Download Zero products of Toeplitz operators

ZERO PRODUCTS OF TOEPLITZ OPERATORS
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
Abstract. We prove that the product of finitely many Toeplitz
operators on the Hardy space is zero if and only if at least one of
the operators is zero. We use some new vector-valued techniques
which not only lead to a vector-valued version of this result but
also appear to be needed in the scalar case.
Introduction
A Toeplitz operator Tu with the symbol u is defined as the operator
of multiplication by u followed by an appropriate projection. Toeplitz
operators acting on Hardy spaces have very simple and natural matrix
representations via infinite Toeplitz matrices, i.e., they have constant
entries on the diagonals parallel to the main one. These operators also
have many natural relationships with other relevant mathematical objects such as sequence spaces, reproducing kernels, maximal ideals in
Banach algebras, Wiener-Hopf processes, the commutant of the shift
operator, etc. The algebra generated by Toeplitz operators is of particular importance and has generated a lot of interest in the recent
years.
It is easy to check that a Toeplitz operator is the zero operator if and
only if its symbol is zero. Likewise, the product of two such operators
is zero if and only if the symbol of one of them is zero [3]. The following
natural and basic conjecture about finite products of Toeplitz operators
has been well-known for a long time: If a product of n Toeplitz operators
is the zero operator then at least one of these operators must be zero.
This was shown to hold for three operators by Axler in the 1970’s
(unpublished) but the method used in [3] becomes quite complicated
for handling more operators. Although the question has received some
Date: 29 January, 2009.
2000 Mathematics Subject Classification. 47B35, 30H05.
The second author was supported by MICINN grant MTM2006-14449-C02-02
and partially by MTM2007-30904-E and CSD2006-00032 (i-MATH), Spain, as well
as by the European ESF Network HCAA (“Harmonic and Complex Analysis and
Applications”).
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
attention, only recently has the conjecture been verified for n = 5 and
n = 6 by Guo [7] and Gu [6] respectively.
In the present paper we show that the above conjecture is true in general and actually holds in the appropriate sense in the vector-valued
case as well. In fact, the vector-valued approach has not been considered here merely for the purpose of greater generality but is crucial in
our proof even in the scalar case.
In order to state our result, we first fix the notation. Denote by dσ =
dθ/(2π) the normalized arc length measure on the unit circle T and
by H p the usual Hardy space of the circle. The space (H p )m is the
set of all vector-valued functions f : T → Cm , written as a column
matrix f = (f 1 , f 2 , . . . f m )t , where each f j ∈ H p . An r × m Toeplitz
operator matrix is an r × m operator-matrix from (H p )m into (H p )r
whose entries are Toeplitz operators. Such an operator is defined via
an r ×m matrix-valued function u with entries in L∞ (T) called symbol .
As in the scalar case, the corresponding Toeplitz operator-matrix Tu is
defined via the Szegő projection:
Z
(uf )(z)
Tu f (ζ) = P+ (uf )(ζ) =
dσ(z), f ∈ (H p )m ,
T 1 − z̄ζ
where the product uf is to be understood in the standard matrix
P product sense; that is, the i-th component is given by (Tu f )i = j Tuij f j .
The values f (z) above denote, as is usual, the nontangential boundary
values of f which exist a.e. on T and belong to Lp (T, Cm ). Throughout
the paper we assume that p > 1, so that the Toeplitz operator-matrices
considered above are bounded on (H p )m .
We shall use the standard notations H ∞ (Mr×m (C)), N (Mr×m (C)) for
the matrices whose entries are bounded analytic functions, or meromorphic functions in the Nevanlinna class on D. Similarly L∞ (T, Mr×m (C))
denotes the matrices with essentially bounded measurable entries. We
can now state our main result.
Theorem A. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)). If Tu1 · · · Tun = 0
then at least one symbol uk , 1 ≤ k ≤ n, satisfies det uk = 0 a.e..
This is a natural generalization of the conjecture for the vector-valued
case since there are even constant non-zero matrices whose product is
the zero matrix. The theorem easily extends to the finite-rank products
of finitely many Toeplitz operators: in order for the rank of the product
to be finite, again at least the symbol of one of the factors must be
singular a.e..
ZERO PRODUCTS OF TOEPLITZ OPERATORS
3
Here is an outline of our method, presented in the scalar case m = 1.
We exploit the close relationship of kernels of Toeplitz operators to the
so-called nearly invariant subspaces [10]. These are subspaces of H p
with the property that each zero of an element in the space which is not
a common zero can be divided out without leaving the space. It is wellknown that the kernel of a Toeplitz operator is nearly invariant (see
also Lemma 2.1 below). Almost conversely, every nearly invariant subspace is contained in a Toeplitz kernel. However, the situation is more
complicated for the kernels of finite products of Toeplitz operators.
Our starting point is the observation that if u1 , . . . , un ∈ L∞ (T) then
(see Lemma 2.2 below) the subspace
N = {(f, Tun f, . . . , Tu2 · · · Tun f )t : f ∈ ker Tu1 · · · Tun }
is a closed nearly invariant subspace of (H p )n , which brings into the
picture the vector-valued case in a natural way. With this idea in
mind, we consider nearly invariant subspaces in the vector-valued case
and prove in Section 1 that such subspaces are always contained in
the kernel of an appropriate Toeplitz operator-matrix which we denote
here by V .
If we assume that Tu1 Tu2 · · · Tun = 0, the equality TV N = {0} leads to
TV (f, Tun f, . . . , Tu2 · · · Tun f )t = 0 ,
f ∈ Hp .
Now we can use an approximate identity argument, Lemma 2.3 (the
result goes back to Douglas [5], see also [2]) to show that
V (f, un f, . . . , u2 · · · un f )t = 0 a.e. ,
f ∈ Hp .
The conjecture would now follow if, under the assumption that none
of the symbols uk is zero, we could choose V to be an n × n matrixvalued function invertible on a set of positive measure on T. Even if we
cannot conclude the proof at this early stage, the analysis of this type
of situation is essentially involved in our argument. It requires several
