Unitary interpolants and factorization indices of matrix functions

20 Pages·0.23 MB·English

by  R. B. Alexeev

UNITARY INTERPOLANTS AND FACTORIZATION
INDICES OF MATRIX FUNCTIONS
1
R.B. ALEXEEV AND V.V. PELLER
0
0
2

n
a
Abstract. For an n×n bounded matrix function Φ we study unitary inter-
J
polants U, i.e., unitary-valued functions U such that Uˆ(j) = Φˆ(j), j < 0. We
6
are looking for unitary interpolants U for which the Toeplitz operator T is
2 U
Fredholm. We give a new approach based on superoptimal singular values and

] thematic factorizations. We describe Wiener–Hopf factorization indices of U in
A
termsofsuperoptimalsingularvaluesofΦandthematic indicesofΦ−F,where
F F is a superoptimal approximation of Φ by bounded analytic matrix functions.
.
h The approachessentially relies on the notion of a monotone thematic factoriza-
t tion introduced in [AP]. In the last section we discuss hereditary properties of
a
m unitary interpolants. In particular, for matrix functions Φ of class H∞+C we
study unitary interpolants U of class QC.
[

1
v
1. Introduction
4
2
2
1 A Hankel operator defined on the Hardy class H2(Cn) of Cn-valued functions
0
has infinitely many different symbols. If Φ is a symbol of such a Hankel operator,
1
0 then Φ − Q is a symbol of the same Hankel operator for any bounded analytic
/ matrix function Q. A natural problem arises for a Hankel operator to choose a
h
t symbol that satisfies certain nice properties. For example, an important problem is
a
m to choose a symbol that has minimal L∞-norm. In this paper we consider another
: important problem to choose a symbol that takes unitary values (such symbols are
v
called unitary-valued).
i
X Recall that for a bounded M -valued function Φ (we denote by M the space
m,n m,n
ar ofm×nmatrices)ontheunitcircleTtheHankel operatorHΦ : H2(Cn) → H−2(Cm)
with symbol Φ is defined by
H f d=ef P Φf, f ∈ H2(Cn),
Φ −
where P is the orthogonal projection onto H2(Cm) d=ef L2(Cm)⊖H2(Cm).
− −
Certainly, when we discuss the problem of finding unitary-valued symbols we
have to assume that m = n, i.e., Φ ∈ L∞(M ). If U is a symbol of H , then the
n,n Φ
The second author is partially supported by NSF grant DMS 9970561.
1
Fourier coefficients of U satisfy
Uˆ(j) = Φˆ(j), j < 0.
A unitary-valued matrix function U satisfying this condition is called a unitary
interpolant of Φ.
A matrix analog of Nehari’s theorem says that
kHΦk = distL∞ Φ,H∞(Mm,n) (1.1)
(see [Pa]). Here for a bounded Mm,n-valu(cid:0)ed function F(cid:1)we use the notation
def
kFkL∞ = esssupkF(ζ)kMm,n,
ζ∈T
where the norm of a matrix in M is its operator norm from Cn to Cm.
m,n
Recall that by Hartman’s theorem, H is compact if and only if
Φ
Φ ∈ (H∞ +C)(M ) (see e.g., [N]), where
m,n
H∞ +C d=ef {f +g : f ∈ H∞, g ∈ C(T)},
while the essential norm kH k (i.e., the distance from H to the set of compact
Φ e Φ
operators) can be computed as follows:
kHΦke = distL∞ Φ,(H∞ +C)(Mm,n)
(see e.g., [Sa] for the proof of this formu(cid:0)la for scalar function(cid:1)s, in the matrix-valued
case the proof is the same).
It follows from (1.1) that for a matrix function Φ ∈ L∞(M ) to have a unitary
n,n
interpolant it is necessary that kH k ≤ 1.
Φ
If ϕ is a scalar function such that kH k ≤ 1, then a unitary interpolant of
ϕ
ϕ exists if H has two different symbols whose L∞-norms are at most 1 (the
ϕ
Adamyan–Arov–Krein theorem [AAK2]).
Dym and Gohberg studied in [DG1] the problem of finding unitary interpolants
for matrix functions with entries in a Banach algebra X of continuous functions
on T that satisfy certain axioms (their axioms are similar to axioms (A1)–(A4) in
§5). They showed that if Φ belongs to the space X(M ) of M -valued functions
n,n n,n
with entries in X and kH k ≤ 1, then Φ has a unitary interpolant U of the same
Φ
class X(M ). Moreover, the negative indices of a Wiener–Hopf factorization of
n,n
U are uniquely determined by Φ while the nonnegative indices can be arbitrary.
Note, however, that we were not able to understand their argument in the proof
of Lemma 4.10. In [DG2] the problem of finding unitary interpolants was studied
in a more general situation. Another approach to this problem was given by Ball
in [B].
In this paper we propose a different approach based on so-called thematic fac-
torizations introduced in [PY1]. We state our results in different terms. We obtain
information about the Wiener–Hopf indices of unitary interpolants of Φ in terms of
2
the so-called superoptimal singular values of Φ and indices of thematic factoriza-
tions of Φ−F, where F is a superoptimal approximation of Φ by bounded analytic
matrix functions (see §2).
We are interested in unitary interpolants U of Φ such that the Toeplitz operator
T is Fredholm (i.e., invertible modulo the compact operators).
U
Recall that for Ψ ∈ L∞(M ) the Toeplitz operator T : H2(Cn) → H2(Cm) is
m,n Ψ
defined by
T f = P Ψf, f ∈ H2(Cn),
Ψ +
where P is the orthogonal projection onto H2(Cm).
+
By Simonenko’s theorem [Si] (see also [LS]), the symbol Ψ ∈ L∞(M ) of a
n,n
Fredholm Toeplitz operator T admits a Wiener–Hopf factorization:
Ψ
zd0 0 ··· 0
0 zd1 ··· 0
Ψ = Q∗2 ... ... ... ... Q1,
 0 0 ··· zdn−1 
 
