Periodic GMP Matrices

0.45 MB·

by  Benjamin Eichinger

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 066, 19 pages
Periodic GMP Matrices(cid:63)
Benjamin EICHINGER
Institute for Analysis, Johannes Kepler University, Linz, Austria
E-mail: [email protected]
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. WerecallcriteriaonthespectrumofJacobimatricessuchthatthecorresponding
isospectraltorusconsistsofperiodicoperators. MotivatedbythoseknownresultsforJacobi
matrices, we define a new class of operators called GMP matrices. They form a certain
6
1 Generalization of matrices related to the strong Moment Problem. This class allows us
0 to give a parametrization of almost periodic finite gap Jacobi matrices by periodic GMP
2 matrices. Moreover, due to their structural similarity we can carry over numerous results

l fromthedirectandinversespectraltheoryofperiodicJacobimatricestotheclassofperiodic
u
GMP matrices. In particular, we prove an analogue of the remarkable “magic formula” for
J
this new class.
7
Key words: spectraltheory; periodicJacobimatrices; basesofrationalfunctions; functional

]
models
P
S 2010 Mathematics Subject Classification: 30E05; 30F15; 47B36; 42C05; 58J53
.
h
t
a
m 1 Introduction
[

We start by recalling some known facts from the spectral theory of Jacobi matrices; see [19,
3
v Chapter 5]. Let dσ be a real scalar compactly supported measure and {P (x)} the corre-
+ n n≥0
3 sponding orthonormal polynomials, which we obtain by orthonormalizing the monomials
0
3
7 1, x, x2, ....
0
. It is easy to see that they obey
1
0
6 xP (x) = a P (x)+b P (x)+a P (x), a > 0,
n n n−1 n n n+1 n+1 n
1
:
v that is, the multiplication by the independent variable in the basis {P (x)} has the matrix
n n≥0
i
X
 
