Kernels of vector-valued Toeplitz operators

0.3 MB·English

by  Chevrot Nicolas

KERNELS OF VECTOR-VALUED TOEPLITZ OPERATORS
NICOLASCHEVROT
0
1
0 Abstract. LetS betheshiftoperatorontheHardyspaceH2 andletS∗ be
2 itsadjoint. AclosedsubspaceF ofH2issaidtobenearlyS∗-invariantifevery
element f ∈F with f(0)=0 satisfies S∗f ∈F. In particular, the kernels of
n
ToeplitzoperatorsarenearlyS∗-invariantsubspaces. Hittgavethedescription
a
J ofthesesubspaces. TheyareoftheformF =g(H2⊖uH2)withg∈H2 and
uinner,u(0)=0. Averyparticularfactisthattheoperatorofmultiplication
3 by g acts as an isometry on H2⊖uH2. Sarason obtained a characterization
2 of the functions g which act isometrically on H2⊖uH2. Hayashi obtained

thelinkbetweenthesymbolϕofaToeplitzoperator andthefunctionsg and
]
A u to ensure that a given subspace F = gKu is the kernel of Tϕ. Chalendar,
Chevrot andPartingtonstudiedthenearlyS∗-invariantsubspacesforvector-
F
valuedfunctions. Inthispaper, weinvestigatethegeneralization ofSarason’s
h. andHayashi’sresultsinthevector-valued context.
t
a
m
[
1. Introduction

1
To begin this section, we presentthe scalarresults of Hitt, SarasonandHayashi
v
0 which will be generalized throughout this paper.
1 WedenotebyH2 theclassicalHardyspaceofanalyticfunctionsontheunitdisc
2 D,andbyH2(Cm)theCm-vector-valuedHardyspaceconsistingofmcopiesofH2.
4
The shift S is the operatorofmultiplication bythe variablez andS is its adjoint.
. ∗
1 The (closed)S -invariantsubspacesofH2 are calledmodel subspaces. They are of
∗
0 the form K =H2 uH2, where u is an inner function.
0 u ⊖
For ϕ L , the Toeplitz operator with symbol ϕ is defined by T f :=p (ϕf),
1 ∈ ∞ ϕ +
: where p+ is the orthogonalprojection from L2 onto H2.
v
Hitt [Hit88] introduced the nearly S -invariant subspaces:
i ∗
X
Definition 1.1. A closed subspace of H2 is said to be a nearly S -invariant
∗
r F
a subspace if every element f with f(0)=0 satisfies S∗f .
∈F ∈F
Inparticular,thekernelofaToeplitzoperatorisanearlyS -invariantsubspace.
∗
Hitt obtained the complete description of this spaces:
Theorem 1.2 (Hitt, 1988). Let be a non-trivial nearly S -invariant subspace.
∗
F
Letgbetheuniqueunit-normfunctionin ,positiveattheorigin,thatisorthogonal
F
to zH2. Then there exists an inner function u vanishing at zero such that, for
F ∩
Date:june2009.
1991 Mathematics Subject Classification. Primary: 47B32, 30D55 Secondary: 46C07,
46E40,47B35.
Key words and phrases. Toeplitz operators, de Branges–Rovnyak spaces, vector-valued
functions.
The author is very grateful to Pr. Thomas Ransford for his advice and to Pr. Andreas
Hartmann.
1
2 CHEVROT
all f , there exists a unique f K and f =gf . Furthermore, f = f .
0 u 0 2 0 2
∈F ∈ k k k k
In other words, multiplication by g acts isometrically on K .
u
Two questions arise.
(1) The first one was already posed by Sarason in [Sar88] where he made this
remark: "The latter theorem leaves mysterious the relation between the
function g and the space K . Given a function g of unit norm in H2, what
u
are the S -invariant subspaces K that can arise with g in Hitt’s theorem?
∗ u
"
(2) Which nearly S -invariant subspaces are kernels of Toeplitz operators.
∗
Sarason obtained the following answer to the first question:
Theorem 1.3 (Sarason,1988). Let g be an outer function of unit norm, and u an
inner function with u(0)=0. We define two analytic functions on the disc:
1 2π eiθ+z f(z) 1
f(z):= g(eiθ)2 dθ and b(z):= − .
2π Z eiθ z| | f(z)+1
0 −
Then the following statements are equivalent:
(1) multiplication by g acts isometrically from K to ;
u
F
(2) bH2 uH2 (i.e. b=ub );
0
⊂
(3) Ku ⊂(1−TbT¯b)1/2H2.
