morrison–Woodbury formula (cf. [8, Section 2.1.3]) to prove the theorem. � Theorem 2.2. Let Ã be defined as in Theorem 2.1 and A be the transfer matrix of A. Then we have Ã(z) = A(z). Proof. First, we represent B(~p, ~q) as a one-dimensional perturbation of a lower diagonal matrix. Applying the Sherman–Morrison–Woodbury formula again, leads to a representation of Ã in terms of M0. Using that M0 is a lower triangular matrix, we obtain Ã(z) = A0(z)a(z; pg, qg), where A0(z) = I − [ 〈(M1 − z)−1u1~p, u1~p 〉〈(M1 − z)−1u1~q, u1~p 〉〈(M1 − z)−1u1~p, u1~q 〉〈(M1 − z)−1u1~q, u1~q 〉 ] j. Using again that all M ′js are lower triangular matrices, we find that Aj−1(z) = Aj(z)a(z, cg+1−j ; pg−j , qg−j), where Aj−1(z) = I − [ 〈(Mj − z)−1uj~p, uj~q 〉〈(Mj − z)−1uj~q, uj~p 〉〈(Mj − z)−1uj~p, uj~q 〉〈(Mj − z)−1uj~q, uj~q 〉 ] j. � Remark 2.3. 1. Note that A is normalized such that detA = 1. 2. For a general (non periodic) GMP matrix the only difference in (2.4) is that in the right- hand side a0 is replaced by ‖~p1‖ and r+ by r (1) + , which is the resolvent function related to the shifted GMP matrix S−(g+1)ASg+1. This explains the name transfer matrix. 3. Unlike Jacobi matrices, we consider the relation between the resolvent function of the initial and the g + 1-shifted GMP matrix, since this shift preserves its structure. Applying the same calculations to r− leads to the following theorem. Periodic GMP Matrices 11 Theorem 2.4. For a periodic GMP matrix A, let A−(z) = [ A−11(z) A−12(z) A−21(z) A−22(z) ] be the transfer matrix of r−, i.e., r−(z) = r−(z)A−11(z) + A−12(z) r−(z)A−21(z) + A−22(z) . (2.5) Then it is of the form A−(z) = a−(z,∞; pg, qg)a−(z, cg; pg−1, qg−1) · · · a−(z, c1; p0, q0), where a−(z,∞; p, q) = [ 0 −1 p p z−pq p ] and a−(z, c; p, q) = I − 1 c− z [ q p ] [ q p ] j. Proof. The proof is the same as for r+, but in order to extract a−(z,∞; p, q) in the first step, one has to write the “mirrored” B-block as a one-dimensional perturbation of an upper triangular matrix. � The following corollary will be an important ingredient in the proof of Theorem 1.8. Corollary 2.5. With the notation from above the entries of A and A− are related by A11(z) = A−11(z), A12(z) = −A−21(z), A21(z) = −A−12(z), A22(z) = A−22(z). Proof. This follows by the relation A(z) = [ 1 0 0 −1 ] A−(z)∗ [ 1 0 0 −1 ] . � Clearly, (2.4) is a quadratic equation for a2 0r+. Let ∆A(z) = trA(z) and V (z) = A11(z) − A22(z). Using the normalization of A, we see that for all z ∈ C+ a2 0r+(z) = 1 2A21(z) ( V (z) + √ ∆A(z)2 − 4 ) , (2.6) where one takes the branch of the square root with √ ∆A(z)2 − 4 = ∆A(z) + O ( 1 ∆A(z) ) near z =∞. Lemma 2.6. The function r−1 − is the second solution of (2.4). That is, 1 r−(z) = 1 2A21(z) ( V (z)− √ ∆A(z)2 − 4 ) , Proof. Due to (2.5), we have r−(z) = A−22r− − A−12 −A−21r− + A−11 (z). Using Corollary 2.5, we see that r−1 − is a solution of (2.4). That r−1 − is distinct form a2 0r+ on C+ follows since Im r−1 − < 0, while