steps which will be briefly discussed below. They are based on two
kernel inclusion theorems for products of Toeplitz operator-matrices
proved in Section 3.
The reasoning described here can be applied to the partial products
u1 · · · uk , k < n and we begin with the case when k = 2. It can be
easily verified that under our assumption we must have u1 u2 6= 0 (this
is actually not needed in our proof). Then the simplest variant of
our Theorem 3.1 asserts that there exists a bounded measurable 2 × 2
matrix-valued function V2 , invertible a.e. on T, such that
N2 = {(f, Tu2 f )t : f ∈ ker Tu1 Tu2 } ⊂ ker TV2 ,
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
or equivalently,
{(f, f )t : f ∈ ker Tu1 Tu2 } ⊂ ker TV2 TU2 ,
where
µ
U2 (z) =
1
0
0 u2 (z)
¶
.
If n ≥ 3 this means that
{(f, f )t : f ∈ ker Tu1 Tu2 Tu3 } ⊂ ker TV2 TU2 TU3 ,
where
µ
U3 (z) =
u3 (z)
0
0
u3 (z)
¶
.
The point is that Theorem 3.1 applies to ker TV2 TU2 TU3 as well, and if
we assume without loss of generality that Tu2 · · · Tun 6= 0 we can start
an inductive process which is described in Theorem 3.2. The outcome
of this process is that the diagonal in (H p )(n−1)! = (ker Tu1 · · · Tun )(n−1)!
satisfies the inclusion
(n−1)!
(H p )d
⊂ ker TVn TW1 · · · TWn−1 ,
where Vn is a bounded measurable (n − 1)! × (n − 1)! matrix-valued
function, invertible a.e. on T, and the product W1 · · · Wn−1 has the
form
µ
¶
J(z)
0
W1 (z) · · · Wn−1 (z) =
,
A(z) B(z)
where A, B, J are bounded measurable matrix-valued functions with J
∈ L∞ (M(n−2)!×(n−2)! (C)) diagonal and non-zero. We then use again the
approximate identity argument mentioned above to obtain
Vn (z)W1 (z) · · · Wn−1 (z)(f (z), f (z), . . . f (z))t = 0,
a.e. on T ,
for all f ∈ H p , which leads to the contradiction J = 0 and concludes
the proof. In the vector-valued case we actually obtain a somewhat
stronger result than Theorem A. In Section 4 we prove that if each of
the symbols uk , 1 ≤ k ≤ n, is invertible on a set of positive measure
on T then the product Tu1 · · · Tun satisfies
dim{Tu1 · · · Tun f (λ) : f ∈ (H p )m } = m
for all λ ∈ D except possibly a discrete subset.
Acknowledgments. We would like to thank N. K. Nikolskiı̆ for helpful discussions, S. Shimorin for pointing out some initial errors in the
first draft, and especially D. Sarason for a careful reading of the first
draft and many useful comments and suggestions. Part of this work
ZERO PRODUCTS OF TOEPLITZ OPERATORS
5
was done while the first author was visiting the Department of Mathematics at the Universidad Autonóma de Madrid. He wants to thank the
department for its hospitality and a stimulating working environment.
1. Nearly invariant subspaces of vector-valued functions
Throughout the paper we shall denote by ζ the identity function on
D, ζ(z) = z. Following Sarason [10] we call a closed subspace N of
H p nearly invariant if for every f ∈ N and every λ ∈ D which is a
f
zero of f but not a common zero of N we have that ζ−λ
∈ N . Nearly
invariant subspaces generalize Toeplitz kernels and play an important
role in many other problems. The main result about such subspaces is
the characterization obtained by Hitt [8] and Sarason [10] for the case
when p = 2. Their theorem asserts that a non-zero nearly invariant
subspace N of H 2 is either invariant for the unilateral shift, or it has
the form
(1.1)
N = θN F Kθ
where θN is the common inner divisor of N , F is an outer function,
θ is inner, and, as usual Kθ = H 2 ª θH 2 . In addition, multiplication
by θN F is an isometry from Kθ onto N . The parameters F, θ are
not uniquely determined by N . However F can be chosen to be the
−1
extremal function of θN
N for the origin, that is, the unique solution
of the extremal problem
−1
F (0) = sup{Ref (0), f ∈ θN
N , kf k = 1}.
An immediate consequence of this result is that a nontrivial nearly
invariant subspace N = θN F Kθ is contained in the kernel of the nonzero Toeplitz operator with symbol ζθθN F /F . Indeed, any function f
is such a space N can be written as f = θN F g for some g ∈ H 2 which
is orthogonal to θH 2 ; hence for all n ≥ 0 we get
hTζθθN F /F f, ζ n i = hP (ζF θg), ζ n i = hζF θg, ζ n i = hg, θF ζ n+1 i = 0,
which implies that TζθθN F /F f = 0. Thus, we can state the following
result which is the starting point for the present section.
Proposition 1.1. A nontrivial nearly invariant subspace of H 2 is either invariant for the unilateral shift Mζ on H 2 , Mζ f = ζf , or it is
contained in the kernel of a non-zero Toeplitz operator.
We should point out here that the above proposition can be proved
directly without use of the powerful theorem of Hitt and Sarason (see
also Theorem 1.1 below). The purpose of this section is to extend this
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
result to the vector-valued case. To this end, fix a positive integer m
and p > 1. If M is a subspace of (H p )m and λ ∈ D we define the fiber
dimension of M at λ by
dimM (λ) = dim{f (λ) : f ∈ M}.
It is a simple exercise to show that for all α ∈ D except possibly a
discrete subset we have
(1.2)
dimM (α) = max dimM (λ).
λ∈D
Moreover, if
max dimM (λ) = k
λ∈D
then we can find f1 , . . . , fk ∈ M such that the matrix with columns
f1 (λ), . . . , fk (λ) has rank k. Since the minors are analytic functions of
λ, the claim follows.
The same argument shows that there is a relatively open dense subset
of k-tuples (f1 , . . . , fk ) ∈ Mk with
_
dim {f1 (λ), . . . , fk (λ)} = k
for all λ ∈ D \ Λ, where Λ is a discrete subset of D which may depend
on f1 , . . . , fk . For further reference, let us record the following simple
observation regarding the maximal fiber dimension. As is usual, by
the kernel of a matrix we shall mean the kernel of the induced linear
operator on Cm .
Proposition 1.2. Let M be a subspace of (H p )m with
k = max dimM (λ) < m.
λ∈D
∞
Then there exists u ∈ H (Mm×m (C)) and a discrete set Λ ⊂ D with
dim ker u(λ) = k for all λ ∈ D \ Λ, such that M ⊂ ker Tu .
Proof. Let λ0 ∈ D and (f1 , . . . , fk ) ∈ Mk with
_
dim {f1 (λ0 ), . . . , fk (λ0 )} = k,
or, equivalently,
rank(fjl (λ0 )) 1≤j≤k = k,
1≤l≤m
l
where f denotes the l-th component of f . Assume without loss of
generality that the first k rows of this matrix are linearly independent.
Then there exists a discrete set Λ ⊂ D such that
det(fjs (λ))1≤j,s≤k 6= 0
ZERO PRODUCTS OF TOEPLITZ OPERATORS
7
for all λ ∈ D \ Λ, since the above determinant is easily seen to be an
analytic function of λ ∈ D. On the other hand, for every f ∈ M, λ ∈ D
and l > k we have