TheanswertothesecondquestionisgivenbyHayashiin[Hay86,Hay85,Hay90]
and Sarason found an alternative proof in [Sar94a]. This answer is expressed in
terms of exposed points of the unit ball of H1, also called rigid functions.
Before stating Hayashi’s result, we need some definitions. With the previous
notation, let = gK be a nearly S -invariant space and let b be the function
u ∗
F
associatedto g as in Theorem1.3. Because log(1 b2) is integrable,we can build
anouterfunction asuchthat a2+ b2 =1a.e.on−T|.|Then(b,a)iscalledacorona
| | | |
pair (or pair) associated to g. Thanks to Theorem 1.3, b=ub . If is the kernel
0
F
ofaToeplitz operator,then (b ,a)is acoronapair associatedto the outerfunction
0
g := a/(1 b ). Some pairs, called special pairs, verify an additional property
0 0
−
which will be precisely defined in section 5. Admitting this, we can reformulate
Hayashi’s result as follows (see also [Sar94a]):
Theorem 1.4 (Hayashi, 1985). The subspace =gK is the kernel of a Toeplitz
u
F
operator if and only if the pair (b ,a) is special and g2 is rigid.
0 0
We would like to generalize the previous theorems to vector-valued functions.
The paper is organized as follows. In section 2, we define the vector- or matrix-
valued objects: we recall the inner-outer matricial factorization, we comment on
the generalization of Theorem 1.2, and we recall the definition of de Branges–
Rovnyak spaces, the vector-valuedanalogue of H(b):=(1−TbT¯b)1/2H2 appearing
in Theorem 1.3.
In section 3, we transcribe Sarason’s approach to the vectorial case. We build
the analogue of the functions b and u. Thanks to de Branges–Rovnyakspaces, we
obtain the matricial version of Theorem 1.3. The matrices do not commute, so we
need to modify the original scalar proof given by Sarason. An example illustrates
this kind of problem.
In section 4, we would like to describe the kernels of Toeplitz operators. We
begin with some examples. This allows us to illustrate the difficulties due to the
KERNELS OF VECTOR-VALUED TOEPLITZ OPERATORS 3
dimension, and to establish some notation. We then investigatethe descriptions of
kernels of Toeplitz operators of finite dimension.
Finally, in section 5, we obtain the full description of the kernels of Toeplitz
operators. We establish the desired generalization of Hayashi’s Theorem.
2. Hardy spaces of vector-valued functions
2.1. Inner-outer factorization. As usual with Hardy spaces,we identify a func-
tion with its radial limits.
Let F,G be two subspaces of Cm of dimension r. Nikolskii, in [Nik02] page 14,
calls Θ H (F G) an inner function if its boundary values Θ(ξ) are surjective
∞
isometri∈es for a.e→. ξ T.
It will be more co∈nvenient to say that Θ H (Cm Cm) is an inner function
∞
if its boundary values Θ(ξ) are partial isom∈etries for a→.e. ξ T, with kernel and
range independent of ξ a.e. in T. In other words, an inner∈function is a square-
matrix-valued function such that there exist two subspaces F,G of Cm with the
same dimension r for which ΘF H∞(F G) is an inner function in the sense of
Nikolskii. The rank of Θ(ξ) is|eq∈ual to r f→or a.e. ξ T.
∈
Herearetwoexamplesofinnerfunctionsofrank2. Thefirstonewillbediscussed
later (see Theorem 3.7). Let θ be an inner scalar function and a,b K verifying
zθ
a2+ b2 =1 a.e.on T. Define ϕ H (C2 C2) and Θ H (C∈3 C3) by the
∞ ∞
| | | | ∈ → ∈ →
following formulae:
C C a 0 b
F=C0, G:=C0, ϕ:=(cid:18)θa¯b −θa¯b(cid:19) and Θ:=θ0¯b 00 −θ0a¯.
     
Both ϕ,Θ are inner of rank 2. Note that ϕ H (Cm Cm) is inner of rank m if
∞
∈ →
and only if detϕ is inner.
Recall the Beurling–Lax Theorem [Lax59] : If a closed subspace H2(Cm)
M ⊂
is invariant by the shift, then there exists an inner function Θ such that =
ΘH2(Cm). This description is unique up to multiplication by an unitary matMrix.