Im a2 0r+ > 0 on C+. � 12 B. Eichinger Lemma 2.7. The resolvent function of a periodic GMP matrix A admits the following repre- sentation: R(z) = 1 2a2 0 √ ∆A(z)2 − 4 [ −2a2 0A21(z) a0V (z) a0V (z) 2A12(z) ] + 1 2a0 [ 0 1 1 0 ] . Proof. This is a consequence of (2.3) and Lemma 2.6. � Proof of Theorem 1.8. We have σac(A) = {z ∈ C : ∆A(z) ∈ [−2, 2]}. Pure point spectrum can only appear at poles of Aij . But since −Res ck∆A(z) > 0, this is not possible. � 3 Functional models 3.1 Definitions We start with a uniformization of the domain C \ E. That is, there exists a Fuchsian group Γ and a meromorphic function z : D→ C \ E with the following properties: (i) ∀ z ∈ C \ E ∃ ζ ∈ D : z(ζ) = z, (ii) z(ζ1) = z(ζ2) ⇔ ∃ γ ∈ Γ: ζ1 = γ(ζ2). We fix z by the normalization condition z(0) =∞ and lim ζ→0 (ζz)(ζ) > 0. By Γ∗ = {α \|α : Γ→ R/Z such that α(γ1γ2) = α(γ1) + α(γ2)} we denote the group of characters of Γ. Note that Γ is equivalent to the fundamental group of C \ E and therefore Γ∗ ∼= Rg/Zg. Let H2 = H2(D) denote the standard Hardy space of the disk. For α ∈ Γ∗ we define the Hardy space of character automorphic functions by H2(α) = { f ∈ H2 : f ◦ γ = e2πiα(γ)f, γ ∈ Γ } . Fix z0 ∈ C \ E and consider the associated orbit orb(ζ0) = z−1(z0) = {γ(ζ0) : γ ∈ Γ}. The Blaschke product bz0(ζ) with zeros at z−1(z0) is called the Green function of the group Γ; cf. [17]. It is normalized so that bz0(0) > 0 if z0 6= ∞ and (zb∞)(0) > 0. It is related with the standard Green function G(z, z0) of C \ E by log 1 \|bz0(ζ)\| = G(z(ζ), z0). The function bz0 is character automorphic. We denote the corresponding character by µz0 . We define kα(ζ, ζ0) = kαζ0(ζ) as the reproducing kernels of H2(α), i.e., 〈f, kαζ0〉 = f(ζ0), ∀ f ∈ H2(α). If z0 =∞, we use the abbreviations b(ζ) = b∞(ζ), µ = µ∞ and kα(ζ) = kα0 (ζ). 3.2 Functional models for Jacobi matrices In this section we recall some results from the theory of finite gap Jacobi matrices. In particular, we will use functional models to explain the appearance of a polynomial Tp in the theory of periodic Jacobi matrices. Theorem 3.1 ([23]). The system of functions eαn(ζ) = bn(ζ) kα−nµ(ζ)√ kα−nµ(0) Periodic GMP Matrices 13 (i) forms an orthonormal basis in H2(α) for n ∈ Z≥0 and (ii) forms an orthonormal basis in L2(α) for n ∈ Z, where L2(α) = { f ∈ L2(T) : f ◦ γ = e2πiα(γ)f, γ ∈ Γ } . Proof. We only prove (i). H2(α) can be decomposed into H2(α) = {eα0 } ⊕H2 0 (α), (3.1) where H2 0 (α) = {f ∈ H2(α) : f(0) = 0}. Since H2 0 (α) = bH2(α− µ), iterating the previous step leads to H2(α) = {eα0 } ⊕ { beα−µ0 } ⊕ { b2eα−2µ 0 } ⊕ · · · . It is easy to see that this system is complete. � The following theorem describes a one-to-one correspondence between Γ∗ and J(E). Theorem 3.2 ([23]). The multiplication operator by z in L2(α) with respect to the basis {eαn} from Theorem 3.1 is the following Jacobi matrix J = J(α): zeαn = an(α)eαn−1 + bn(α)eαn + an+1(α)eαn+1, where an(α) = A(α− nµ), A(α) = (zb)(0) √ kα(0) kα+µ(0) and bn(α) = B(α− nµ), B(α) = (zb)(0) b′(0) + { (kα)′ (0) kα(0) − (kα+µ) ′ (0) kα+µ(0) } + (zb)′ (0) b′(0) . This Jacobi matrix J(α) belongs to J(E). Thus, we have a map from Γ∗ to J(E). Moreover, this map is one-to-one. Remark 3.3. Since S−1J(α)S = J(α − µ), we see that J(E) consists of p-periodic Jacobi matrices if and only if pµ = 0Γ∗ . Note that pµ = 0Γ∗ implies that bp(γ(ζ)) = bp(ζ). Therefore, the function, Φ(z), defined by Φ(z) := Φ(z(ζ)) = bp(ζ) is single valued. Since (i) \|Φ\| < 1 in C \ E and \|Φ\| = 1 on E, (ii) Φ has a zero of multiplicity p at infinity, it is not hard to show that Φ is the function given by (1.5). Moreover, this definition implies that for all α ∈ Γ∗ Tp(J(α)) = Sp + S−p, which proves one direction of the magic formula. 14 B. Eichinger 3.3 Functional models for periodic GMP matrices We have already mentioned in the introduction that the Ahlfors function, Ψ, will serve as an analogue of Φ for general finite gap sets. Due to Lemma 1.2 and its proof, we obtain the following properties of Ψ: (i) \|Ψ\| < 1 in C \ E and \|Ψ\| = 1 on E. (ii) Ψ(z) = 0⇔ z ∈ {c1, . . . , cg} ∪ {∞}. All this implies that log 1 \|Ψ(z)\| = G(z) + g∑ j=1 G(z, cj). Therefore, Ψ(z(ζ)) = b(ζ) g∏ j=1 bcj (ζ). In particular, µ + ∑g j=1 µcj = 0Γ∗ . We define the per- mutation π by demanding that cj ∈ (aπ(j),bπ(j)). Moreover, we fix generators of the group Γ, {γj}gj=1, where γj corresponds to a closed curve, which starts at ∞ and passes through the gap (aπ(j),bπ(j)); cf. [19, Chapter 9, Section 6]. Due to the symmetry of C+ and C− we can choose a fundamental domain, F ⊂ D, of the group, which is symmetric w.r.t. ζ 7→ ζ. Choosing ζj ∈ (F ∩ z−1(cj)), we have ζj = γj(ζj). Let us fix these ζj ∈ D. The relation (1.10) together with the factorization of Ψ into Blaschke products suggests to consider the following counterpart of (3.1). Let βn = α− n∑ k=1 µck and ηn = √ exp(−2πiβn(γn+1)). Then H2(α) = {kαζ1 , . . . , k α ζg , k α} ⊕ΨH2(α) = {fα0 } ⊕ · · · ⊕ {fαg } ⊕ΨH2(α), where fα0 = η0 kαζ1√ kαζ1(ζ1) , fα1 = η1 bc1k α−µc1 ζ2√ k α−µc1 ζ2 (ζ2) , . . . , fαg = ∏g j=1 bcjk α+µ√ kα+µ(0) . Remark 3.4. The unimodular factors ηn are chosen such that the matrix representation of multiplication by z w.r.t. this basis, i.e., the corresponding GMP matrix is real. Note that the square root of e−2πiβn(γn+1) is defined up to the choice of a multiplicative constant ±1. Thus, in fact to a given character α we associate 2g bases and therefore GMP matrices. This is the reason for the identification (pj , qj)→ (−pj ,−qj) in Theorem 1.5. In order to prove completeness of the system in L2(α) we show the following lemma. Lemma 3.5. Let θ 6= 1 be a character automorphic inner function with character χ. Then ∞⋂ n=1 { θ−nH2(α+ nχ) }⊥ = {0}. Proof. In the following we will use that L2(α) H2(α) = ΘH2 0 (ν − α), where Θ is the inner part of the derivative of b; cf. [23]. It is character automorphic and we denote its