 l
f (λ) f1l (λ) . . . fkl (λ)
 f 1 (λ) f11 (λ) . . . fk1 (λ) 
=0
det 
..
.. 
 ...
.
...
. 
f k (λ) f1k (λ) . . . fkk (λ)
which implies that for each such l we have
(1.3)
l
f (λ) +
k
X
as (λ)f s (λ) = 0,
s=1
where the coefficients as (·) are independent of f and belong to the
Nevanlinna class on D (each of them being a quotient of two H p/k
functions, by generalized Hölder’s inequality). This can easily be seen
from the expansion by the first column. There exists a function F ∈
H ∞ such that F as ∈ H ∞ , 1 ≤ s ≤ k, and which vanishes at most on a
discrete exceptional set which does not depend on f . Indeed,Qwriting
each as = gs /hs with gs , hs ∈ H ∞ it suffices to choose F = ks=1 hs .
Then the lower-triangular block matrix
u = (uls )1≤l,s≤m
with

 F as , if 1 ≤ s ≤ k < l
F,
if l = s > k
uls =

0,
otherwise
satisfies the conditions in the statement. Indeed, it is obvious that
Tu f = uf since u ∈ H ∞ (Mm×m (C)) and one easily verifies that uf is
the zero vector in view of (1.3). Again by (1.3) and the non-vanishing
property of F , it is also quite evident that for each λ ∈ D \ Λ we have
dim ker u(λ) = k since this kernel is formed precisely by those vectors
f = (f 1 (λ), f 2 (λ), . . . , f m (λ)) ∈ Cm for which f l (λ) = 0 whenever
k < l ≤ m.
¤
Let us now turn to nearly invariant subspaces of vector-valued functions.
Definition 1.1. A closed subspace N of (H p )m is called nearly invariant if for every f ∈ N and every λ ∈ D with f (λ) = 0 and
1
f ∈ N.
dimN (λ) = maxα∈D dimN (α) we have that ζ−λ
As pointed out above, we want to extend Proposition 1.1 to this context. The only complication which occurs in the vector-valued case is
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
to find the appropriate interpretation for the dichotomy given by that
proposition. This appears naturally, since there are nearly invariant
subspaces which are shift-invariant in certain components but not in
others.
In the proof of the next result we shall make use of the well known jump
theorem for Cauchy transforms (see, for example, [4, § 2.4]). Given
h ∈ L1 (T, Cm ) its Cauchy transform is defined by
Z
h(z)
Ch(λ) =
dσ(z), λ ∈ C \ T.
T z −λ
If Ci h(λ), Ce h(λ) denote the nontangential limits of Ch from the inside
and outside of the unit disc respectively, then these limits exist σ-a.e.
and the jump theorem asserts that
Ci h(λ) − Ce h(λ) = λ̄h(λ)
a.e. on T.
In what follows we shall use the notation N ⊥ for the annihilator of the
closed subspace N of H p . When p = 2, we mean by this the usual
orthogonal complement of N in H 2 while for p 6= 2 it will denote the
annihilator in the dual space of N , a subspace of the dual space of
(H p )m . That is, if we denote by x · y the scalar product of the vectors
x, y ∈ Cm then saying that g ∈ N ⊥ means that
Z
g(z) · h(z)dσ(z) = 0
T
for all h ∈ N . See [9, Chapter 7] for a description of the dual of H p
when m = 1; the description is similar in the vector-valued case.
Proposition 1.3. Let N be a nontrivial nearly invariant subspace of
(H p )m .
(i) For every λ ∈ D with dimN (λ) = maxα∈D dimN (α) we have
dimN (λ) = dim N /(N ∩ (ζ − λ)N ) .
(ii) If α ∈ C \ D then
dim N /(N ∩ (ζ − α)N ) ≤ max dimN (λ) .
λ∈D
Proof. (i) Let λ be a value for which the maximum fiber dimension is
attained. It follows from the definition of a nearly invariant subspace
that a function f ∈ N vanishes at such λ if and only it can be written as
f = (ζ − λ)g for some g ∈ N , hence N ∩ (ζ − λ)N = {f ∈ N : f (λ) =
ZERO PRODUCTS OF TOEPLITZ OPERATORS
9
0}. The functional of point evaluation at λ is linear, hence by standard
linear algebra
dim N /{f ∈ N : f (λ) = 0} = dim{f (λ) : f ∈ N }
and therefore
dim N /(N ∩ (ζ − λ)N ) = dimN (λ) .
(ii) Let k = maxλ∈D dimNW
(λ), let λ0 ∈ D with dimN (λ0 ) = k, and let
f1 , . . . , fk ∈ N with dim {f1 (λ0 ), . . . , fk (λ0 )} = k. Then there is a
discrete, possibly void, subset Λ of D and a fixed minor of order k of
the determinant with column vectors f1 (λ), . . . , fk (λ) which does not
vanish on D \ Λ. If λ ∈ D \ Λ and g ∈ N we deduce, as above, that
there exist scalars cj (λ, g), 1 ≤ j ≤ k such that
(1.4)
g(λ) =
k
X
cj (λ, g)fj (λ)
j=1
and consequently by near-invariance,
Ã
!
k
X
1
(1.5)
Rλ g =
g−
cj (λ, g)fj ∈ N .
ζ −λ
j=1
The coefficients cj (λ, g) can be calculated from (1.4) with help of the
inverse of the matrix corresponding to the minor considered above.
It is important to note that the entries of this matrix are functions
in H p , hence the entries of its inverse are meromorphic functions in
the Nevanlinna class on the unit disc. Thus by inserting zero column
vectors in this matrix if necessary, we obtain that there exists a k × mmatrix A(λ), λ ∈ D \ Λ of rank k such that
(c1 (λ, g), . . . , ck (λ, g))t = A(λ)g(λ),
λ ∈ D \ Λ,
where the entries of A(λ) extend to meromorphic functions in the
Nevanlinna class on D. In particular, cj (·, g) , 1 ≤ j ≤ k are meromorphic functions in the Nevanlinna class on D and (1.4) continues to
hold a.e. on T. Now let α ∈ C \ D and let h ∈ (N ∩ (ζ − α)N )⊥ . Note
that
(ζ − α)Rλ g = g −
k
X
cj (λ, g)fj + (λ − α)Rλ g ∈ N ∩ (ζ − α)N
j=1
hence
Z
(z − α)Rλ g(z) · h(z)dσ(z) = 0,
T
λ ∈ D \ Λ,
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
or equivalently,
(1.6) C((ζ − α)g · h)(λ) =
k
X
cj (λ, g)C((ζ − α)fj · h),
λ ∈ D \ Λ.
j=1
By taking nontangential limits we then have
i
(1.7)
C ((ζ − α)g · h)(λ) =
k
X
cj (λ, g)Ci ((ζ − α)fj · h) ,
j=1
a.e. on T. On the other hand, the jump theorem gives
Ce ((ζ − α)g · h)(λ) = Ci ((ζ − α)g · h)(λ) − λ̄(λ − α)g · h(λ)
(1.8)
a.e. on T. From (1.6), (1.7) and (1.8) it follows that if g1 , . . . , gl ∈ N
and h ∈ (N ∩ (ζ − α)N )⊥ then
(Ce ((ζ − α)g1 · h)(λ), . . . , Ce ((ζ − α)gl · h)(λ))t = Γkl (λ)x(λ) ,
where
Γkl (λ) = (cj (λ, gi )) 1≤j≤k ,
1≤i≤l
and x(λ) is the k-dimensional vector with components
xj (λ) = Ci ((ζ − α)fj · h)(λ) − λ̄(λ − α)fj · h(λ) ,
e
e
1 ≤ j ≤ k.
t
Thus all vectors (C (g1 · h)(λ), . . . , C (gl · h)(λ)) lie in the range of the