Next, we recall the notion of outer vector-valued function. The outer scalar
functions are cyclic vectors for the shift. For g H2(Cm), we define , the small-
∈ G
est S invariant subspace containing g, by := span(Skg :k N). Thanks to
Beurl−ing–Lax theorem, there exists Θ, innerGwith of 1, such tha∈t = ΘH2(Cm).
We say that g is outer if Θ is a constant matrix. Then = H2(ΘG(0)Cm). It will
be useful to write G := Θ(0)Cm. Finally, the function gGis a cyclic vector for S in
H2(G).
We extend this construction to define the outer matrix-valued functions. Let
g ,...,g H2(Cm), with r m, be a independent family of vector-valued func-
1 r
tions. Let∈G H2(Cr Cm≤) be the rectangular matrix-valued functions where
∈ →
the columns are (g ) . In this case we write G = [g ,...,g ]. It is said to be
ℓ ℓ r 1 r
outer if := span(Sk≤g :ℓ r,k N) = ΘH2(Cm), where Θ is a constant partial
ℓ
isometryGof rank r. Then, w≤e will∈write G=Θ(0)Cm, dimG=r, and =H2(G).
Due to the rank theorem, there exists an unitary mapping Θ :Cr GG. To G, we
0
associate G˜ H2(Cr Cr) such that G := Θ G˜. This allows us t→o translate the
0
∈ →
properties of square-matrix-valuedfunctions to rectangular ones.
For more details about inner-outer factorization of square matrix-valued func-
tionswithdeterminantdifferentfromzero,see[KK97]. Inparticularthe Definition
4 CHEVROT
5.3 in [KK97] of Beurling left outer function coincides with that of outer given
above. The Smirnov–Nevanlinna class +(Cm Cm) of square matrix-valued
N →
functions is the set of all matrices with entries in the scalar Smirnov–Nevanlinna
class. The Definition 3.1 in [KK97] of outer function in +(Cm Cm) is that E
N →
is outer if det E is outer in +. The authors shows that all definitions of outer
functions are equivalent in HN2(Cm Cm). Theorem 5.4 in [KK97] says that,
given a function F in +(Cm Cm→), detF(z) 0, there exist functions F inner
i
N → 6≡
and F outer (resp. F ,F ) , unique up to a unitary matrix, such that F = F F
o i′ o′ i o
(resp. F = F F ). Furthermore, Theorem 3.1 of [KK97] will be useful later: Let
o′ i′
E +(Cm Cm) an outer square-matrix-valued function. Then det(z)=0 for
all∈zND and→E 1 +(Cm Cm). 6
−
∈ ∈N →
2.2. Nearly S -invariant subspaces of H2(Cm). Thenextresultis thedescrip-
∗
tion of the nearly S -invariant subspaces of H2(Cm). For more details, see [CCP].
∗
Theorem 2.1. Let H2(Cm) be a non-trivial nearly S -invariant subspace.
∗
F ⊂
Let (g ,...,g ) be a orthonormal basis of
1 r
W := zH2(Cm) ⊥.
F ∩ F ∩
Thenr :=dim W mandthereexist(cid:8)anintegerr ,1(cid:9) r r,andU H (Cr
′ ′ ∞
Cr) inner, rank U≤=r , such that ≤ ≤ ∈ →
′
=[g ,...,g ] H2(Cr) UH2(Cr) =GK .
1 r U
F ⊖
(cid:0) (cid:1)
For all f , there exists an unique f K such that f = Gf . Furthermore,
0 U 0
∈ F ∈
kf0kH2(Cr) =kfkH2(Cm).
Because the columns of G form an orthonormal basis of W, the norm of G
H2(Cr Cm)is1. Foranyh H2(Cr),wedefineT htobetheFourierprojectio∈n
G
of the L→1(Cm) function Gh on∈H2(Cm). It is an unbounded operator, but, as in
the scalar case, it is an isometry on K .
U
2.3. De Branges–Rovnyak spaces. Now, we will recall the definition and the
main properties of de Branges–Rovnyak spaces. For more details, see the first
chapter of [Sar94b]. Let H and H be two Hilbert spaces and B (H ,H) be
1 1
a bounded operator. We define (B) to be the range space BH ∈wiLth the inner
1
product that makes B be a coisoMmetry on H:
∀f,g ∈H1∩(kerB)⊥, hBf,BgiM(B) :=hf,giH1.