 
where d ,··· ,d ∈ Z are Wiener–Hopf indices, and Q and Q are functions
0 n−1 1 2
invertible in H2(M ). Here we start enumeration with 0 for technical reasons.
n,n
Clearly, we can always arrange the indices in the nondecreasing order:
d ≤ d ≤ ··· ≤ d
0 1 n−1
inwhich casetheindices d areuniquely determinedbythefunctionΨ. Uniqueness
j
follows easily from the following well-known identity
dimKerT = −d (1.2)
Ψ j
{j:Xdj<0}
applied to the matrix functions zjΨ, j ∈ Z.
Let U be a unitary-valued function on T. It is well known that the Toeplitz
operator T is Fredholm if and only if
U
kHUke < 1, and kHU∗ke < 1. (1.3)
Indeed, suppose that T is Fredholm. Clearly, kH k = kH k for any j ∈ Z.
U zjU e U e
Multiplying U by zN if necessary, we may assume that KerT is trivial, and so T
U U
is left-invertible. It follows now from the obvious equality
kH fk2 +kT fk2 = kfk2, f ∈ H2(Cn),
U 2 U 2 2
that kH k < 1, and so kH k < 1. Applying the same reasoning to U∗, we find
U U e
that kHU∗ke < 1.
Suppose now that (1.3) holds. It is easy to see that
TU∗TU = I −HU∗HU, TUTU∗ = I −HU∗∗HU∗,
3
Since kHUke < 1 and kHU∗ke < 1, it follows that TU∗TU and TUTU∗ are invertible
modulo the compact operators which implies that T is Fredholm.
U
Obviously, if U is a unitary interpolant of Φ, then H = H . Therefore for Φ
U Φ
to have a unitary interpolant U with Fredholm T it is necessary that kH k < 1.
U Φ e
Throughout this paper we assume that this condition is satisfied.
We conclude the introduction with the following fact.
Lemma 1.1. Let U be a unitary-valued matrix function in L∞(M ) such that
n,n
kH k < 1. Suppose that U admits a Wiener–Hopf factorization of the form
U e
U = Q∗ΛQ ,
2 1
where Q and Q are invertible in H2(M ) and Λ is a diagonal matrix function
1 2 n,n
with diagonal terms zdj. Then T is Fredholm and
U
kHUke = kHU∗ke.
Proof. Multiplying U by z¯N for a sufficiently large N if necessary, we may
assume that all exponents d are nonpositive. Let us show that in this case T has
j U
dense range. Suppose that g ∈ H2(Cn) and g ⊥ RangeT . Let f be a polynomial
U
in H2(Cn). We have
0 = (T f,g) = (Uf,g) = (Q f,Λ∗Q g).
U 1 2
Clearly, Λ∗ ∈ H∞(M ) and the set of functions of the form Q f is dense in
n,n 1
H2(Cn). Hence, Λ∗Q g = 0, and so g = 0 which proves that T has dense range.
2 U
In [Pe3] it was proved that if T has dense range in H2(Cn), then the operator
U
HU∗∗HU∗ is unitarily equivalent to the restriction of the operator HU∗HU to the
subspace
H2(Cn)⊖{f ∈ H2(Cn) : kH fk = kfk }
U 2 2
(see also [PK] where this was proved in the scalar case). Clearly, the condition
kH k < 1 implies that the above subspace has finite codimension. Therefore
U e
kHU∗ke = kHUke < 1 and as we have already observed, this implies that TU is
(cid:4)
Fredholm.
To simplify the notation, for a matrix function Φ and a space X of functions on
T we can write Φ ∈ X when all entries of Φ belong to X, if this does not lead to
confusion.
In §2 we collect necessary information on superoptimal approximation and the-
matic (as well as partial thematic) factorizations. In particular, we define the
important notion of a monotone (partial) thematic factorization that was intro-
duced in [AP] and state the theorem on the invariance of indices of monotone
(partial) thematic factorizations obtained in [AP].
In §3 we study unitary interpolants of matrix functions Φ ∈ L∞(M ) satisfying
n,n
kH k < 1. We state the main results in terms of the superoptimal singular values
Φ e
4
of Φ and the indices of a monotone (partial) thematic factorization of Φ−F, where
F is a best approximation of Φ by bounded analytic functions.
Finally,weshowin§4howtoapplytheresultsof§3tostudyunitaryinterpolants
of class X for various function spaces X. In particular we show in §4 that if
Φ ∈ (H∞+C)(M ), then all unitary interpolants U of Φ satisfying the condition
n,n
kHU∗ke < 1 belong to the class
QC d=ef {f ∈ H∞ +C : f¯∈ H∞ +C}.
2. Superoptimal approximation and thematic indices
This section is an introduction to superoptimal approximation and thematic