linear map induced by Γkl (λ). Note that this matrix does not depend
on the choice of h ∈ (N ∩ (ζ − α)N )⊥ and clearly, at any point λ ∈ T
where it is defined, its rank is at most k. This means that if l > k,
g1 , . . . , gl ∈ N and h1 , . . . , hl ∈ (N ∩ (ζ − α)N )⊥ the vectors
(Ce ((ζ − α)g1 · hi )(λ), . . . , Ce ((ζ − α)gl · hi )(λ))t ,
1≤i≤l
are linearly dependent, or equivalently,
det(Ce ((ζ − α)gj · hi )(λ))1≤i,j≤l = 0
a.e. on T. Then these determinants vanish identically on C \ D since
they belong to the Nevanlinna class of C \ D. In particular,
µZ
¶
det
gj · hi dσ
= det(Ce ((ζ − α)gj · hi )(α))1≤i,j≤l = 0 ,
T
1≤i,j≤l
and we obtain a contradiction which completes the proof.
¤
Given a nearly invariant subspace of (H p )m , the ordered pair
(max dimN (λ), max dim N /(N ∩ (ζ − λ)N ))
λ∈D
λ∈C\D
will be called the index of N . It is easy to see that N is invariant for
the shift operator Mζ if and only if it has index (k, 0) for some k ≤ m.
ZERO PRODUCTS OF TOEPLITZ OPERATORS
11
The aim of this section is to give a vector-valued version of Proposition 1.1. Of course, for a nearly invariant subspace N 6= {0} of (H p )m
with index (k, l), where k < m, Proposition 1.2 immediately provides a
choice of a Toeplitz operator-matrix Tu with N ⊂ ker Tu . In this case
we can choose u ∈ H ∞ (Mm×m (C) with rank u(λ) = m − k a.e. on T.
The remaining case k = m is the most interesting one and the rest of
this section is devoted to this case (see next Theorem).
For further purposes let us record the definition and some simple properties of the generalized backward shifts used in this proof. Given
fj ∈ (H p )m , 1 ≤ j ≤ k we shall use throughout the notation
f = (f1 , . . . , fk ) ∈ [(H p )m ]k
Since the dimension of f is always clear from the context it will not be
specified in the notation. Also, we shall frequently identify [(H p )m ]k
with (H p )mk .
Let N be a nearly invariant subspace of (H p )m with index (k, l), let
λ0 ∈ D with dimN (λ0 ) = k and let fj ∈ N , 1 ≤ j ≤ k such that
_
dim {f1 (λ0 ), . . . , fk (λ0 )} = k.
As we have seen in the proof of Proposition 1.3 (formula (1.4)), if λ ∈ D
with
_
dim {f1 (λ), . . . , fk (λ)} = k
then for every f ∈ N we can write
f (λ) =
k
X
cj (λ, f )fj (λ) ,
j=1
where the coefficients cj (λ, f ) are given by
(1.9)
(c1 (λ, f ), . . . , ck (λ, f ))t = A(λ)f (λ)
with
A(·) ∈ N (Mk×m (C)),
rank(A(λ)) = k
for all λ ∈ D outside of a discrete exceptional set. The matrix-valued
function A is uniquely determined by (1.9) and satisfies
A(λ)fj (λ) = (δ1j , . . . , δkj )t ,
1 ≤ j ≤ k.
For such λ ∈ D the linear operator defined by
P
f − kj=1 cj (λ, f )fj
,
(1.10)
Rλ,f f =
ζ −λ
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ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
maps N into itself. By the closed graph theorem Rλ,f is bounded. For
our purposes, the crucial property of these operators is that for every
α∈C
(1.11) (ζ −α)Rλ,f f = f −
k
X
cj (λ, f )fj +(λ−α)Rλ,f f ∈ N ∩(ζ −α)N .
j=1
We also note that if α, λ ∈ D are such that
_
_
dim {f1 (α), . . . , fk (α)} = dim {f1 (λ), . . . , fk (λ)} = k
then
1
A(λ)(Rα,f f )(λ) =
λ−α
Ã
A(λ)f (λ) −
k
X
!
cj (α, f )A(λ)fj (λ)
,
j=1
and from above we obtain that
cj (λ, f ) − cj (α, f )
(1.12)
cj (λ, Rα,f f ) =
, 1 ≤ j ≤ k.
λ−α
We can now turn to the vector-valued version of Proposition 1.1.
Theorem 1.1. Let N be a nontrivial nearly invariant subspace of
(H p )m with index (k, l), where l > 0. Then there exists a non-zero
l × m-Toeplitz operator-matrix Tu with N ⊂ ker Tu . The symbol matrix
u can be chosen to have the form
u(λ) = B(λ)A(λ)
a.e. on T, where A ∈ N (Mk×m (C)) has rank k a.e. on T and B ∈
H ∞ (Ml×k (C)) has rank l a.e. on T.
Proof. Let λ0 ∈ D with dimN (λ0 ) = k and let λ1 ∈ C \ D with
dim N /N ∩ (ζ − λ1 )N = l .
Recall that by Proposition 1.3 (ii) we have l ≤ k. Denote by
[g] = g + (ζ − λ1 )N ∩ N
the coset of g ∈ N in (ζ −λ1 )N ∩N and use again the simple argument
mentioned at the beginning of the section to conclude that both sets
of tuples
n
o
_
(f1 , . . . , fk ) ∈ N k : dim {f1 (λ0 ), . . . , fk (λ0 )} = k
and
n
(g1 , . . . , gk ) ∈ N k : dim
_
o
{[g1 ], . . . , [gl ]} = l
ZERO PRODUCTS OF TOEPLITZ OPERATORS
13
are open and dense in N k . Then we can choose f1 , . . . , fk ∈ N such
that
_
_
dim {f1 (λ0 ), . . . , fk (λ0 )} = k, dim {[f1 ], . . . , [fl ]} = l .
Clearly, there exist h1 , . . . , hl ∈ (N ∩ (ζ − λ1 )N )⊥ such that
µZ
¶
det
fj · hr dσ
6= 0 .
T
1≤r,j≤l
Let f = (f1 , . . . , fk ) and let Rλ,f be the operator defined in (1.10). By
(1.11) we have
Z
(ζ − λ1 )Rλ,f f · hr dσ = 0 , 1 ≤ r ≤ l ,
T
for all λ ∈ D \ Λ, where Λ is a discrete subset of D. This gives
C((ζ − λ1 )f · hr )(λ) =
k
X
cj (λ, f )C((ζ − λ1 )fj · hr ),
1 ≤ r ≤ l,
j=1
and after taking nontangential limits
i
C ((ζ − λ1 )f · hr )(λ) =
k
X
cj (λ, f )Ci ((ζ − λ1 )fj · hr ),
1 ≤ r ≤ l,
j=1
a.e. on T. Recall also that cj (., f ) are Nevanlinna functions with
k
X
cj (λ, f )fj (λ) = f (λ) ,
j=1
a.e. on T. Then by an application of the jump theorem we obtain
e
C ((ζ − λ1 )f · hr )(λ) =
k
X
cj (λ, f )Ce ((ζ − λ1 )fj · hr )(λ),
1 ≤ r ≤ l,
j=1
a.e. on T. If A is the k × m matrix given by (1.9) and
B1 (λ) = (Ce ((ζ − λ1 )fj · hr )(λ)) 1≤j≤k
1≤r≤l
then the last equality can be rewritten as
(Ce ((ζ − λ1 )f · h1 )(λ), . . . , Ce ((ζ − λ1 )f · hl )(λ))t = B1 (λ)A(λ)f (λ)
a.e. on T. Finally, by an argument similar to the ones used before,
we can find a function F ∈ H ∞ such that F B1 ∈ H ∞ (Ml×k (C)) and
F B1 A ∈ L∞ (T, Mk×m (C)). Set
u = F B1 A
to obtain the desired result.
¤
14
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
2. Near-invariance and kernels of products of Toeplitz
operator-matrices
The starting point and motivation for the work in this section is the following observation essentially related to the commutator of a Toeplitz
operator-matrix with a backward shift. For simplicity we shall consider
throughout the case of quadratic m × m-matrices
Lemma 2.1. If u ∈ L∞ (T, Mm×m (C)) and f ∈ (H p )m has the zero
λ ∈ D then