For a contraction B, the inclusion is a contraction from (B) to H. The comple-
mentary space (B) is defined to be (Id BB )1/M2 . In the particular case
H ∗
H M −
where B is the multiplication by an inn(cid:0)er function B, th(cid:1)en (B) = BH2(Cm)
and (B) = K . In this case, the inner products of (B)Mand (B) coincide
B
H M H
with the H2 inner product and these two spaces are really complementary spaces
in the H2 sense. In this article, H and H will be Hardy spaces like H2(Cm) or
1
closed subspaces of H2(Cm) isometrically equivalent to H2(Cr), and B will be the
multiplication by a matrix B in the unit ball of H (Cr Cm).
∞
The reproducing kernels in H2(Cm) are k u := 1 →u for λ D and u Cm.
λ 1 λ¯z ∈ ∈
Thus, for all f H2(Cm), the reproducing kernels ve−rify
∈
f,kλu 2 = f(λ),u Cm.
h i h i
KERNELS OF VECTOR-VALUED TOEPLITZ OPERATORS 5
Becausethe inclusionfrom (B)to H2 iscontractive,deBranges–Rovnyakspaces
H
have kernel functions, and a simple calculation shows that
Id B(z)B(λ)
kλBu:= r −1 λ¯z ∗u and hf,kλBuiH(B) =hf(λ),uiCm.
−
GivenasymbolB, wewrite (B)(resp. (B))insteadof (T )(resp. (T )).
B B
M H M H
3. Toeplitz operators acting as an isometry on a model space
In this section, we verify that the tools used by Sarason [Sar88] can be applied
to matrix-valued functions.
3.1. A matricial intertwining. Let (g ) be an orthogonalbasis of W and let
ℓ ℓ r
G H2(Cr Cm) be the matrix-valued fu≤nction [g ,...,g ].
1 r
∈We denot→e by H2(Cm,µ ) the Hardy space of vector-valued functions with the
G
norm
1 2π
q 2 := G(eiθ)q 2 dθ.
k kH2(Cm,µG) 2π Z k kCm
0
Remember that = span(Skg : ℓ r,k 0). Let f = Gq be in H2(Cm). This
ℓ
forces q to be inGH2(Cm,µ ). The m≤easure≥µ will play a role in section 5.
G G
WewouldliketobuildfromGthefunctionsF andB,theanaloguesofthoseap-
pearinginTheorem1.3. After, wewillshowthatTIdr BTG∗ isancoisometryfrom
G toH(B),orequivalently,q 7→(TIdr−BTG∗G)q isan−coisometryfromH2(Cm,µG)
to (B). As a consequence, we will obtain the following equality, the key of the
H
proof of the generalization of Theorem 1.3 :
Idr−TBTB∗ =(TIdr−BTG∗)(TIdr−BTG∗)∗.
We begin by defining F the analytic function on the disc by
1 2π eiθ+z
z D, F(z):= G(eiθ) G(eiθ)dθ.
∗
∀ ∈ 2π Z eiθ z
0 −
Note that, if G = UG, where U is inner of rank m, then the functions F and F
′ ′
are the same. Because the (g ) form an orthogonal basis of W, the coefficient
k k r
F(0) is g ,g . So, F(0) =≤Id . For z D, let u Cr be an eigenvector of
i,j i j H2 r 0
h i ∈ ∈
the matrix F(z ). Then Re( F(z )u,u ) is a Poisson integral, so the real parts of
0 0
h i
the eigenvalues of F(z ) are G(z )u 2 0. This implies that the moduli of the
0 0
k k ≥
eigenvalues of F(z )+Id are greater than 1, so F(z )+Id is invertible.
0 r 0 r
Next, we define B, the matrix-valued Herglotz integral of µ , by
G
B(z):=(F(z)+Id ) 1(F(z) Id ).
r − r
−
Because F(0)=Id , the function B vanishes in zero. For all u Cr,
r
∈
(F(z) Id )u 2 = F(z)u 2+ u 2 2Re F(z)u,u .
r
k ± k k k k k ± h i
Then, because Re F(z)u,u 0,
h i≥
(F(z)+Id )u 2 (F(z) Id )u 2
r r
k k ≥k − k
and B lies in the unit ball of H (Cr Cr). We can therefore consider (B).