character by ν. Let f ∈ ⋂∞ n=1 { θ−nH2(α+ nχ) }⊥ . Thus, for all n ∈ N 0 = 〈f, θ−nh〉 = 〈θnf, h〉, ∀h ∈ H2(α+ nχ). Periodic GMP Matrices 15 Therefore, θnf ∈ H2(α+nχ)⊥ = ΘH2 0 (ν − α− nχ). Hence, there exists gν−α−nχ ∈ H2 0 (ν −α− nχ) such that Θf = θngν−α−nχ, i.e., Θf ∈ ∞⋂ n=1 θnH2(ν − α− nχ). For g ∈ ⋂∞ n=1 θ nH2(ν − α− nχ) and arbitrary ζ0 ∈ D, we see that \|g(ζ0)\|2 = ∣∣〈g, θnθn(ζ0)kν−α−nχζ0 〉 ∣∣2 ≤ ‖g‖2\|θ(ζ0)\|2nkν−α−nχζ0 (ζ0). Since kν−α−nχζ0 (ζ0) is uniformly bounded in n from above, the right-hand side converges to zero as n→∞ and hence g(ζ0) = 0. � Theorem 3.6. The system of functions fαn = fαn (ζ; c1, . . . , cg) = Ψmfαj , n = (g + 1)m+ j, 0 ≤ j ≤ g, (i) forms an orthonormal basis in H2(α) for n ∈ Z≥0 and (ii) forms an orthonormal basis in L2(α) for n ∈ Z. Proof. By construction, this system is orthogonal. The completeness of the second system follows by the previous lemma. � Proof of Theorem 1.5. We show that A(α) is a GMP matrix. Clearly, A(α) is g+1-periodic. Let Pα− be the projection onto L2(α) H2(α). Then we have Pα−(zfαn ) = pn(α)fα−1, j = 0, . . . , g. Since z is self-adjoint and A(α) has constant block-coefficients, this shows that A(~p) has the right structure.We have pn(α) = 〈zfαn , fα−1〉 = (zb)(0)ηn n∏ k=1 bck(0)kβn(0, ζn+1)√ kβnζn+1 (ζn+1)kα+µ(0) for 0 ≤ n ≤ g. Since kα(ζ̄n) = kα(ζn, 0), we obtain kβn(0, ζn+1) = kβn(ζn+1, 0) = kβn(γn+1(ζn+1), 0) = η2 nk βn(0, ζn+1). Thus, pn(α) are real. In particular, pg(α) > 0. We define qm(α) = − ηm √ kα+µ(0)kβm+µ(ζm+1) m∏ k=1 bck(0)b(ζm+1)kβm+µ(0) √ kβmζm+1 (ζm+1) Let n > m. Note that zfαn has a simple poles at z−1(∞). Therefore, zfαn − (zfαn b)(0)kα+µ kα+µ(0)b ∈ H2(α). Using this and the fact that (zfαn )(ζm+1) = 0, we obtain 〈zfαn , fαm〉 = − ηnηm√ kβnζn+1 (ζn+1) √ kβmζm+1 (ζm+1) × (zb)(0) n∏ k=m+1 bck(0)kβnζn+1 (0) kβm+µ(0) kβm+µ(ζm+1) b(ζm+1) = pn(α)qm(α), 16 B. Eichinger by definition. Using in addition that b(ζ) = b(ζ) we obtain in the same way as before that kβm+µ(ζm+1, 0) b(ζm+1) = ηm 2kβm+µ(ζm+1, 0) b(ζm+1) . Hence qm(α) ∈ R. Finally, we consider the diagonal terms, i.e., 〈zfαn , fαn 〉 = cn+1 − (zb)(0)kβnζn+1 (0) kβnζn+1 (ζn+1)kβn+µ(0) kβn+µ(ζn+1) b(ζn+1) = cn+1 + pn(α)qn(α). The structure of the resolvents given in Definition 1.4 follows by the conformal invariance of the Ahlfors function. More specifically, if w = wj = 1 cj−z , then Ψj(w) := Ψ(z) is the Ahlfors function in the w-plane. The given ordering C generates the specific ordering Cj = { 1 cj+1 − cj , . . . , 1 cg − cj , 0, 1 c1 − cj , . . . , 1 cj−1 − cj } and the multiplication by w is again a periodic GMP matrix (up to an appropriate shift). That is, S−j(cj −A(α,C))−1Sj ∈ A(Ej ,Cj), where Ej = {y = 1 cj−x : x ∈ E}. This shows that A(α) ∈ GMP(C). Hence, A(α) ∈ A(E). Now, we turn to the map A(E,C) → Γ∗. To A ∈ A(E,C), we associate the resolvent func- tions a2 0r+ and r−1 − . Due to (2.6), Lemma 2.6 and Theorem 1.8, A is reflectionless on E. Hence, we can apply the construction