f
Tu f (z) − Tu f (λ)
Tu
(z) =
.
ζ −λ
z−λ
Proof. We have
Z
f
ζf
Tu
dσ
(z) =
ζ −λ
T (1 − ζλ)(1 − ζz)
Z
Z
1
1
Tu f (z) − Tu f (λ)
f
f
=
dσ −
dσ =
z − λ T 1 − ζz
z − λ T 1 − ζλ
z−λ
and the result follows.
¤
As an immediate application of the lemma we obtain that the kernel
of such an operator is a nearly invariant subspace of (H p )m . Then by
Theorem 1.1 we obtain that ker Tu ⊂ ker Tv where v is a matrix as
given in Theorem 1.1.
The above result has an immediate generalization for kernels of products of Toeplitz operators which will be given below.
Given u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , we define the operator
Tu : (H p )m → (H p )mn by
(2.13)
Tu f = (f t , (Tun f )t , . . . (Tu2 · · · Tun f )t )t ,
f ∈ (H p )m .
Note that the operator Tu1 does not appear in this definition.
Lemma 2.2. If u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n then
Tu ker Tu1 · · · Tun
is a closed nearly invariant subspace of (H p )mn .
Proof. The fact that the subspace is closed is obvious by the continuity
of Toeplitz operators. To see the near-invariance, let f ∈ ker Tu1 · · · Tun
and λ ∈ D with
f (λ) = Tun f (λ) = . . . = Tu2 · · · Tun f (λ) = 0 .
ZERO PRODUCTS OF TOEPLITZ OPERATORS
15
Then using repeatedly Lemma 2.1 we get
Tu · · · Tun f
f
Tu f
f
= n , . . . , Tuj · · · Tun
= j
ζ −λ
ζ −λ
ζ −λ
ζ −λ
for all j ≥ 2. Also,
Tun
Tu1 · · · Tun
Thus,
f
Tu · · · Tun f − Tu1 · · · Tun f (λ)
= 1
= 0.
ζ −λ
ζ −λ
Tu f
f
= Tu
ζ −λ
ζ −λ
and
f
∈ ker Tu1 · · · Tun .
ζ −λ
which completes the proof.
¤
The main result of this section will be a refinement of Lemma 2.2. In
order to state our theorem we need the following notations.
We shall use throughout multi-indices i = (i1 , . . . , il ) where the integers
i1 , . . . , il satisfy i1 < i2 < . . . < il . For s ≥ l we denote by Isi =
(aij )1≤i,j≤s the s × s-matrix with entries
aij = 1 if i = j = iν , 1 ≤ ν ≤ l,
and aij = 0 otherwise.
We shall keep the usual notation for the identity matrix, that is, if
s = l and i0 = (1, . . . , l) we write Isi0 = Is .
Next, if R is a subspace of Cm , a set J ⊂ {1, . . . , m} will be called
a coordinate basis for R if the functionals of evaluation of the j-th
coordinate, j ∈ J , form a basis in the algebraic dual of R.
Finally, if M is a closed subspace of (H p )m and z ∈ T, consider the
vector space Mz ⊂ Cm consisting of all nontangential limits f (z),
f ∈ M (whenever these limits exist). It is not difficult to verify that
dim Mz = max dimM (λ)
λ∈D
a.e. on T. Indeed, let λ0 ∈ D with
dimM (λ0 ) = max dimM (λ) = r .
λ∈D
Then there exist f1 , . . . , fr ∈ M with f1 (λ0 ), . . . , fr (λ0 ) linearly independent. As we have seen before, for every λ ∈ D except possibly for a
discrete set, f1 (λ), . . . , fr (λ) are linearly independent and again since
determinants formed with the components of these vectors are functions in the Nevanlinna class of the unit disc it follows that f1 ,. . . ,fr
are linearly independent a.e. on T. Moreover, if f ∈ M then f can be
16
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
expressed as a linear combination of the functions fj , 1 ≤ j ≤ m with
coefficients in the Nevanlinna class, which shows that
_
Mz = {f1 (z), . . . , fr (z)} a.e. .
The main result of this section is the following theorem. Recall that σ
denotes the normalized Lebesgue measure on the unit circle.
Theorem 2.1. Let u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , R a fixed
subspace of Cm such that
M = {f ∈ ker Tu1 · · · Tun : f (λ) ∈ R, λ ∈ D} 6= {0} ,
and let γ ≥ 0 be an integer with
σ({z ∈ T : dim ker((u1 · · · un )(z)|Mz ) ≤ γ}) > 0 .
Let i = (i1 , . . . , iM ), M ≤ mn, be a multi-index with the following
properties:
(i) {i1 , . . . , iM } ∩ {1, . . . , m} is a coordinate basis for R;
(ii) maxλ∈D dimImn
i T M (λ) = maxλ∈D dimTu M (λ) .
u
i
Then Imn
Tu M is a closed nearly invariant subspace of (H p )mn with
index (k, k ∗ ), where k = maxλ∈D dimTu M (λ) and
k − γ ≤ k∗ ≤ k .
At this stage, the use of the subspace R may appear as an unnecessary
complication, but this more general form of the theorem will be used
in full strength later. The proof is based on the following lemma which
is a quantitative version of a result on the symbol homomorphism on
the Toeplitz algebra; for m = 1 and p = 2 this goes back to Douglas
(see [5] or [2]). We have included a proof for the sake of completeness.
For a general Cm -valued function h on D we denote by nt − limλ→z h(λ)
the nontangential limit of h at z ∈ T whenever this limit exists. The
nontangential lim sup is defined in a similar way.
Lemma 2.3. If u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and f ∈ (H p )m then
¶
Z µ
f
dσ(ξ)
2
nt − lim (1 − |λ| )
Tu1 . . . Tun
(ξ)
λ→z
1 − λζ
1 − λξ
T
= u1 (z) . . . un (z)f (z)
a.e. on T.
Proof. We proceed by induction. The case when n = 1 is a direct consequence of Fatou’s theorem. We actually have a stronger estimate which
ZERO PRODUCTS OF TOEPLITZ OPERATORS
17
will be used in the sequel. If P+ denotes the usual Szegő projection
then for almost every z ∈ T we have
µ
¶
µ
¶
f
1
u1 f
1
Tu1
(ξ)−
u1 (z)f (z) = P+
−
u1 (z)f (z) (ξ)
1 − λζ
1 − λξ
1 − λζ 1 − λζ
a.e. on T. Then by the M. Riesz theorem we obtain
(2.14)
2 p−1
(1 − |λ| )
°p
µ
¶
Z °
°
°
f
1
° dσ(ξ)
°Tu1
(ξ)
−
u
(z)f
(z)
1
° m
°
1
−
λζ
1
−
λξ
T
C
Z
dσ(ξ)
≤ Cp (1 − |λ|2 )p−1 ku1 (ξ)f (ξ) − u1 (z)f (z)kpCm
,
|1 − λξ|p
T
where Cp > 0 depends only on p. It is not difficult to see that the right
hand side converges nontangentially to zero as λ → z for almost every
z ∈ T. This follows by standard estimates from [11, p. 57] adapted
to the disk instead of the half-plane. Namely, it is shown there that
Fatou’s theorem holds for a wide class of kernels bounded by a constant
multiple of a certain integrable function; our kernel actually equals such
a function.
Now assume that the statement holds for some n ≥ 1. For u1 , . . . , un+1 ∈
L∞ (T, Mm×m (C)), f ∈ (H p )m and x ∈ Cm write
À
¶
µ
Z ¿
f
1
Tu1 · · · Tun+1
x
dσ(ξ)
(ξ),
1 − λζ
1 − λξ Cm
T