∞
→ H
Lemma 3.1. For all u,v Cm we have:
∈
Gk u,Gk v = kB(Id B(w) ) 1u,kB(Id B(z) ) 1v .
h w z iH2 h w r − ∗ − z r − ∗ − iH(B)
6 CHEVROT
Proof. For all u,v Cm, we express the inner product Gk u,Gk v in terms
w z H2
∈ h i
of F:
1 2π 1
hGkwu,GkzviH2 =2π Z (1 w¯eiθ)(1 ze iθ)hG(eiθ)u,G(eiθ)viCmdθ
0 − − −
1 1 e iθ+w¯ eiθ +z
= − + G(eiθ)u,G(eiθ)v Cm dθ
2π(1 w¯z)Z 2(cid:20)e iθ w¯ eiθ z(cid:21)h i
− − − −
1
= (F(w)∗ +F(z))u,v Cm.
2(1 w¯z)h i
−
Because (Id +B(z)) and (Id B(z)) 1 commute,
r r −
−
F(w) +F(z)=(Id B(z)) 1(Id +B(z))+(Id +B(w) )(Id B(w) ) 1
∗ r − r r ∗ r ∗ −
− −
=2(Id B(z)) 1[Id B(z)B(w) ](Id B(w) ) 1.
r − r ∗ r ∗ −
− − −
Finally, we interpret Gk u,Gk v in term of inner product of kB(z):
h w z iH2 w
1
hGkwu,GkzviH2 =2(1 w¯z)h(F(w)∗+F(z))u,viCm
−
1
= (Idr B(z))−1[Idr B(z)B(w)∗](Idr B(w)∗)−1u,v Cm
1 w¯zh − − − i
−
=h(Idr −B(z))−1kwB(z)(Idr−B(w)∗)−1u,viCm
=hkwB(z)(Idr−B(w)∗)−1u,(Idr−B(z)∗)−1viCm
= kB(Id B(w) ) 1u,kB(Id B(z) ) 1v .
h w r− ∗ − z r− ∗ − iH(B)
(cid:3)
The following lemma is useful in connection with de Branges–Rovnyak spaces
([Sar94b] I-5):
Lemma 3.2 (Douglas’s criterion). Let H, H and H be Hilbert spaces, and let
1 2
A : H H, B : H H be contractions. We define (A) := AH and
1 2 1
→ → M
(B):=BH . Then (A)= (B) is equivalent to AA =BB .
2 ∗ ∗
M M M
Remember that = span(Skg :ℓ r,k N) = span(Gk u:u Cr,w D)
ℓ w
and that Θ is an innGer function such th≤at =∈ΘH2(Cm). (When G i∈s outer,∈Θ is
G
a constant unitary-matrix)
Lemma 3.3.
(1) For all u∈Cm, TIdr−BTG∗ maps Gkwu to kwB(Idr −B(w)∗)−1u.
(2) If G is outer, then TIdr−BTG∗ is an isometry from G onto H(B).
(3) If G=GiGo, with Gi inner and Go outer, then TIdr BTG∗ is a coisometry
−
of to (B) with null space K .
G H Gi ∩G
(4) Define M(TIdr−BTG∗):=TIdr−BTG∗G. Equipped with the inner product
hTIdr−BTG∗h1,TIdr−BTG∗h2i:=hh1,h2i2 ∀h1,h2 ∈G∪(kerTIdr−BTG∗)⊥,
M(TIdr−BTG∗) coincides with the de Branges–Rovnyak space H(B).
KERNELS OF VECTOR-VALUED TOEPLITZ OPERATORS 7
Proof. (1) We begin by computing the range of Gkwu by TIdr BTG∗:
−
h(TIdr−BTG∗)Gkwu,kzviH2 =h(Idr −B(z))TG∗G(z)kw(z)u,viCm
= TG∗Gkwu,kz(Idr B∗)v H2
h − i
= Gkwu,Gkz(Idr B(z)∗)v H2
h − i
= kB(Id B(w) ) 1u,kBv
h w r − ∗ − z iH(B)
= kB(Id B(w) ) 1u,k v .
h w r − ∗ − z iH2
Therefore, (TIdr−BTG∗) sends Gkwu to kwB(Idr−B(w)∗)−1u.
(2) The inner product of two functions in is equal to the inner product of
G
their images in (B):
H
hTIdr−BTG∗Gkwu,TIdr−BTG∗GkzviH2 =hkwB(Idr −B(w)∗)−1u,kzB(Idr −B(z)∗)−1viH(B)
= Gk u,Gk v .
w z H2
h i
Because Gis outer, the functions Gk u span , reducedto H2(G) andthe
w
G
result follows.
(3) ThedefinitionofBdoesnotdependonGi,soTIdr−BTG∗ =TIdr−BTG∗oTG∗i.