b a 0
r 0 1
a a1 b1 a2 
 
J+ = 0 ... ... ...,
 
... ...
where |a |,|b | ≤ C for C such that dσ has support [−C,C]. Matrices of this sort are called
n n +
one-sided Jacobi matrices. In general, we call an operator one-sided if it is an operator on
(cid:96)2+ = (cid:96)2(Z≥0) and correspondingly two-sided if it acts on (cid:96)2 = (cid:96)2(Z). Moreover, let {en}n∈Z
denote the standard basis of (cid:96)2 and (cid:96)2 = (cid:96)2(cid:9)(cid:96)2 with the classical embedding of (cid:96)2 into (cid:96)2. By
− + +
(cid:90)
(cid:10)(J −z)−1e ,e (cid:11) = dσ+(x),
+ 0 0
x−z
(cid:63)ThispaperisacontributiontotheSpecialIssueonOrthogonalPolynomials,SpecialFunctionsandApplica-
tions. The full collection is available at 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 (cid:96)2. One can show that there exists a polynomial, T , of degree p
− p
such that the spectrum, E, of J is given by
1) E = T−1([−2,2]), (1.1)
p
2) all critical points of T (i.e., zeros of T(cid:48)) are real and 3) |T (c)| ≥ 2 for all critical points c;
p p p
cf. [19, Theorem 5.5.25]. We define the resolvent functions by
rJ(z) = (cid:10)(J −z)−1e ,e (cid:11), rJ(z) = (cid:10)(J −z)−1e ,e (cid:11), z ∈ C .
− − −1 −1 + + 0 0 +
In the periodic case they can be given explicitly in terms of the orthogonal polynomials and
they satisfy
1
= a2rJ(x+i0), for almost all x ∈ E. (1.2)
rJ(x+i0) 0 −
+
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([−2,2]) the isospectral torus (finite gap class) of Jacobi matrices, J(E), by
p
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) = (cid:8)J(α): α ∈ Rg/Zg(cid:9), (1.4)
where the map α (cid:55)→ 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 T , one can define a function Φ(z) by
p
1
T (z) = Φ(z)+ , z ∈ C\E. (1.5)
p
Φ(z)
This is possible since T (z) ∈ C\[−2,2] for z ∈ C\E, which is the image of the Joukowski map
p
ζ (cid:55)→ ζ + 1. Fix α ∈ Γ∗ and let H2(α) be the Hardy space of character automorphic functions
ζ
on C\E with character α and L2(α) the corresponding space of character automorphic square
integrable functions. The decomposition
H2(α) = K (α)⊕ΦH2(α)
Φ
defines a natural basis {eα}∞ of H2(α) and respectively {eα}∞ a basis of L2(α) with the
n n=0 n n=−∞
properties that eα = Φeα and K (α) = {eα,...,eα }. The multiplication by z in the basis
n+p n Φ 0 p−1
{eα}∞ is a Jacobi matrix J(α). Moreover, if α runs through Γ∗ we obtain J(E), i.e., this
n n=−∞
construction proves (1.4). Note that eα = Φeα means that Φ is the symbol of Sp, where S
n+p n
Periodic GMP Matrices 3
is the right shift on (cid:96)2, i.e., Se = e . Since zΦ(z) = Φ(z)z implies J(α)Sp = SpJ(α), the
n n+1
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) ⇐⇒ T (J) = Sp+S−p.
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 T with the properties 1), 2) and 3). In
p
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
g
(cid:91)
E = [b ,a ]\ (a ,b ), g ∈ N, a < b < a . (1.6)
0 0 j j j j j+1
j=1
The corresponding bases {eα} can be defined as well, which leads exactly in the same way to
n
aparametrizationofJ(E)byΓ∗. Moreover, itgivesexplicitformulasforthecoefficientsofJ(α)
by means of continuous functions A, B on Γ∗, i.e.,
a (α) = A(α−nµ), b (α) = B(α−nµ), (1.7)
n n
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, T , with the properties 1), 2) and 3), the
p
characterizationoftheisospectraltorusintermsofthemagicformulaisonlypossibleforspectra
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 |T (c)| > 2 for all critical
p
points c, which is a strong restriction on E. The idea of GMP matrices is to substitute the
polynomial T by a rational function. In [6], we carried out this idea for the simplest case,
p
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, ∆ , of the form
E
λ
1
∆ (z) = λ z+c + , λ ,λ > 0, (1.8)