¶ À
¶
µ
µ
Z ¿
1
f
=
x (ξ)
dσ(ξ)
Tu2 · · · Tun+1
(ξ), Tu∗1
1 − λζ
1 − λζ
T
Cm
¶
µ
¶
À
µ
Z ¿
1
1
f
∗
=
x (ξ) −
u1 (z)x
dσ(ξ)
Tu2 · · · Tun+1
(ξ), Tu∗1
1 − λζ
1 − λζ
1 − λξ
T
Cm
À
µ
¶
Z ¿
1
f
∗
+
u1 (z)x
dσ(ξ)
Tu2 · · · Tun+1
(ξ),
1 − λζ
1 − λξ
T
Cm
= J1 (λ, z) + J2 (λ, z) .
Use Hölder’s inequality to conclude that
°
°
° f °
°
(1 − |λ| )|J1 (λ, z)| ≤ (1 − |λ| )kTu2 · · · Tun+1 k °
° 1 − λζ °
p
°q
µ
¶1/q
µZ °
¶
°
°
1
1
∗
°Tu∗
° dσ(ξ)
,
×
(z)x
x
(ξ)
−
u
1
° 1 1 − λζ
° m
1
−
λξ
T
C
2
2
18
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
+ 1q = 1. By the standard estimates mentioned above we have
°
°
° f °
2 1/q °
° <∞
nt − lim sup(1 − |λ| ) °
1 − λζ °p
λ→z
a.e. on T. Using also (2.14) and the same estimates we obtain that
°q
µ
¶1/q
µZ °
¶
°
°
1
1
2 1/p
∗
°
°
=0
nt−lim (1−|λ| )
°Tu∗1 1 − λζ x (ξ) − 1 − λξ u1 (z)x° m dσ(ξ)
λ→z
T
C
where
that
1
p
a.e. on T, i.e. nt − limλ→z (1 − |λ|2 )J1 (λ, z) = 0 a.e. on T. By the
induction hypothesis we have that
nt − lim (1 − |λ|2 )J2 (λ, z) = hu2 (z) · · · un+1 (z)f (z), u∗1 (z)xi
λ→z
a.e. on T and the result follows.
¤
We are now ready to prove our theorem.
Proof of Theorem 2.1. Note that by assumption (i) on the multi-index
i, if
(2.15)
{i1 , . . . , iM } ∩ {1, . . . , m} = {i1 , . . . , il }
then every coordinate of a vector f ∈ M can be written as a linear combination of the coordinates f i1 , . . . , f il with constant coefficients independent of f . This, together with the continuity of Toeplitz
i
operator-matrices, implies that Imn
Tu M is closed.
i
To see that Imn
Tu M is nearly invariant let λ0 ∈ D be such that
dimImn
i T M (λ0 ) = max dimI i T M (λ) = max dimTu M (λ)
u
mn u
λ∈D
λ∈D
(here we are using assumption (ii)) and let f ∈ M be such that
i
Imn
Tu f (λ0 ) = 0. Assumption (i) on i implies that Tu f (λ0 ) = 0 hence
by Lemma 2.2 and its proof we have
Tu f
f
= Tu
∈ Tu ker Tu1 · · · Tun .
ζ − λ0
ζ − λ0
The fact that
f (α)
∈ R , α ∈ D \ {λ0 } and f 0 (λ0 ) ∈ R
α − λ0
is obvious, hence
f
ζ−λ0
∈ M. Thus,
i
f
Tu f
Imn
i
i
Tu
Tu M ,
= Imn
∈ Imn
ζ − λ0
ζ − λ0
i
Tu M is nearly invariant.
which shows that Imn
ZERO PRODUCTS OF TOEPLITZ OPERATORS
19
Let us now estimate k ∗ . Let λ0 ∈ D \ {0} such that
i
i
i
dim Imn
Tu M/(Imn
Tu M ∩ (1 − λ0 ζ)Imn
Tu M) = k ∗
and consider a basis of cosets [g1 ], . . . , [gk∗ ] in this quotient space. Then
there exists a discrete subset Λ of D such that [g1 ], . . . , [gk∗ ] is a basis
i
i
i
Tu M) for every λ ∈ D \ Λ. Thus
in Imn
Tu M/(Imn
Tu M ∩ (1 − λζ)Imn
i
if λ ∈ D \ Λ and g ∈ Imn Tu M there exist scalars a1 (λ), . . . ak∗ (λ) such
that
!
Ã
k∗
X
1
i
(2.16)
Gλ =
aν (λ)gν ∈ Imn
Tu M .
g+
1 − λζ
ν=1
It is easily seen that aν extend to meromorphic functions in the Nevanlinna class on D. To verify this we use (2.16) to obtain
i
i
λ(1−λ0 ζ)Gλ = (λ−λ0 )Gλ +λ0 (1−λζ)Gλ ∈ Imn
Tu M∩(1−λ0 ζ)Imn
Tu M
that is,
Z
λ
(1 − λ0 ζ)Gλ · hdσ = 0 ,
T
i
i
whenever h ∈ (Imn
Tu M ∩ (1 − λ0 ζ)Imn
Tu M)⊥ and λ ∈ D. If we now
choose functions
i
i
hµ ∈ (Imn
Tu M ∩ (1 − λ0 ζ)Imn
Tu M)⊥ ,
such that
1 ≤ µ ≤ k∗
Z
gν · hµ dσ = δµν ,
T
this leads to the linear system of equations
B(λ)(a1 (λ), . . . ak∗ (λ))t = (x1 (λ), . . . , xk∗ (λ))t ,
λ ∈ D,
where the matrix B(λ) = (bµν (λ))1≤µ,ν≤k∗ is given by
Z
gν · hµ
dσ , 1 ≤ µ, ν ≤ k ∗ , λ ∈ D ,
bµν (λ) = λ (1 − λ0 ζ)
1 − λζ
T
and
Z
g · hµ
xµ (λ) = −λ (1 − λ0 ζ)
dσ, λ ∈ D, 1 ≤ µ ≤ k ∗ .
1 − λζ
T
We have B(λ0 ) = λ0 Ik∗ , and the entries bµν (λ) of B(λ) are complex
conjugates of analytic functions of λ in the Nevanlinna class on D. This
implies that there exists a discrete subset Λ1 of D such that det B(λ) 6=
0 for all λ ∈ D \ Λ1 . Since the components xµ , 1 ≤ µ ≤ k ∗ , are also
complex conjugates of analytic functions of λ in the Nevanlinna class
on D, the claim follows immediately from the equality
(a1 (λ), . . . ak∗ (λ))t = B −1 (λ)(x1 (λ), . . . , xk∗ (λ))t ,
λ ∈ D \ Λ1 .
20
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
Denote by Ts h, m < s ≤ mn, the corresponding components of Tu h.
Each Ts is a 1 × m operator-matrix whose entries are finite sums of
products of Toeplitz operators of the form
Tsµ
(s)
X NY
=
j
Tvijµ ,
1 ≤ µ ≤ m.
i=1
Also, let Tes (z) be the 1 × m matrix-valued functions with entries
Tesµ (z) =
(s)
X NY
j
vijµ (z) ,
1 ≤ µ ≤ m.
i=1
i
i
If g, g1 , . . . , gk∗ are as above, write g = Imn
Tu f and gν = Imn
Tu fν with
∗
f, fν ∈ M, 1 ≤ ν ≤ k , and let
∗
fλ = f +
k
X
aν (λ)fν .
ν=1
Then (2.16) can be written as
1
i
i
Imn
Tu fλ = Imn
Tu F λ ,
1 − λζ
fλ (z)
with Fλ ∈ M. Note that for each z ∈ D we have 1−λz
, Fλ (z) ∈ R and
since the first dim R components of i form a coordinate basis of R we
conclude that
fλ (z)
= Fλ (z) , z ∈ D .
1 − λz
Thus for a.e. ξ ∈ T we have
µ
¶
fλ
(2.17)
0 = Tu1 · · · Tun
(ξ) ,
1 − λζ
and if iM > m then
(2.18)
1
(Ts fλ )(ξ) =
1 − λξ
µ
fλ
Ts
1 − λζ
¶
(ξ) ,
whenever s is an entry of i with s > m.
2
Multiply these equalities by 1−|λ|
, integrate on T with respect to σ
1−λξ
and let λ approach z ∈ T nontangentially. The limit of the left-hand
side of (2.17) is, of course, zero. From the definition of fλ and the fact
that the coefficients aν have nontangential limits a.e. on T, on the left
ZERO PRODUCTS OF TOEPLITZ OPERATORS
21
hand side of the equations contained in (2.18), in the limit we obtain
by Poisson’s formula
Z
k∗
X
1 − |λ|2
nt − lim
(Ts fλ )(ξ)dσ(ξ) = Ts f (z) +
aν (z)Ts fν (z)
λ→z T |1 − λξ|2
ν=1
a.e. on T. On the right-hand sides of (2.17) and (2.18) we use Lemma 2.3
and the above remark on the coefficients aν to obtain for a.e. z ∈ T
µ
¶
Z
fλ
1 − |λ|2
nt − lim
Tu1 · · · Tun
(ξ)dσ(ξ)
λ→z T 1 − λξ
1 − λζ
Ã
!
k∗
X
= (u1 · · · un )(z) f (z) +
aν (z)fν (z)
ν=1
in (2.17), and
µ
¶
Z
k∗
X
1 − |λ|2
fλ
nt−lim
Ts
(ξ)dσ(ξ) = Tes (z)f (z)+
aν (z)Tes (z)fν (z)
λ→z T 1 − λξ
1 − λζ
ν=1
in (2.18). Assume first that iM > m and let j = (1, . . . , m, il+1 , . . .),
where l is given by (2.15). Our argument applied to the equations
(2.17) and (2.18) yields
Ã
!
k∗
X
j
j
(2.19) W (z) Imn
Tu f (z) +
aν (z)Imn
Tu fν (z) = 0 , a.e. on T ,
ν=1
where



