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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008
Iterated Aluthge transforms: a brief survey
Jorge Antezana*, Enrique R. Pujals† and Demetrio Stojanoff*
* Partially supported by CONICET (PIP 4463/96), Universidad de La Plata (UNLP 11 X472) and ANPCYT (PICT03-09521).
† Partially supported by CNPq
Dedicated to the memory of Mischa Cotlar
Abstract. Given an r × r complex matrix T, if is the polar decomposition of T, then the Aluthge transform is defined by
![n Δ (T )](/img/revistas/ruma/v49n1/1a0410x.png)
![0 Δ (T ) = T](/img/revistas/ruma/v49n1/1a0412x.png)
![n n- 1 Δ (T ) = Δ (Δ (T ))](/img/revistas/ruma/v49n1/1a0413x.png)
![n ∈ ℕ](/img/revistas/ruma/v49n1/1a0414x.png)
![{Δn (T )}n∈ℕ](/img/revistas/ruma/v49n1/1a0415x.png)
2000 Mathematics Subject Classification. Primary 37D10. Secondary 15A60.
Key words and phrases. Aluthge transform, stable manifold theorem, similarity orbit, polar decomposition.
Let be a Hilbert space and
a bounded operator defined on
whose (left) polar decomposition is
. The Aluthge transform of
is the operator defined by
![]() | (1) |
This transform was introduced in [1] by Aluthge, in order to study p-hyponormal and log-hyponormal operators. Roughly speaking, the idea behind the Aluthge transform is to convert an operator into another operator which shares with the first one some spectral properties but it is closer to being a normal operator.
The Aluthge transform has received much attention in recent years. One reason is its connection with the invariant subspace problem. Jung, Ko and Pearcy proved in [15] that has a nontrivial invariant subspace if an only if
does. On the other hand, Dykema and Schultz proved in [10] that the Brown measure is preserved by the Aluthge transform. Another reason is related with the iterated Aluthge transform. Let
and
for every
. In [16] Jung, Ko and Peacy raised the following conjecture:
Conjecture 1. The sequence of iterates converges, for every matrix
.
This paper intends to give a brief survey on different properties of the Aluthge transform, making special emphasis on those results related with Conjecture 1, which was originally stated for operators on Hilbert spaces, and remains open for finite factors.
We begin the article with a historical summary that helps to explain the connection of the Aluthge transform with the invariant subspace problem and to describe some results that motivated and suggested that the conjecture might be true for operators on Hilbert spaces. Nevertheless, some couterexamples were found in this setting. We will expose one of them with some detail, which is particularly interesting because it shows an operator such that the sequence
does not converge even in the weak operator topology.
In the second part of the article, we summarize two works ([6] and [8]) which contain a proof of a positive answer to Conjecture 1 and some results on the regularity of the limit function. In these papers a new approach, based on techniques from dynamical systems, is introduced. The most important result used is the so-called stable manifold theorem for pseudo-hyperbolic systems (briefly described in Appendix A). Using this dynamical approach Conjecture 1 is firstly proved in [6], for every diagonalizable matrix. In the second article [8], using again a dynamical approach, combined this time with geometrical arguments in order to manage some technical difficulties, Conjecture 1 is completely solved. Although we shall not describe it, we also refer to the reader to the work by Huajun Huang and Tin-Yau Tam [13], where some related results are shown using different techniques
We also include a section with some open problems regarding the continuity of the limit function and the convergence for some particular operators acting on infinite dimensional space. Finally, we add two appendices where we give the precise statements of the stable manifold theorem, and describe the geometrical properties of similarity and unitary orbits of matrices.
Notation. Throughout this paper denotes the algebra of complex
matrices,
the group of all invertible elements of
, and
the group of unitary operators. We denote
is normal
. If
, we denote by
the diagonal matrix with
in its diagonal.
Given ,
denotes the spectrum of
,
the vector of eigenvalues of
(counted with multiplicity), and
the spectral radius of
. We shall consider the space of matrices
as a real Hilbert space with the inner product defined by
. The norm induced by this inner product is the Frobenius norm, that is denoted by
. For
and
, by means of
we denote the distance between them, with respect to the Frobenius norm. If
is a Hilbert space,
denotes the algebra of bounded operators on
.
One of the most challenging problems in operator theory is the invariant subspace problem (ISP from now on). This problem states that every operator in has a non trivial invariant subspace. It is a property that has every operator in a (complex) finite dimensional space because of the existence of eigenvectors. It is not difficult to see that the ISP also has a positive answer if the underlying Hilbert space is not separable. However, for separable Hilbert spaces, the problem is still open.