E 0 0 0 1
c −z
1
such that ∆−1([−2,2]) = E. By a linear change of variable we may assume that c = 0. This
E 1
suggests to consider matrices obtained by orthonormalizing the family of functions
1 (−1)2
1, − , x, , ..., (1.9)
x x2
for a given real compactly supported measure dσ . Denoting this basis by ϕ , we call the
+ n
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 Szeg˝o discussed orthogonal
polynomials, ψ , w.r.t. a measure, dµ, supported on the unit circle and showed that there are
n
constants {α }∞ in D, called Verblunsky coefficients, so that
n n=0
(cid:112)
1−|α |2ψ (z) = zψ (z)−α znψ (1/z).
n n+1 n n n
DuetoVerblunsky[22], whodefinedtheminanothercontext, themap dµ (cid:55)→ {α }isone-to-one
n
and onto all of D∞. Recent developments are due to Cantero, Moral and Vel´azques [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 ∆ the
E
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) ⇔ ∆ (A) = S2+S−2.
E
Todealwitharbitraryfinitegapsetsoftheform(1.6), onefirsthastogeneralize(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
(cid:12) (cid:12)
Cap (E) = (cid:12) lim zΨ(z)(cid:12) (the so-called analytic capacity), among all functions, which vanish at
a
z→∞
infinity and are bounded by one in modulus.
Lemma 1.2. The function
1
∆ (z) := Ψ(z)+ (1.10)
E
Ψ(z)
is a rational function of the form
g
(cid:88) λj
∆ (z) = λ z+c + , (1.11)
E 0 0
c −z
j
j=1
with λ > 0, j ≥ 0, c ∈ (a ,b ), j ≥ 1 and
j j j j
g
(cid:91)
E = [b ,a ]\ (a ,b ) = ∆−1([−2,2]). (1.12)
0 0 j j E
j=1
In fact, if we demand that Im∆ (z) > 0 for Imz > 0 and lim ∆ (z) = ∞, ∆ is the unique
E E E
z→∞
rational function with the property (1.12).
Periodic GMP Matrices 5
Proof. Due to [16], we have
(cid:118)
(cid:117) g
1−Ψ(z) (cid:117)(cid:89) z−aj
= (cid:116) =: G(z).
1+Ψ(z) z−b
j
j=0
Therefore,
1 1+G2(z)
∆ (z) := Ψ(z)+ = 2 ,
E Ψ(z) 1−G2(z)
is of the form P (z)/Q (z), where m ≤ g. Like in [15, Chapter VII], we see that ImG2(z) > 0
g+1 m
for Imz > 0, which then clearly also holds for ∆ . Therefore, G2 is increasing on the interval
E
(a ,b ) and has a zero at a and a pole at b . Hence, there is exactly one pole of ∆ in each
j j j j E
gap. SinceG2(∞) = 1, thereisalsoapoleatinfinity. Toproveuniquenesslet∆(z)bearational
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) = ∞ already implies
z→∞
that ∆ is of the form (1.11). Since ∆(cid:48)(x) > 0 on R\{c ,...,c }, ∆−1([−2,2]) = E implies
1 g
∆(a ) = 2 and ∆(b ) = −2 for j ≥ 0.
j j
This defines ∆ uniquely. (cid:4)
Let C = {c ,...,c } be a collection of distinct real points and dσ a measure such that
1 g −
the points c don’t belong to its support. Like SMP matrices, we define a one-sided GMP
k
matrix, A , as the matrix of the multiplication operator w.r.t. the basis obtained by orthonor-
−
malizing the family of functions
1 1 1 1
1, , , ..., , x, , ....
c −x c −x c −x (c −x)2
g g−1 1 g
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 c does not belong to the support of the measure, we can also consider
k
multiplication by 1 . Hence, if the first block of A corresponds to the basis
c −x −
k
(cid:20) (cid:21)
1 1
1, ,..., (1.13)
c −x c −x
1 g
then the linear change of variable y = 1 leads to the basis related to
c −x
k
(cid:20) (cid:21)
1 1 1
− , ,...,1,..., , (1.14)
y y(c )−y y(c )−y
1 g
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(cid:126)∈ Cg+1, (p(cid:126))∗ is a (g+1)-dimensional row vector.
ThenotationT− denotestheuppertriangularpartofamatrixT (excludingthemaindiagonal),
and T+ = T −T− is its lower triangular part (including the main diagonal).
Firstofall,theGMPclassdependsonanorderedcollectionofdistinctpointsC={c ,...,c }.
1 g
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(cid:126)−1) B((cid:126)p−1) A(p(cid:126)0) 
 A∗(p(cid:126)0) B((cid:126)p0) A(p(cid:126)1) 
 