factorizations. We refer the reader to [PY1], [PY2], [PT], and [AP] for more
detailed information.
It is a well-known fact [Kh] that if ϕ is a scalar function of class H∞ +C, then
there exists a unique function f ∈ H∞ such that
distL∞(ϕ,H∞) = kϕ−fkL∞.
ForamatrixfunctionΦ ∈ L∞(M )wesaythatamatrixfunctionF ∈ H∞(M )
m,n m,n
is a best approximation of Φ by bounded analytic matrix functions if
kΦ−FkL∞ = distL∞(Φ,H∞(Mm,n).
However, consideration of diagonal matrices makes it obvious that for matrix func-
tions we can have uniqueness only in exceptional cases.
To define a superoptimal approximation of a matrix function Φ ∈ L∞(M ) by
m,n
bounded analytic matrix functions, we consider the following sets:
Ω = {F ∈ H∞(M ) : F minimizes t d=ef esssupkΦ(ζ)−F(ζ)k};
0 m,n 0
ζ∈T
def
Ω = {F ∈ Ω : F minimizes t = esssups (Φ(ζ)−F(ζ))}.
j j−1 j j
ζ∈T
Recall that for a matrix A ∈ M the jth singular value s (A) is defined by
m,n j
s (A) = inf{kA−Rk : rankR ≤ j}, j ≥ 0.
j
Functions in F ∈ Ω are called superoptimal approximations of Φ by
min{m,n}−1
analytic functions, or superoptimal solutions of Nehari’s problem. The numbers t
j
are called superoptimal singular values of Φ.
In [PY1] it was shown that any matrix function Φ ∈ (H∞ + C)(M ) has a
m,n
unique superoptimal approximation by bounded analytic matrix functions. An-
other approach to the uniqueness of a superoptimal approximation was found
later in [T]. Those results were extended in [PT] to the case of matrix functions
5
Φ ∈ L∞(M ) such that the essential norm kH k of the Hankel operator H is
m,n Φ e Φ
less than the smallest nonzero superoptimal singular value of Φ.
A matrix function Φ ∈ L∞(M ) is called badly approximable if
m,n
distL∞(Φ,H∞(Mm,n)) = kΦkL∞.
It is called very badly approximable if the zero matrix function is a superoptimal
approximation of Φ.
Recall that a nonzero scalar function ϕ ∈ H∞ + C is badly approximable if
and only if it has constant modulus almost everywhere on T, belongs to QC, and
indT > 0. For continuous ϕ this was proved in [Po] (see also [AAK1]). For the
ϕ
general case see [PK]. More generally, if ϕ is a scalar function in L∞ such that
kH k < kH k, then ϕ is badly approximable if |ϕ| is constant almost everywhere
ϕ e ϕ
on T, T is Fredholm, and indT > 0.
ϕ ϕ
Let us now define a thematic matrix function. Recall that a function
F ∈ H∞(M ) is called inner if F∗(ζ)F(ζ) = I almost everywhere on T (I
m,n n n
stands for the identity matrix in M ). F is called outer if FH2(Cn) is dense in
n,n
H2(Cm). Finally, F is called co-outer if the transposed function Ft is outer.
An n×n matrix function V, n ≥ 2, is called thematic (see [PY1]) if it is unitary-
valued and has the form
V = v Θ ,
where the matrix functions v ∈ H∞(C(cid:0)n) and Θ(cid:1)∈ H∞(M ) are both inner and
n,n−1
co-outer. Note that if V is a thematic function, then all minors of V on the first
column (i.e., all minors of V of an arbitrary size that involve the first column of
V) belong to H∞ ([PY1]). If n = 1 the thematic functions are constant functions
whose modulus is equal to 1.
Let now Φ ∈ L∞(M ) be a matrix function such that kH k < kH k. It
m,n Φ e Φ
follows from the results of [PT] that Φ is badly approximable if and only if it
admits a representation
su 0
Φ = W∗ V∗,
0 Ψ
(cid:18) (cid:19)
where s > 0, V and Wt are thematic functions, u is a scalar unimodular function
(i.e., |u(ζ)| = 1 for almost all ζ ∈ T) such that T is Fredholm, indT > 0,
u u
and kΨkL∞ ≤ s. Note that in this case s must be equal to kHΦk. In the case
Φ ∈ (H∞ +C)(M ) this was proved earlier in [PY1].
m,n
Suppose now that kH k is less than the smallest nonzero superoptimal singular
Φ e
value of Φ and let m ≤ n. It was proved in [PT] that Φ is very badly approximable
if and only if Φ admits a thematic factorization, i.e.,
Φ = W∗W∗···W∗ DV∗ V∗ ···V∗ (2.1)
0 1 m−1 m−1 m−2 0
6
where an m×n matrix function D has the form
s u 0 ··· 0 0 ··· 0
0 0
0 s u ··· 0 0 ··· 0
1 1
D =  ... ... ... ... ... ... ... ,
 0 0 ··· s u 0 ··· 0 
 m−1 m−1 
 