ButTIdr−BTG∗o sendsTG∗iG isometricallyto H(B), whichis densein Go,so
we get the result by continuation. Moreover, kerTIdr−BTG∗|G =kerTG∗i ∩
=K .
G Gi ∩G
(4) ThelastsentenceallowsustoidentifythetwodeBranges–Rovnyakspaces:
M(TIdr−BTG∗):=TIdr−BTG∗G =H(B)=M(cid:16)(Idr −TBTB∗)1/2(cid:17).
(cid:3)
Theorem 3.4. As operators on H2(Cr), we have (TIdr BTG∗)(TIdr BTG∗)∗ =
− −
Idr TBTB∗.
−
ePqruooivfa.leTnhtetoDo(TuIgdlra−sBcrTiGte∗r)i(oTnI,dLr−emBTmGa∗)3∗.2=, iImdprl−iesTtBhTaBt∗Mas(ToIpderr−aBtoTrGs∗o)n=HH2((CBr))(cid:3).is
3.2. A matricial version of Sarason’s theorem. With the previous notation,
Theorem 3.5. Let G = [g ,...,g ] H2(Cr Cm) and let U H (Cr Cr)
1 r ∞
∈ → ∈ →
be inner of rank r vanishing at zero. Then
TG|KU is an isometry ⇐⇒TB∗KU ={0}⇐⇒BH2(Cr)⊂UH2(Cr).
The proof follows Sarason’s ideas, with modifications to bypass the fact that
TB∗KU might not be a subspace of KU.
Proof. The last equivalence is obvious.
Suppose that TB∗KU ={0}. Then TGh =TGTIdr−B∗h for all h∈KU. Thanks
ntoorLmemofmaT3.h3,2(T:Idr−BTG∗)(TIdr−BTG∗)∗ = Idr −TBTB∗, and we compute the
k G kH2
kTGhk2H2 =kTGTIdr−B∗hk2H2 =h(TIdr−BTG∗)(TIdr−BTG∗)∗h,hi
= (Idr TBTB∗)h,h
h − i
= h,h = h 2 .
h iH2 k kH2
Thus, T acts as an isometry on K .
G U
8 CHEVROT
Conversely, suppose that T is an isometry. Let h K . Lemma 3.3 and
G|KU ∈ U
Tk(TThGeIdoTrrI−edmBr−TB3G.∗4∗h)ak(Ts2HsI2edrr=t−tBhh(TaIGtd∗rT)−I∗d=rT−BIBTdTBrG∗−∗)h:T,GBhTi→HB2∗H,ow(nBhHo)si2es(dCaercv)oe.ilsoIotpmmfoeeltlnrotyw,issa:nthdat
TGh 2 TGTB∗h,TGh TGh,TGTB∗h + TGTB∗h 2 = h 2 TB∗h 2.
k k −h i−h i k k k k −k k
Using the hypothesis T h = h , we get
G
k k k k
(3.1) TGTB∗h,TGh + TGh,TGTB∗h = TB∗h 2+ TGTB∗h 2.
h i h i k k k k
Now, with Sarason’strick,we will showthat TGTB∗h=0, for h KU. Remem-
ber that U(0) = 0, so Cr KU and because B(0) = 0, we get T∈B∗v = 0 for all
v Cm. With c C and ⊂v Cm, h+cv stays in KU and TB∗(h+cv) = TB∗h.
∈ ∈ ∈
Replacing h by h+cv in the equality 3.1, we have:
TGTB∗h,TG(h+cv) + TG(h+cv),TGTB∗h = TB∗h 2+ TGTB∗h 2.
h i h i k k k k
This is equivalent to
2Re (c TGTB∗h,TGv )= TB∗h 2+ TGTB∗h 2 2Re TGTB∗h,TGh .
h i k k k k − h i
This holds for all c C, so necessarily
∈
(3.2)
Re TG∗TGTB∗h(0),v =0 and TB∗h 2+ TGTB∗h 2 2Re TG∗GB∗h,h =0.
h i k k k k − h i
Thefirstequalityholdsforallv Cm,soTG∗TGTB∗h(0)=0. ReplacinghbyS∗kh,
∈
whichstaysinKU,we deduce thatTG∗TGTB∗S∗kh(0)=0andso TG∗TGTB∗h=0.
This implies that TB∗h kerTG or TGTB∗h kerTG∗. We denote f = TGTB∗h.