... ... ...
such that
(cid:126)p = (p(cid:126),(cid:126)q) ∈ R2g+2, A(p(cid:126)) = δ p(cid:126)∗, B((cid:126)p) = ((cid:126)qp(cid:126)∗)−+(p(cid:126)(cid:126)q∗)++C˜,
g
and
 
c1 p(j) q(j)
C˜ =  ... , p(cid:126)j =  0... , (cid:126)qj =  0... , p(gj) > 0.
 cg  (j) (j)
p q
0 g g
We call {(cid:126)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 {c −A}g , for
k k=1
1 ≤ k ≤ g, are invertible, and moreover S−k(c −A)−1Sk are also of the class A. To abbreviate
k
we write A ∈ GMP(C).
We call a GMP matrix one-block periodic or simply periodic if (cid:126)p =(cid:126)p for all j ∈ Z.
j
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) = K (α)⊕ΨH2(α),
Ψ
leads to a new basis {fα}∞ such that fα = Ψfα, K (α) = {fα,...,fα} and the multipli-
n n=−∞ n+p n Ψ 0 g
cation by z is a GMP matrix. Note that the property fα = Ψfα shows that the corresponding
n+p n
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) = (cid:8)A(α,C): α ∈ Rg/Zg(cid:9),
where A(α,C) is the multiplication by the independent variable w.r.t. the basis {fα}. The map
n
α (cid:55)→ A(α,C) is one-to-one up to the identification (p ,q ) (cid:55)→ (−p ,−q ) for j = 0,...,g−1.
j j j j
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) ⇐⇒ ∆ (A) = Sg+1+S−(g+1).
E
Periodic GMP Matrices 7
ThiswasoneofthemainobservationswhichallowedYuditskiiin[25]togeneralizetheKillip–
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 a = (cid:107)p(cid:126)(cid:107), e˜ = 1 P Ae and A = P AP∗, P = I −P . Then
0 0 a0 + −1 ± ± ± + −
(cid:20) (cid:21)
A 0
A = − +a ((cid:104)·,e (cid:105)e˜ +(cid:104)·,e˜ (cid:105)e ).
0 A 0 −1 0 0 −1
+
Definition 1.7. Let A ∈ GMP(C) be a periodic GMP matrix with coefficients p(cid:126) and (cid:126)q. Let
p = (cid:2)p q(cid:3) ∈ R2. We introduce the matrix functions
(cid:20) (cid:21)
0 −p
a(z;p) = a(z,∞;p,q) = , (1.15)
1 z−pq
p p
and
(cid:20) (cid:21) (cid:20) (cid:21)
1 p (cid:2) (cid:3) 0 −1
a(z,c;p) = I − p q j, j = .
c−z q 1 0
Then the product
A(z) = a(z,c ;p )a(z,c ;p )···a(z,c ;p )a(z;p )
1 0 2 1 g g−1 g
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(cid:126) and (cid:126)q. Then A
has purely absolutely continuous spectrum, which is given by
σ(A) = σ (A) = (cid:8)z ∈ C: ∆A(z) ∈ [−2,2](cid:9).
ac
If for a given set E A ∈ A(E,C), then we show in Lemma 3.9 that indeed ∆ = ∆A. This
E
allows us to explain an alternative definition of GMP matrices. We define for a periodic GMP
matrix with generating coefficients p(cid:126) and (cid:126)q
(cid:40)k−2 g−1 (cid:41)
(cid:89) (cid:89)
Λ ((cid:126)p) = −tr a(c ,c ;p )p p∗ j a(c ,c ;p )a(c ;p ) , (1.17)
k k m+1 m k−1 k−1 k m+1 m k g
m=0 m=k
where by definition Λ ((cid:126)p) = −Res trA(z).
k ck
Let us consider the last non-zero diagonal of ∆ (A), i.e., ∆ (A) for 0 ≤ j ≤ g. By
E E j,g+1+j
definition of GMP matrices, (c − A)−1 > 0 for 1 ≤ k ≤ g, whereas (c − A)−1 =
k k−1,g+k l k−1,g+k
A = 0 for l (cid:54)= k. Moreover, (c −A)−1 = 0 for 1 ≤ k ≤ g and A > 0. Thus,