u ,··· ,u are unimodular scalar functions such that the operators T are Fred-
0 m−1 uj
holm, indT > 0, {s } is a nonincreasing sequence of nonnegative num-
uj j 0≤j≤m−1
bers,
I 0 I 0
j j
W = , V = , 1 ≤ j ≤ m−1, (2.2)
j 0 W˘ j 0 V˘
j j
(cid:18) (cid:19) (cid:18) (cid:19)
and Wt,W˘ t,V ,V˘ are thematic matrix functions, 1 ≤ j ≤ m − 1. Moreover, in
0 j 0 j
this case the s are the superoptimal singular values of Φ: s = t , 0 ≤ j ≤ m−1.
j j j
The indices k of the thematic factorization (2.1) (thematic indices) are defined in
j
case t 6= 0: k = indT . Note that this result was established earlier in [PY1] in
j j uj
the case Φ ∈ (H∞ +C)(M ).
m,n
It also follows from the results of [PT] that if r ≤ min{m,n} is such that
t < t , kH k < t and F is a matrix function in Ω , then Φ−F admits a
r r−1 Φ e r−1 r−1
factorization
t u 0 ··· 0 0
0 0
0 t u ··· 0 0
1 1
Φ−F = W∗···W∗  ... ... ... ... ... V∗ ···V∗,
0 r−1 r−1 0
 0 0 ··· t u 0  (2.3)
 r−1 r−1 
 0 0 ··· 0 Ψ 
 