W (z) = 

u1 · · · un (z) 0 0 . . .
−Teil+1 (z) 1 0 . . .
..
.
...
e
−TiM (z) 0 0 . . .
0
0
..
.



.

1
Here the zeros in the first row are 1 × m zero matrices, while the rest
of zeros and ones are scalars.
Let
j
K = {Imn
Tu f : f ∈ M}
and note that maxλ∈D dimK (λ) = k. As was done in the text preceding
the statement of Theorem 2.1, define the vector space Kz ⊂ Cm of all
nontangential limits g(z), g ∈ K (whenever these limits exist). Then
(2.19) shows that dim W (z)Kz ≤ k ∗ .
On the other hand, from the special form of the matrix W (z) we see
j
Tu fα (z)} is a basis for ker(W (z)|Kz ) then {fα (z)} is linearly
that if {Imn
22
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
independent in ker((u1 · · · un )(z)|Mz ), hence
dim ker(W (z)|Kz ) ≤ dim ker((u1 · · · un )(z)|Mz ) .
By assumption, there exists a set of positive measure E ⊂ T such
that dim ker(u1 · · · un (z)|Mz ) ≤ γ on E. If we choose z ∈ E so that
dim Kz = k we obtain that
k ∗ ≥ dim W (z)Kz ≥ k − γ
and the result follows. The case when iM ≤ m is easier. The above
argument applies with j = (1, . . . , m) and W (z) = u1 · · · un (z). The
proof is now complete.
¤
3. Two Kernel Inclusion Theorems
We want to apply Theorem 1.1 to the nearly invariant subspaces considered in the previous section. This will lead to two kernel inclusion
theorems for products of finitely many Toeplitz operator-matrices.
Our first result is essentially a direct application of both Theorem 1.1
and Theorem 2.1. We make use of the following notation. Given
u1 , . . . , un ∈ L∞ (T, Mm×m (C)) we consider for j ≥ 2 the mn × mn
diagonal block-matrices
Uj = diag(Im , . . . , Im , uj , . . . , uj ) ,
where the block Im occurs n − j + 1 times. Note that the matrix-valued
function U1 which can, of course, be defined similarly, is not needed in
the sequel.
Theorem 3.1. Let u = (u1 , . . . , un ) ∈ (L∞ (T, Mm×m (C)))n , let R be
a fixed subspace of Cm such that
M = {f ∈ ker Tu1 · · · Tun : f (λ) ∈ R, λ ∈ D} 6= {0} ,
γ ≥ 0 an integer with
σ({z ∈ T : dim ker((u1 · · · un )(z)|Mz ) ≤ γ}) > 0 ,
and let r = dim R, m = maxλ∈D dimM (λ).
Then there exists a multi-index i with the properties stated in Theorem 2.1 and V ∈ L∞ (T, Mmn×mn (C)) with
rank(V ) ≥ mn − r + m − γ
a.e. on T
such that
Mnd ⊂ ker TV Imn
TU2 · · · TUn ,
i
n
where Md stands for the diagonal in Mn .
ZERO PRODUCTS OF TOEPLITZ OPERATORS
23
Proof. Let k = maxλ∈D dimTu M (λ) We will show that r + k − m ≤ mn
and construct the multi-index i = (i1 , . . . , ir+k−m ) as follows.
We start with λ0 ∈ D such that
dimM (λ0 ) = m ,
max
g1 ,...,gmn ∈M
rank(Tu gj (λ0 ))1≤j≤mn = k .
Let f1 , . . . fm ∈ M be such that f1 (λ0 ), . . . , fm (λ0 ) are linearly independent and let the indices 1 ≤ i1 < . . . < im ≤ m be chosen so
that det(fsil (λ0 ))1≤s,l≤m 6= 0. We can find f1 , . . . fm ∈ M as above and
fm+1 , . . . fmn such that
rank(Tu fj (λ0 ))1≤j≤mn = k .
This is because the sets of tuples (f1 , . . . , fm ) and (g1 , . . . , gmn ) as above
are easily seen to be open and dense in Mm and Mmn respectively.
If m < k, there exist j1 , . . . , jk−m which satisfy j1 < . . . < jk−m ≤ mn
and the corresponding rows of (Tu fj (λ0 ))1≤j≤mn are linearly independent. Obviously, j1 > m, which shows that mn − m > k − m, hence
r + k − m ≤ mn (whenever m < k). If m = k, the inequality is obvious.
Finally, note that the functionals of evaluation of the il -th coordinate,
1 ≤ l ≤ m, on R must be linearly independent because of the linear
independence of f1 (λ0 ), . . . fm (λ0 ). Then by a straightforward inductive
procedure, it is easy to see that there exist im+1 , . . . , ir such that im <
im+1 < . . . < ir ≤ m and {i1 , . . . , ir } is a coordinate basis for R. Now
define (i1 , . . . , ir+k−m )
½
(i1 , . . . , ir , j1 , . . . , jk−m ) , if m < k
i=
(i1 , . . . , ir ) ,
if m = k
which obviously also satisfies conditions (i) and (ii) of Theorem 2.1.
i
We now apply Theorem 2.1 to obtain that Imn
Tu M is a closed nearly
p mn
∗
invariant subspace of (H ) with index (k, k ), where k − γ ≤ k ∗ ≤
i
k. Suppress the zero entries of the vectors in the subspace Imn
Tu M
p k+r−m
. By
and identify it with a nearly invariant subspace N of (H )
Theorem 1.1 applied to N there exists
v ∈ L∞ (T, Mk∗ ×(k+r−m) (C)),
rank(v) = k ∗ ,
such that N ⊆ ker Tv . We then add mn − k ∗ zero rows to the matrix
v in arbitrary positions. If mn − k − r + m = 0, we have obtained
the desired matrix V . If mn − k − r + m > 0, add mn − k − r + m
zero columns so as to make it a square mn × mn matrix w. After
permuting some columns in w if necessary, we may assume that if the
l-th zero column in w is one of the newly added zero columns, then
i
is zero. Note that
the entry in the place (l, l) on the diagonal of Imn
24
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
mn−k ≥ mn−k−r+m. Finally, in these newly added zero columns we
place exactly one entry = 1 in each one of the newly added rows off the
diagonal. Note that this is possible because mn − k ∗ ≥ mn − k − r + m.
Let V the matrix obtained in this way.
Note that in either case rank (V ) = k ∗ +mn−k−r+m ≥ mn−r+m−γ.
i
i
Also, it is easy to verify that V Imn
= wImn
.
Finally, by construction we

f

Tun f

TV Imn
i
..

.

i
have Tw Imn
Tu M = {0} hence




Tu2 · · · Tun f
 
I 0 ... 0 0
I 0 ... 0
0
..
 0 I . . . 0 0   ...
.
.
.
.



= TV Imn
i
..  × 
 ...
...
.
0 0 . . . Tu3 0
0 0 . . . 0 Tu2
0 0 . . . 0 Tu3

  
I 0 ... 0 0
f
 0 Tun . . . 0 0   f 
··· × 
× .  ,
.. 
 ...
...
.   .. 
0
0
. . . 0 Tun