k−1,g+k k g,2g+1 g,2g+1
on the last non-zero diagonal of ∆ (A) only one of the summands is non-zero. Note that the
E
relation ∆ (z) = ∆A(z) implies λ = Λ ((cid:126)p). The previous consideration, the magic formula
E k k
and this identity yield
Λ ((cid:126)p)(c −A)−1 = 1 for 1 ≤ k ≤ g.
k k k−1,g+k
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 (cid:126)p. Then A ∈ GMP(C) if and
only if
Λ ((cid:126)p) > 0 for 1 ≤ k ≤ g.
k
Proof. The proof is based on the idea that one can find the entries of the inverse matrices
(c −A)−1 explicitly; cf. [25, Lemma 3.2 and Theorem 3.3] (cid:4)
k
The relation ∆ (z) = ∆A(z) finally leads to an algebraic description for A(E,C).
E
Theorem 1.10. Let A ∈ GMP(C) be periodic with generating coefficients(cid:126)p. Then A ∈ A(E,C)
if and only if
g−1
1 (cid:88)
p = , q = −c −λ p q , Λ ((cid:126)p) = λ for k = 1,...,g, (1.18)
g g 0 0 j j k k
λ
0
j=1
where Λ ((cid:126)p) is defined as in (1.17).
k
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. {fα} and
n
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 {a ,b }g . Its
j j j=0
transfer matrix is defined by
AJ(z) = a(z,a ,b /a )a(z,a ,b /a )···a(z,a ,b /a ),
1 0 1 2 1 2 p p−1 p
where
(cid:34) (cid:35)
0 −a
j
a(z,a ,b /a ) = ,
j j−1 j 1 z−bj−1
aj aj
is defined as in (1.15). Moreover, the discriminant is given by T (z) = trAJ(z). This is not
p
a notationalconflict, but trAJ(z) isindeed the polynomial in(1.1). The spectrumofJ is purely
absolutely continuous and
σ(J) = σ (J) = T−1([−2,2]), (2.1)
ac p
cf. [19, Chapter 5]. One way of proving this is to write J as
(cid:20) (cid:21)
J 0
J = − +a ((cid:104)·,e (cid:105)e +(cid:104)·,e (cid:105)e ), (2.2)
0 J 0 −1 0 0 −1
+
where {e ,e } spans a cyclic subspace for J. From this representation it is easy to deduce that
0 −1
the matrix resolvent function RJ(z), defined by
(cid:20)(cid:104)(J −z)−1e ,e (cid:105) (cid:104)(J −z)−1e ,e (cid:105)(cid:21)
RJ(z) = −1 −1 0 −1 ,
(cid:104)(J −z)−1e ,e (cid:105) (cid:104)(J −z)−1e ,e (cid:105)
−1 0 0 0
Periodic GMP Matrices 9
admits the representation
(cid:20)rJ(z)−1 a (cid:21)−1
RJ(z) = − 0 ,
a rJ(z)−1
0 +
where rJ(z) = (cid:104)(J − z)−1e ,e (cid:105). Using what is called coefficient stripping in [19,
± ± −1±1 −1±1
2 2
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 ,e˜ } span
−1 0
a cyclic subspace for A. Like in (2.2), we can represent A as
(cid:20) (cid:21)
A 0
A = − +a ((cid:104)·,e (cid:105)e˜ +(cid:104)·,e˜ (cid:105)e ),
0 A 0 −1 0 0 −1
+
where a = (cid:107)p(cid:126)(cid:107). Hence, defining
0
(cid:20)(cid:104)(A−z)−1e ,e (cid:105) (cid:104)(A−z)−1e˜ ,e (cid:105)(cid:21)
R(z) = −1 −1 0 −1 ,
(cid:104)(A−z)−1e ,e˜ (cid:105) (cid:104)(A−z)−1e˜ ,e˜ (cid:105)
−1 0 0 0
we obtain
(cid:20)r (z)−1 a (cid:21)−1
R(z) = − 0 , (2.3)
a r (z)−1
0 +
where
r (z) = (cid:104)(A −z)−1e ,e (cid:105), r (z) = (cid:104)(A −z)−1e˜ ,e˜ (cid:105).
− − −1 −1 + + 0 0
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 (cid:126)x ∈ Rg+1, we define
 