 
in which the V and W have the form (2.2), the Wt,W˘ t,V ,V˘ are thematic ma-
j j 0 j 0 j
trix functions, the u are unimodular functions such that T is Fredholm and
j uj
indTuj > 0, and kΨkL∞ ≤ tr and kHΨk < tr.
Factorizations of the form (2.3) with a nonincreasing sequence {t } are
j 0≤j≤r−1
called partial thematic factorizations. If Φ − F admits a partial thematic fac-
torization of the form (2.3), then t ,t ,··· ,t are the largest r superoptimal
0 1 r−1
singular values of Φ, and so they do not depend on the choice of a partial thematic
factorization.
It was observed in [PY1] that the indices of a thematic factorization are not
uniquely determined by the matrix function but may depend on the choice of a
thematic factorization. On the other hand it follows from the results of [PT] that
under the conditions kH k < kH k and t > kH k , the sum of the indices
Φ e Φ r−1 Φ e
of a (partial) thematic factorization that correspond to all superoptimal singular
values equal to a positive specific value is uniquely determined by the function Φ
itself. Note that for H∞ +C matrix functions this was proved earlier in [PY2].
7
In[AP]thenotionofamonotone(partial)thematicfactorizationwasintroduced.
It plays a crucial role in this paper. A (partial) thematic factorization is called
monotone if for any positive superoptimal singular value t the thematic indices
k ,k ,··· ,k thatcorrespondtoallsuperoptimalsingularvaluesequaltotsatisfy
r r+1 s
k ≥ k ≥ ··· ≥ k .
r r+1 s
It was shown in [AP] that under the conditions kH k < kH k and
Φ e Φ
t > kH k , Φ − F admits a monotone partial thematic factorization of the
r−1 Φ e
form (2.3). Moreover, it was established in [AP] that the indices of a monotone
partial thematic factorization are uniquely determined by Φ and do not depend on
the choice of a partial thematic factorization.
3. Unitary interpolants and Wiener–Hopf indices
We study in this section the problem of finding unitary interpolants for a square
matrix function. In other words, given Φ ∈ L∞(M ), the problem is to find a
n,n
unitary-valued function U such that Φ−U ∈ H∞(M ). For such a problem to
n,n
be solvable, the function Φ must satisfy the obvious necessary condition
kHΦk = distL∞ Φ,H∞(Mn,n) ≤ 1. (3.1)
However, it is well known that this con(cid:0)dition is not s(cid:1)ufficient even for scalar func-
tions (see [AAK1]).
We assume that the matrix function Φ satisfies the following condition
kHΦke = distL∞ Φ,(H∞ +C)(Mn,n) < 1. (3.2)
andweareinterested inunitaryinter(cid:0)polantsU ofΦsucht(cid:1)hattheToeplitzoperator
T is Fredholm. We show that under conditions (3.1) and (3.2) the function Φ has
U
such a unitary interpolant. Clearly, (3.2) is satisfied if Φ ∈ (H∞ +C)(M ).
n,n
As we have already observed in the introduction, a unitary interpolant U of a
matrix function Φ satisfying (3.2) is the symbol of a Fredholm Toeplitz operator
if and only if
kHU∗ke < 1. (3.3)
We describe all possible indices of Wiener–Hopf factorizations of such unitary
interpolantsofΦintermsofsuperoptimalsingularvaluesandindicesofamonotone
partial thematic factorization of Φ−Q, where Q is a best approximation of Φ by
bounded analytic matrix functions.
Let us first state the results. As we have explained in §2, if Φ satisfies conditions
(3.1) and (3.2) and Q ∈ H∞(M ) is a best approximation of Φ by bounded ana-
n,n
lyticmatrixfunctions, thenΦ−Qadmitsamonotonepartialthematicfactorization
8
of the form
u 0 ··· 0 0
0
0 u ··· 0 0
1
Φ−Q = W∗···W∗  ... ... ... ... ... V∗ ···V∗,
0 r−1 r−1 0
 0 0 ··· u 0 
 r−1 
 0 0 ··· 0 Ψ 
 