of [23, Sections 3 and 4] to obtain α ∈ Γ∗. Due to uniqueness of the associated character α, this map is one-to-one, up to the identification (pk, qk) 7→ (−pk,−qk). � 3.4 The magic formula and parametrization of A(E,C) Let us turn to the proof of the magic formula and Theorem 1.10. The following lemma describes the coefficients of ∆A in terms of A. Lemma 3.7. Let A ∈ GMP(C) be a periodic GMP matrix with generating coefficients ~p and ∆A(z) its discriminant, cf. (1.16). Then ∆A is a rational function with simple poles at ck and infinity, i.e., ∆A(z) = d0 + ν0z + g∑ k=1 νk z − ck . Moreover, the coefficients are given by ν0 = 1 pg , d0 = −qg − ν0 g−1∑ j=1 pjqj , νk = Λk(~p) for k = 1, . . . , g, where Λk is defined in (1.17). Proof. Considering the residues of ∆A at ck and infinity we obtain the coefficients νk, for k = 0, . . . , g. Thus it remains to show the formula for d0. To this end, we write A(z) = Az +B + g∑ j=1 1 cj − z Cj , Periodic GMP Matrices 17 where the constant term is given by B = [ 0 −pg 1 pg −qg ] − 1 pg g∑ j=1 [ 0 p2 j−1 0 pj−1qj−1 ] Since ∆A = trA, we also obtain the expression for d0. � Remark 3.8. Note that due to the definition of GMP matrices νk > 0 for k = 0, . . . , g. Thus, ∆A maps the upper half plane into itself. Lemma 3.9. Let E be a finite gap set and ∆E the corresponding function from Lemma 1.2. Let A ∈ GMP(C) be periodic. Then the following are equivalent: (i) A ∈ A(E,C), (ii) ∆A(z) = ∆E(z), (iii) ∆A(A) = ∆E(A). Proof. Let ∆A(z) = ∆E(z). By Theorem 1.8, we obtain that σ(A) = { z ∈ C : ∆A(z) ∈ [−2, 2] } = {z ∈ C : ∆E(z) ∈ [−2, 2]} = E. Thus, A ∈ A(E,C). On the other hand, if A ∈ A(E,C), then {z ∈ C : ∆A(z) ∈ [−2, 2]} = E. By the previous remark and the uniqueness of ∆E , we obtain ∆A(z) = ∆E(z). Hence, (i) ⇐⇒ (ii). (i) =⇒ (ii) is clear. On the last non-zero diagonal of ∆A(A), i.e., ∆A(A)j,g+1+j for j = 0, . . . , g, only one of the summands is non-vanishing. With the notation from the previous lemma, (iii) yields (νk − λk)(ck −A)−1 k−1,g+k = 0, (λ0 − ν0)Ag,2g+1 = 0. Hence, ∆A(z) = ∆E(z). � Proof of Theorem 1.6. Let A ∈ A(E,C). Due to Proposition 1.5, there exists α ∈ Γ such that A = A(α). Hence, (1.10) is the magic formula in terms of functional models. Let A ∈ GMP(C) satisfy ∆E(A) = Sg+1 + S−(g+1). Năıman’s lemma (cf. [19, Lemma 8.2.4]) implies that A is periodic. Since by definition A ∈ A(σ(A),C), we obtain that ∆A(A) = Sg+1 + S−(g+1) = ∆E(A). By Lemma 3.9 A ∈ A(E,C). � Proof of Theorem 1.10. Let A ∈ A(E,C). Due to Lemma 3.9, ∆A(z) = ∆E(z) and by Lemma 3.7, the coefficients of A satisfy (1.18). On the other hand, if the coefficients satis- fy (1.18), then ∆A(z) = ∆E(z) and hence A ∈ A(E,C). � 3.5 The Jacobi flow on A(E,C) Finally, we would like to explain briefly the idea of the so-called Jacobi flow on GMP matrices, which was another main tool in proving the Killip–Simon theorem for general system of intervals. There is an obvious map F : A(E)→ J(E) defined by FA(α) = J(α). The question is, how to find the coefficients of J(α) in terms of coefficients of A(α). 