x
0
.
sk(cid:126)x =  .. .
x
g−k
Moreover, let the M ’s be upper triangular matrices such that
k
(cid:20) (cid:21)
M 0
B(p(cid:126),(cid:126)q)−p(cid:126)((cid:126)q)∗ = M(p(cid:126),(cid:126)q) := M = 1 +(−p(cid:126)q +(cid:126)qp )δ∗
0 0 0 g g g
and
(cid:20) (cid:21)
M 0
M = k+1 +(−s p(cid:126)q +s (cid:126)qp )(s δ )∗ for 1 ≤ k ≤ g−1.
k 0 c k g−k k g−k k g−k
g+1−k
Theorem 2.1. Let
(cid:34) (cid:35)
(cid:20)R(z,p,p) R(z,g,p)(cid:21) (cid:104)(B(p(cid:126),(cid:126)q)−z)−1p(cid:126),p(cid:126)(cid:105) (cid:104)(B(p(cid:126),(cid:126)q)−z)−1(cid:126)δ ,p(cid:126)(cid:105)
= g .
R(z,p,g) R(z,g,g) (cid:104)(B(p(cid:126),(cid:126)q)−z)−1p(cid:126),(cid:126)δ (cid:105) (cid:104)(B(p(cid:126),(cid:126)q)−z)−1(cid:126)δ ,(cid:126)δ (cid:105)
g g g
Let A be a periodic GMP matrix, r (z) the resolvent function of A , i.e.,
+ +
r (z) = (cid:10)(A −z)−1e˜ ,e˜ (cid:11).
+ + 0 0
10 B. Eichinger
Then we have
a2r (z)A˜ (z)+A˜ (z)
a2r (z) = 0 + 11 12 , (2.4)
0 + a2r (z)A˜ (z)+A˜ (z)
0 + 21 22
where a2 = (cid:107)p(cid:126)(cid:107)2 and
0
(cid:20)A˜ (z) A˜ (z)(cid:21) 1 (cid:20)R(z,p,p)R(z,g,g)−R(z,p,g)2 −R(z,p,p)(cid:21)
A˜(z) = 11 12 = .
A˜ (z) A˜ (z) R(z,p,g) R(z,g,g) −1
21 22
Proof. We write
(cid:20) (cid:21)
B(p(cid:126),(cid:126)q) 0
A = +a ((cid:104)·,e (cid:105)e˜ +(cid:104)·,e˜ (cid:105)e ),
+ 0 A 0 g 1 1 g
+
where e˜ = Sg+1e˜ and apply the Sherman–Morrison–Woodbury formula (cf. [8, Section 2.1.3])
1 0
to prove the theorem. (cid:4)
Theorem 2.2. Let A˜ be defined as in Theorem 2.1 and A be the transfer matrix of A. Then we
have A˜(z) = A(z).
Proof. First,werepresentB(p(cid:126),(cid:126)q)asaone-dimensionalperturbationofalowerdiagonalmatrix.
Applying the Sherman–Morrison–Woodbury formula again, leads to a representation of A˜ in
terms of M . Using that M is a lower triangular matrix, we obtain
0 0
A˜(z) = A (z)a(z;p ,q ),
0 g g
where
(cid:20)(cid:104)(M −z)−1u p(cid:126),u p(cid:126)(cid:105) (cid:104)(M −z)−1u (cid:126)q,u p(cid:126)(cid:105)(cid:21)
A (z) = I − 1 1 1 1 1 1 j.
0 (cid:104)(M −z)−1u p(cid:126),u (cid:126)q(cid:105) (cid:104)(M −z)−1u (cid:126)q,u (cid:126)q(cid:105)
1 1 1 1 1 1
Using again that all M(cid:48)s are lower triangular matrices, we find that
j
A (z) = A (z)a(z,c ;p ,q ),
j−1 j g+1−j g−j g−j
where
(cid:20)(cid:104)(M −z)−1u p(cid:126),u (cid:126)q(cid:105) (cid:104)(M −z)−1u (cid:126)q,u p(cid:126)(cid:105)(cid:21)
A (z) = I − j j j j j j j. (cid:4)
j−1 (cid:104)(M −z)−1u p(cid:126),u (cid:126)q(cid:105) (cid:104)(M −z)−1u (cid:126)q,u (cid:126)q(cid:105)
j j j j j 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-
(1)
hand side a is replaced by (cid:107)p(cid:126) (cid:107) and r by r , which is the resolvent function related to
0 1 + +
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.
−