 
where kΨkL∞ ≤ 1 and kHΨk < 1 (here r is the number of superoptimal singular
valuesofΦequalto1; itmaycertainlyhappenthatr = 0inwhichcaseΨ = Φ−Q).
We denote by k , 0 ≤ j ≤ r −1, the thematic indices of the above factorization.
j
Theorem 3.1. Let Φ be a matrix function in L∞(M ) such that kH k < 1.
n,n Φ e
Then Φ has a unitary interpolant U satisfying kHU∗ke < 1 if and only if kHΦk ≤ 1.
If U is a unitary interpolant of a matrix function Φ and conditions (3.1), (3.2),
and (3.3) hold, we denote by d , 0 ≤ j ≤ n − 1, the Wiener–Hopf factorization
j
indices of U arranged in the nondecreasing order: d ≤ d ≤ ··· ≤ d . Recall
0 1 n−1
that the indices d are uniquely determined by the function U.
j
Theorem 3.2. Let Φ be a matrix function in L∞(M ) such that kH k ≤ 1
n,n Φ
and kH k < 1. Let r be the number of superoptimal singular values of Φ equal
Φ e
to 1. Then each unitary interpolant U of Φ satisfying kHU∗ke < 1 has precisely r
negative Wiener–Hopf indices. Moreover, d = −k , 0 ≤ j ≤ r −1.
j j
In particular, Theorem 3.2 says that the negative Wiener–Hopf indices of a
unitary interpolant U of Φ that satisfies (3.3) are uniquely determined by Φ.
Theorem 3.3. Let Φ and r satisfy the hypotheses of Theorem 3.2. Then for
any sequence of integers {d } satisfying
j r≤j≤n−1
0 ≤ d ≤ d ≤ ··· ≤ d
r r+1 n−1
there exists a unitary interpolant U of Φ such that kHU∗ke < 1 and the nonnegative
Wiener–Hopf factorization indices of U are d ,d ,··· ,d .
r r+1 n−1
Note that Theorem 3.1 follows immediately from Theorem 3.3.
Theorem 3.4. Let Φ satisfy the hypotheses of Theorem 3.2. Then Φ has a
unique unitary interpolant if and only if all superoptimal singular values of Φ are
equal to 1.
Proof of Theorem 3.2. Suppose that U is a unitary interpolant of Φ that
satisfies (3.3). Put G = U −Φ ∈ H∞(M ).
n,n
Let κ ≥ 0. Let us show that if f ∈ H2(Cn), then kH fk = kfk if and only
zκΦ 2 2
if f ∈ KerT .
zκU
Indeed, if f ∈ KerT , then zκUf ∈ H2(Cn). It follows that
zκU −
kH fk = kH fk = kP (zκUf)k = kzκUfk = kfk .
zκΦ 2 zκ(Φ+G) 2 − 2 2 2
9
Conversely, suppose that kH fk = kfk . Then kH fk = kfk . Hence,
zκΦ 2 2 zκU 2 2
zκUf ∈ H2(Cn), and so f ∈ KerT .
− zκU
It follows from formula (1.2) that
dimKerT = −d −κ.
zκU j
{j∈[0,n−X1]:−dj>κ}
By Theorem 4.8 of [AP],
dim{f ∈ H2(Cn) : kH fk = t kfk } = k −κ.
zκΦ 2 0 2 j
{j∈[0,rX−1]:kj>κ}
Hence,
−d −κ = k −κ, κ ≥ 0. (3.4)
j j
{j∈[0,n−X1]:−dj>κ} {j∈[0,rX−1]:kj>κ}
It is easy to see from (3.4) that U has r negative Wiener–Hopf factorization
indices and d = −k for 0 ≤ j ≤ r −1. (cid:4)
j j
To prove Theorem 3.3, we need two lemmas.
Lemma 3.5. Let Ψ ∈ L∞(M ) and kH k < 1. Then for any nonnegative
m,m Ψ
integers d , 0 ≤ j ≤ m−1, there exists a unitary interpolant U of Ψ that admits
j
a representation
u 0 ··· 0
0
0 u ··· 0
1
U = W0∗···Wm∗−1 ... ... ... ... Vm∗−1···V0∗, (3.5)
 0 0 ··· u 
 m−1 
 
where
I 0 I 0
j j
W = , V = , 1 ≤ j ≤ m−1,
j 0 W˘ j 0 V˘
j j
(cid:18) (cid:19) (cid:18) (cid:19)
V , Wt, V˘ , W˘ t are thematic matrix functions, and u ,··· ,u are unimodular
0 0 j j 0 m−1
functions such that the Toeplitz operators T are Fredholm and
uj
indT = −d , 0 ≤ j ≤ m−1.
uj j
Proof of Lemma 3.5. We argue by induction on m. Assume first that m = 1.
Letψ beascalarfunctioninL∞ suchthatkH k < 1. Without lossofgenerality we
ψ
mayassume thatkψk < 1. Consider thefunctionz¯d0+1ψ. Clearly, kH k < 1.
∞ z¯d0+1ψ
It is easy to see that there exists c ∈ R such that
kH k = 1.
z¯d0+1ψ+cz¯
Since H has finite rank, it is easy to see that
cz¯
kH k = kH k < 1 = kH k.
z¯d0+1ψ+cz¯ e z¯d0+1ψ e z¯d0+1ψ+cz¯
10