Then, f kerTG∗ a∈nd there exists q H∈2(Cm,µG) such that f = Gq. The
∈ ∩G ∈
norm of f is f 2 = G∗Gq,q = TG∗f,q =0. Finally, TB∗KU kerTG.
k k h i h i ⊂
The second equality of (3.2) implies the following equivalences:
TB∗h 2+ TGTB∗h 2 2Re TG∗TGTB∗h,h =0
k k k k − h i
TB∗h 2 TGh 2+ TGh 2+ TGTB∗h 2 2Re TG∗TGTB∗h,h =0
⇐⇒k k −k k k k k k − h i
⇐⇒kTB∗hk2−kTGhk2+kTGTIdr−B∗hk2 =0
⇐⇒kTGTIdr−B∗hk2 =kTGhk2−kTB∗hk2.
But we know that TB∗KU ⊂kerTG, so kTGTIdr−B∗hk2 =kTGhk2 and kTB∗hk2 =
0. So we get TB∗KU = 0 as desired.
{ } (cid:3)
Corollary 3.6. The operator TIdr−BTG∗ acts on F as division by G.
Proof. Let Gh . Then, thanks to the last theorem, TB∗h=0. So,
∈F
TId BTG∗Gh=TId bTG∗TGTId B∗h
− − −
and Lemma 3.4 implies that (Id TBTB∗)h=h. (cid:3)
−
In the original proof, Sarason uses the fact that scalar model spaces K (or
u
more generally de Branges spaces [Sar94b] II-7) are stable under the action of T¯b
for every symbol b H . This does not hold for matrix symbols. The inclusion
∞
TB∗KU KU mean∈sthatBUH2(Cr) UH2(Cr),andsoU∗BU H∞(Cr Cr).
⊂ ⊂ ∈ →
This is obvious if B and U commute.
KERNELS OF VECTOR-VALUED TOEPLITZ OPERATORS 9
In this section, we will construct an example in H∞(C2 C2) where TB∗KU is
→
notcontainedin K . The followingcharacterizationof(2 2)-matrix-valuedinner
U
×
functions is due to Garcia, in [Gar06].
Theorem 3.7. Let U H (C2 C2). Then U is inner if and only if U is of the
∞
∈ →
form:
a b
U =(cid:18)θ¯b −θa¯(cid:19)
where θ :=det U is inner, and a,b K verify a2+ b2 =1 a.e. on T.
zθ
∈ | | | |
Garciagivesaninterestingexampleofaninnerfunctionbytakinga:=(1+θ)/2,
b:= i(1 θ)/2 and
− −
1 (1+θ) i(1 θ)
V := − .
2(cid:18) i(1 θ) (1+θ)(cid:19)
− −
Taking θ =z, for example, we notice that the entries are outer scalar functions.
We can look for U =zV which is still inner and vanishing at zero.
b b
Let B = 1 2 H (C2 C2). A calculation shows that U BU is equal
(cid:18)b3 b4(cid:19) ∈ ∞ → ∗
to:
b1+b4+ Re(θ)(b1−b4)+iImθ(b3+b2) b2−b3+ Re(θ)(b3+b2)+iImθ(b1+b4) .
(cid:18)b3−b2+ Re(θ)(b3−b2)+iImθ(−b1+b4) −b1+b4+ Re(θ)(b1+b4)+iImθ(−b3−b2)(cid:19)
If we suppose that b = b and b = b , then
4 1 3 2
− −
Re(θ)b b + Im(θ)b
U BU = 1 2 1 ,
∗ (cid:18) 2b2+ Im(θ)b1 Re(θ)b1 (cid:19)
− −
which is not in H∞(C2 C2) and so TB∗KU KU.
→ 6⊂
Remark 3.8. In [Sar88], Sarason establishes an alternative proof of Theorem 2.1
using Corollary 3.4. This approach could be generalized to the vector-valued case.
4. Kernel of Toeplitz operators
4.1. Some examples. For a nearly S -invariant subspace = GK , we recall
∗ U
that W = ( zH2(Cm)) , and r := dim W m. If Fm = 2, then we have
⊥
F ∩ F ∩ ≤
two ways to build with dim =2.
F F
Example 1. If r=2, G=[g ,g ] and U =zId . Let be the space
1 2 2
F
a(1+z)1/2
,(a,b) C2 .
(cid:26)(cid:18) b(1 z)1/2 (cid:19) ∈ (cid:27)
−
Because f(0)=0 implies f =0, it is S -nearly invariant. We see that W = and
∗
=T C2 with F
G
F
1 (1+z)1/2 0
G(z)= .