×

f
which is what we had to prove.
¤
We now turn to our second kernel inclusion theorem. Although it looks
rather technical, it is a crucial step towards proving our main theorem.
Theorem 3.2. Let v1 , . . . , vs+1 ∈ L∞ (T, Mm×m (C)) with
N = ker Tv1 · · · Tvs+1 6= {0}.
Suppose that v1 6= 0 a.e. on T and that for 1 ≤ l ≤ s + 1 there exists
a set of positive measure El ⊂ T such that vl |El is invertible. Also
assume that for 3 ≤ j ≤ s + 1
(3.20)
max dimTvj ···Tvs+1 N (λ) = m.
λ∈D
If γ0 < m is an integer with dim ker v1 (ξ) ≤ γ0 a.e. on T, then the
diagonal Nds! in N s! satisfies
(3.21)
Nds! ⊂ ker TV Is!m
TW1 · · · TWs ,
i
where V ∈ L∞ (T, Ms!m×s!m (C)) with
rank(V ) ≥ s!m − γ0 ,
ZERO PRODUCTS OF TOEPLITZ OPERATORS
25
a.e. on T, i is a multi-index with at least m entries in {1, . . . , (s−1)!m}
which form a coordinate basis for the diagonal in (C m )s! ,
Ws = diag(vs+1 , . . . , vs+1 ) ∈ L∞ (T, Ms!m×s!m (C)),
and W1 , . . . , Ws−1 are block diagonal matrix-valued functions of the
form
Wj (z) = diag(I(s−1)!m , Wj1 (z), . . . Wjs−1 (z))
with Wjl ∈ L∞ (T, M(s−1)!m×(s−1)!m (C)), 1 ≤ j, l ≤ s − 1.
Proof. We proceed by induction. For s = 2 note that trivially
ker Tv1 Tv2 ⊃ Tv3 N
and, by assumption, maxλ∈D dimTv3 N (λ) = m. Thus, we can apply
Theorem 3.1 to the product Tv1 Tv2 with R = Cm and m = r = m,
γ = γ0 (since v2 is invertible on a set of positive measure) and obtain
(Tv3 N )2d ⊂ ker TV I2m
i TU
with V , U and i as in Theorem 3.1. This clearly yields the statement
in the case s = 2.
Assume the result holds true for some integer s ≥ 2 and let v1 , . . . , vs+2
be as in the statement. Since
Tvs+2 N ⊂ ker Tv1 · · · Tvs+1
it follows easily that v1 , . . . , vs+1 satisfy (3.20), hence by the induction
hypothesis, (3.21) holds for v1 , . . . vs+1 with matrix-valued functions
V, W1 , . . . Ws and multi-index i. We want to apply Theorem 3.1 to the
product
TW1 · · · TWs
T = TV Is!m
i
of s + 1 Toeplitz operator-matrices on (H p )s!m , with R being the diagonal in (Cm )s! . Clearly, r = dim R = m. Moreover, if
M = { (f t , f t , . . . , f t )t ∈ ker T } ,
|
{z
}
s! times
then from the fact that
(Tvs+2 f t , . . . , Tvs+2 f t )t ∈ M
whenever f ∈ ker Tv1 · · · Tvs+2 , we deduce that
m = max dimM (λ) = m.
λ∈D
In order to find an integer γ as in Theorem 3.1 recall that vs+1 is defined
and invertible on a set of positive measure Es+1 ⊂ T. Then for every
z ∈ Es+1 , Ws (z) is invertible and if Mz is defined then Ws (z)Mz is
26
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
contained in (actually equal to) the diagonal in (Cm )s! . But by the
i
conditions on i and W1 , . . . Ws−1 , it follows that Is!m
W1 (z) · · · Ws−1 (z)
i
is injective on Mz , hence Is!m
W1 (z) · · · Ws (z)|Mz is injective. Thus by
hypothesis
i
dim ker V (z)Is!m
W1 (z) · · · Ws (z)|Mz ≤ γ0
a.e. on Es+1 and we can apply Theorem 3.1 with the parameters given
above and γ = γ0 . We obtain that
(s+1)!
(ker Tv1 · · · Tvs+1 )d
⊂ Ms+1
⊂ ker TV1 I i1
d
(s+1)!
TU2 . . . TUs+1 ,
where V1 , Uj and the multi-index i1 are as in the theorem. In particular,
they satisfy the requirements in the statement. This means that for all
f ∈ ker Tv1 · · · Tvs+2 we have
TV1 I i1
(s+1)!m
TU2 . . . TUs+1 (Tvs+2 f t , . . . , Tvs+2 f t )t = 0,
i.e. (3.21) holds for v1 , . . . vs+2 with V = V1 , Wν = Uν+1 , 1 ≤ ν ≤ s,
i = i1 , and the result follows.
¤
4. Main result
We can now prove the main result announced in the introduction. This
will be deduced from the following stronger theorem.
Theorem 4.1. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and assume that for
1 ≤ j ≤ n there exists a set of positive measure Ej ⊂ T such that uj |Ej
is invertible. Then
max dimTu1 ···Tun (H p )m (λ) = m .
λ∈D
Proof. Assume the contrary, i.e.
max dimTu1 ···Tun (H p )m (λ) < m.
λ∈D
Then there exists k ∈ {1, . . . , n} such that
max dimTul ···Tun (H p )m (λ) = m
λ∈D
if l > n − k + 1 and
γ0 = max dimTun−k+1 ···Tun (H p )m (λ) < m .
λ∈D
By Proposition 1.2 there exists u0 ∈ H ∞ (Mm×m (C)) with dim ker u0 (z) =
γ0 a.e. on T such that
Tu0 Tun−k+1 · · · Tun = 0 .
ZERO PRODUCTS OF TOEPLITZ OPERATORS
27
Then, if k > 1, we can apply Theorem 3.2 to the product Tu0 Tun−k+1 · · · Tun
to obtain that
[(H p )m ]k! ⊂ ker TV Ik!m
TW1 · · · TWk
i
with V, W1 , . . . , Wk and the multi-index i as in that statement. Thus
for f ∈ H p and λ ∈ D we have
¶
Z µ
1 − |λ|2 t
1
t t
TV Ik!m
TW1 · · · TWk
(f , . . . , f ) (ξ)
dσ(ξ) = 0 .
i
1 − λζ
1 − λξ
T
Let λ approach nontangentially to z ∈ T and apply Lemma 2.3 to
obtain for a.e. z ∈ T and every f ∈ (H p )m
i
V (z)Ik!m
W1 (z) · · · Wk (z)(f (z)t , . . . , f (z)t )t = 0.
Since dim ker V (z) ≤ γ0 a.e. this implies
i
rank(Ik!m
W1 (z) · · · Wk (z)) ≤ γ0 < m
a.e. on T. On the other hand, i has at least m entries in {1, . . . , k!m}
which form a coordinate basis for the diagonal in (Cm )k! and for 1 ≤
j ≤ k, Wj (z) has the form
Wj (z) = diag(Ik!m , Wj1 (z), . . . Wjn (z)),
so that the rank in question is at least m. This is a contradiction which
concludes the proof.
In the case k = 1, we can still apply Theorem 3.2 to Tu0 Tun TIm =
Tu0 Tun . The argument is completely analogous to the above so we
omit it.
¤
Corollary 4.1. Let u1 , . . . , un ∈ L∞ (T, Mm×m (C)) and assume that
for 1 ≤ j ≤ n there exists a set of positive measure Ej ⊂ T such that
uj |Ej is invertible. Then Tu1 · · · Tun cannot have finite rank.
Proof. Suppose the operator S = Tu1 Tu2 . . . Tun is of rank N . Then
there exist N non-zero Toeplitz operators: Tv1 ,. . . ,TvN such that
(4.22)
W
TvN . . . Tv1 S = 0 .
Indeed, if S((H p )m ) = {f1 , . . . , fN } with f1 ,. . . ,fN ∈ (H p )m , choose
v1 to be the diagonal m × m matrix whose diagonal entries are
( l
ζf1
, if f1l 6= 0
f1l
ull =
1,
otherwise
Then
Tv1 f1 = P (ζf1 ) = 0
28
ALEXANDRU ALEMAN AND DRAGAN VUKOTIĆ
and therefore
Tv1 S((H p )m ) ⊂
_
{Tv1 f2 , . . . , Tv1 fN } .
Hence dim Tv1 S((H p )m ) ≤ N − 1, so we can proceed inductively to
obtain (4.22). Now Theorem 4.1 implies that one of the operators
Tv1 ,. . . ,TvN , Tu1 , . . . , Tun is the zero operator. Since none of the operators Tv1 ,. . . ,TvN is zero by construction, it follows that some Tuν = 0,
which contradicts the assumptions of the statement. This ends the
proof.
¤
One may ask if analogous results hold for Toeplitz operators acting on
Bergman spaces. In the case n = 2, m = 1, p = 2 this has been proved
for harmonic symbols by Ahern and Čučković in [1]. The question
remains open for n ≥ 3; however, we have not been able to adapt our
methods to that context.
References
[1] P. Ahern and Ž. Čučković, A theorem of Brown-Halmos type for
Bergman space Toeplitz operators, J. Funct. Anal. 187 (2001), 200–
210.
[2] J. Barrı́a and P. Halmos, Asymptotic Toeplitz operators, Trans. Amer.
Math. Soc. 273 (1982), no. 2, 621–630.
[3] A. Brown and P. Halmos, Algebraic properties of Toeplitz operators, J.
reine angew. Math. 213 (1963), 89–102.
[4] J. Cima, A. Matheson, and W. T. Ross, The Cauchy Transform, Mathematical Surevys and Monongraphs 125, AMS, Providence, RI, 2006.
[5] R. G. Douglas, Banach algebra techniques in the theory of Toeplitz
operators, Expository Lectures from the CBMS Regional Conference
(Athens, GA 1972.) Conference Board of the Mathematical Sciences
Regional Conference Series in Mathematics, No. 15, American Mathematical Society, Providence, R.I., 1973.
[6] C. Gu, Products of several Toeplitz operators, J. Funct. Anal. 171
(2000), no. 2, 483–527.
[7] K. Y. Guo, A problem on products of Toeplitz operators, Proc. Amer.
Math. Soc. 124 (1996), no. 3, 869–871.
[8] D. Hitt, Invariant subspaces of H 2 of an annulus, Pacific J. Math. 134
(1988), no. 1, 101-120.
[9] P. Koosis, Introduction to Hp spaces, Second edition, Cambridge University Press, Cambridge 1998.
[10] D. Sarason, Nearly invariant subspaces of the backward shift, in: Contributions to operator theory and its applications, (Mesa, AZ, 1987),
481-493, Oper. Theory Adv. Appl. 35, Birkhäuser, Basel, 1988.
ZERO PRODUCTS OF TOEPLITZ OPERATORS
29
[11] E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality,
and Oscillatory Integrals, Princeton University Press, Princeton, NJ
1993.
Alexandru Aleman, Department of Mathematics, Lund University, P.O.
Box 118, S-221 00 Lund, Sweden
E-mail address: [email protected]
Dragan Vukotić, Departamento de Matemáticas & ICMAT, Módulo
C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain
E-mail address: [email protected]