18 B. Eichinger Proposition 3.10. To α ∈ Γ∗, we associate a GMP matrix A(α) with coefficients (~p(α), ~q(α)) and a Jacobi matrix J(α) with coefficients (aj(α), bj(α)). Then we have a0(α) = ‖~p(α)‖ and b−1(α) = pg(α)qg(α). Proof. Let P+(α) be the orthogonal projection onto H2(α). Since fα−1 = eα−1, we see that a0(α) = ‖P+(α)zeα−1‖ = ‖P+(α)zfα−1‖ = ‖p(α)‖ and b−1(α) = 〈eα−1, ze α −1〉 = 〈fα−1, zf α −1〉 = pg(α)qg(α). � Since S−1J(α)S = J(α− µ), one can find all coefficients of J(α) by finding the coefficients of A(α− µ). Definition 3.11. We define the Jacobi flow on A(E,C) as the dynamical system generated by the following map: JA(α) = A(α− µ), α ∈ Γ∗. For a parametric description of this map see [25, Theorem 4.5]. Acknowledgements The author was supported by the Austrian Science Fund FWF, project no: P25591-N25. He would like to thank his advisor Peter Yuditskii for his guidance and help during the preparation of this paper. Finally, he is grateful to the anonymous referees for their remarks that improved the presentation of the paper. References [1] Ahiezer N.I., Orthogonal polynomials on several intervals, Soviet Math. Dokl. 1 (1960), 989–992. [2] Aptekarev A.I., Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Math. USSR Sb. 125(167) (1984), 233–260. [3] Baratchart L., Kupin S., Lunot V., Olivi M., Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties, J. Anal. Math. 114 (2011), 207–253, arXiv:0812.2050. [4] Cantero M.J., Moral L., Velázquez L., Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29–56, math.CA/0204300. [5] Damanik D., Killip R., Simon B., Perturbations of orthogonal polynomials with periodic recursion coefficients, Ann. of Math. 171 (2010), 1931–2010, math.SP/0702388. [6] Eichinger B., Puchhammer F., Yuditskii P., Jacobi flow on SMP matrices and Killip–Simon problem on two disjoint intervals, Comput. Methods Funct. Theory 16 (2016), 3–41. [7] Eichinger B., Yuditskii P., Killip–Simon problem and Jacobi flow on GSMP matrices, arXiv:1412.1702. [8] Golub G.H., Van Loan C.F., Matrix computations, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. [9] Hasumi M., Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Math., Vol. 1027, Springer-Verlag, Berlin, 1983. [10] Heins M., Hardy classes on Riemann surfaces, Lecture Notes in Math., Vol. 98, Springer-Verlag, Berlin – New York, 1969. [11] Hendriksen E., Nijhuis C., Laurent–Jacobi matrices and the strong Hamburger moment problem, Acta Appl. Math. 61 (2000), 119–132. http://dx.doi.org/10.1070/SM1986v053n01ABEH002918 http://dx.doi.org/10.1007/s11854-011-0016-9 http://arxiv.org/abs/0812.2050 http://dx.doi.org/10.1016/S0024-3795(02)00457-3 http://arxiv.org/abs/math.CA/0204300 http://dx.doi.org/10.4007/annals.2010.171.1931 http://arxiv.org/abs/math.SP/0702388 http://dx.doi.org/10.1007/s40315-014-0104-9 http://arxiv.org/abs/1412.1702 