2(cid:18) 0 (1 z)1/2(cid:19)
−
The functions 1(1+z)1/2 and 1(1 z)1/2 are outer in H2 and G is outer, because
2 2 −
its determinant is outer (see section 2.1). Moreover, =H2(C2).
Is the kernel of a Toeplitz operator T ? We wilGl build ϕ L (C2 C2) as
ϕ ∞
F ∈ →
the following. Remark that G(eit) G 1(eit) is diagonal. The diagonal terms are
∗ −
e−12it and e21it. So, G∗G−1 lies in L∞(C2 C2) and then
− →
(4.1) T =p z−32 0 a.e. z T.
z¯G(z)∗G(z)−1 +(cid:18) 0 z−12(cid:19) ∈
−
10 CHEVROT
Every f in F is of the form f =Ge, with e∈C2, and Tz¯G∗G−1f =p+(z¯G∗e)=0.
So is the kernel of Tz¯G∗G−1.
F
Example 2. We modify the previousexample togetanearlyS -invariantsubspace
∗
which is not the kernel of a Toeplitz operator. Let and G be defined by
F
a(1+z) 1 1+z 0
= (a,b) C2 and G(z)= .
F (cid:26)(cid:18) b(1 z) (cid:19) ∈ (cid:27) √2(cid:18) 0 1 z(cid:19)
− −
We will show that if it is the kernel of a Toeplitz operator, it is also the kernel of
the Toeplitz operator with symbol ϕ(eit) := G (eit)U (eit)G 1(eit). This symbol
∗ ∗ −
is a diagonal matrix. The diagonal terms are e 2it and 1 a.e. eit T. But
−
− ∈
a+bz
kerT = : (a,b) C2 = ,
ϕ (cid:26)(cid:18) 0 (cid:19) ∈ (cid:27)6 F
and so fails to be the kernel of a Toeplitz operator.
F
Example 3. Let r=1, G=[g ] and dim K =2. Let be defined by
1 U
F
a+bz
:= : (a,b) C2 .
F (cid:26)(cid:18) 0 (cid:19) ∈ (cid:27)
1
ThisspaceisthenearlyS -invariant =GK withG(z)= andU(z)=z2.
∗ F U (cid:18) 0 (cid:19)
With the notation defined in section 2.1, we have
1 0 1 C
:=span(SkG)= H2(C2), Θ = , and G= .
G (cid:18)0 0(cid:19) 0 (cid:18)0(cid:19) (cid:18)0(cid:19)
Because = H2(G), the function G is outer. As we saw before, G˜ is the (1 1)-
G ×
square matrix such that G=Θ G˜. Here, G˜ =1. Furthermore, =kerT where
0 ϕ
F
G˜ U G˜ 1 0 z¯2 0
ϕ:= ∗ ∗ − = .
(cid:18) 0 1(cid:19) (cid:18)0 1(cid:19)
This example illustrates the problem of dimensions : the interesting part lies in G
which is a subspace of Cm.
4.2. The case r =m. This section treats the particular case where r=m seen in
the Example 1.
First of all, a nearly S -invariant subspace which is the kernel of a Toeplitz
∗
F
operator T has the form = T K with G outer. Remember that the columns
ϕ G U
F
of G H2(Cm Cm) form an orthogonal basis of W := zH2(Cm) ⊥.
∈ → F ∩ F ∩
BecausedetG 0, the inner-outerfactorizationfor matrix-value(cid:8)dfunctions allo(cid:9)ws
6≡
us to factorize with an inner function on the right. Let G be outer, and G
o i
be inner, such that G = G G . Because U(0) = 0, it follows K contains Cm
o i U
and GCm kerT . For all e Cm, we have Ge kerT . So, there exists
ϕ ϕ
H H2(Cm⊂ Cm) such that ϕ∈G = z¯H . But G =∈G G , then ϕG G = z¯H
∗ o i o i ∗
∈ →
and ϕG = z¯H G . Finally, T (G e) = 0, which implies that the columns of G
o ∗ ∗i ϕ o o
form an orthonormal basis of W. A nearly S -invariant subspace which is the
∗
F
kernel of a Toeplitz operator is of the form GK , with G outer.
U
Following Sarason, the first step is to understand what happens if dimker =
F
m = r. A new notion, namely the rigid functions, appears to characterize when
is the kernel of a Toeplitz operator. A scalar function f H1 is said to be
F ∈
rigid if the only functions in H1 which have the same argument are of the form