http://dx.doi.org/10.1007/BFb0071447 http://dx.doi.org/10.1007/BFb0080775 http://dx.doi.org/10.1023/A:1006441812051 http://dx.doi.org/10.1023/A:1006441812051 Periodic GMP Matrices 19 [12] Katsnelson V., On a family of Laurent polynomials generated by 2 × 2 matrices, Complex Anal. Oper. Theory , to appear, arXiv:1507.06101. [13] Killip R., Simon B., Sum rules for Jacobi matrices and their applications to spectral theory, Ann. of Math. 158 (2003), 253–321, math-ph/0112008. [14] Krichever I.M., Algebraic curves and nonlinear difference equations, Russ. Math. Surv. 33 (1978), no. 4, 255–256. [15] Levin B.Ja., Distribution of zeros of entire functions, Translations of Mathematical Monographs, Vol. 5, Amer. Math. Soc., Providence, R.I., 1980. [16] Pommerenke C., Über die analytische Kapazität, Arch. Math. (Basel) 11 (1960), 270–277. [17] Pommerenke C., On the Green’s function of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 409–427. [18] Simon B., CMV matrices: five years after, J. Comput. Appl. Math. 208 (2007), 120–154, math.SP/0603093. [19] Simon B., Szegő’s theorem and its descendants. Spectral theory for L2 perturbations of orthogonal polyno- mials, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. [20] Sodin M.L., Yuditskii P.M., Infinite-dimensional real problem of Jacobi inversion and Hardy spaces of character-automorphic functions, Dokl. Akad. Nauk 335 (1994), 161–163. [21] Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Mono- graphs, Vol. 72, Amer. Math. Soc., Providence, RI, 2000. [22] Verblunsky S., On positive harmonic functions: a contribution to the algebra of Fourier series, Proc. London Math. Soc. S2-38 (1935), 125–157. [23] Volberg A., Yuditskii P., Kotani–Last problem and Hardy spaces on surfaces of Widom type, Invent. Math. 197 (2014), 683–740, arXiv:1210.7069. [24] Widom H., Hp sections of vector bundles over Riemann surfaces, Ann. of Math. 94 (1971), 304–324. [25] Yuditskii P., Killip–Simon problem and Jacobi flow on GMP matrices, arXiv:1505.00972. http://dx.doi.org/10.1007/s11785-016-0568-x http://dx.doi.org/10.1007/s11785-016-0568-x http://arxiv.org/abs/1507.06101 http://dx.doi.org/10.4007/annals.2003.158.253 http://arxiv.org/abs/math-ph/0112008 http://dx.doi.org/10.1070/RM1978v033n04ABEH002503 http://dx.doi.org/10.1007/BF01236943 http://dx.doi.org/10.1016/j.cam.2006.10.033 http://arxiv.org/abs/math.SP/0603093 http://dx.doi.org/10.1112/plms/s2-38.1.125 http://dx.doi.org/10.1112/plms/s2-38.1.125 http://dx.doi.org/10.1007/s00222-013-0495-7 http://arxiv.org/abs/1210.7069 http://dx.doi.org/10.2307/1970862 http://arxiv.org/abs/1505.00972 1 Introduction 2 Direct spectral theory of periodic GMP matrices 3 Functional models 3.1 Definitions 3.2 Functional models for Jacobi matrices 3.3 Functional models for periodic GMP matrices 3.4 The magic formula and parametrization of A(E,C) 3.5 The Jacobi flow on A(E,C) References

Periodic GMP Matrices

Репозитарії

Схожі ресурси