Although the ISP is very difficult to deal with, it was proved that some particular classes of operators have many non-trivial invariant subspaces. One of the most important is the class of normal operators. Indeed, using the functional calculus developed by von Neumann, it can be proved that a normal operator has as many invariant subspaces. This suggested the idea of isolating some properties of normal operators that could be related with the fact of having invariant subspaces. This motivated the definition of hyponormal and p-hyponormal operators. Recall that, given a Hilbert space and
, and operator
is called
-hyponormal (
-hn) if
![p = 1](/img/revistas/ruma/v49n1/1a0467x.png)
![* ∥Tx ∥ ≥ ∥T x ∥](/img/revistas/ruma/v49n1/1a0468x.png)
![x ∈ H](/img/revistas/ruma/v49n1/1a0469x.png)
![T](/img/revistas/ruma/v49n1/1a0470x.png)
![hn](/img/revistas/ruma/v49n1/1a0471x.png)
In 1987, Brown was able to prove that every hyponormal operator whose spectrum has non-empty interior has a non trivial invariant subspace. In 1990, Aluthge considered the possibility of extending this result to p-hyponormal operator and defined what is now called Aluthge transform. The first result that caught the attention on this transformation is summarized in the following statement:
Theorem 2.1 (Aluthge [1]). Let be p-hyponormal. Then
- If
, then
is hn,
- If
, then
is
-hn,
- It holds that
is hn.
Later on, Jung, Ko and Pearcy proved the next result that allowed to extend Brown's result to p-hyponormal operators:
Theorem 2.2 (Jung-Ko-Pearcy [15]). If denotes the lattice of invariant subspaces of a given operator
, then
.
This result led to the first version of Jung-Ko-Pearcy conjecture on the iterated Aluthge transform sequence: The sequence of iterates converges to a normal operator for every
. As soon as they raised this conjecture, many results supporting this conjecture appeared. The following formula for the spectral radius due to Yamazaki (see also Wang [19]) was one of the most important:
Theorem 2.3 (Yamazaki [21]). Given , then
.
However, after several positive partial results, some counterexamples appeared. One of the most interesting was found by Yanahida's [20]. Using a smart selection of weights, Yanahida defines a weighted shift operator whose sequence of iterated Aluthge transforms does not converge, even with respect to the weak operator topology! Let us briefly describe it: let be the canonical basis of
, and
the weighted shift operator defined by
where
Straightforward computations show that is also a weighted shift with weights:
,
. Then, using some tricky estimates, it can be proved that the sequence
does not converge, which implies that the sequence of iterates does not converge in the weak operator topology. After these counterexamples, the conjecture was restricted to matrices and takes the form stated in the introduction. Although the ISP has no sense for matrices, several authors have kept working on the conjecture in this setting because of the following reasons:
- Despite the positive computational evidence, it was surprisingly difficult. For example, very complicated computations were needed to prove the
case (see [3]).
- It would be considered as a first step in order to get a characterization of the operators
for which the sequence
converges (see [14] and the references therein).
- The conjecture remains open in the context of finite von Neumann factors (i.e. II
factors), where the ISP has growing interest (see [10]).
We conclude this section with some results that have been very useful in order to study Conjecture 1. The first result is on the limit points of the iterated sequence. Note that, by Theorem 2.3, the sequence is bounded. Hence, if we restrict our attention to matrices, it has limit points. The following result, independently proved by Ando [2] and Jung, Ko and Pearcy [16], gives more details on them:
Proposition 2.4 (Ando, Jung-Ko-Pearcy). If , the limit points of the sequence
are normal. Moreover, if
is a limit point, then
with the same algebraic multiplicity.
Then, studying the Jordan structure of with respect with the Jordan structure of
, the next reduction of the conjecture was proved in [5]:
Proposition 2.5. If the Aluthge transform sequence converges for every invertible matrix (resp. invertible diagonalizable matrix), then it does for every matrix (resp. diagonalizable matrix).
These two results motivate us to consider the dynamical approach that will be described in the next section. In the next proposition we summarize some easy properties of the Aluthge transform which are necessary to understand this approach.
if and only if
is normal.
for every
.
for every
.
. In [5] it is proved that equality holds only if
.
and
have the same spectrum and characteristic polynomial.
- If
then
(orthogonal decompositions).
We will also use systematically the following result on the regularity properties of on
(see [10] or [6]):
Theorem 2.7. The map is continuous in
and it is of class
in
.
Remark 2.8. The map fails to be differentiable at several non invertible matrices.
Throughout this section, we fixe a matrix . We denote by
, the vector of eigenvalues of
(counted with multiplicity). In the following subsections (3.1 and 3.2) we shall describe briefly the proof of Conjecture 1, following the articles [6], for the diagonalizable case, and [8], for the general case. Let
denote the set of diagonalizable matrices of
.
3.1. The diagonalizable case. As we mentioned in the previous section, Conjecture 1 can be reduced to the invertible case. Since , it holds that
So,
![T](/img/revistas/ruma/v49n1/1a04145x.png)
![S (T )](/img/revistas/ruma/v49n1/1a04146x.png)
![M (ℂ) r](/img/revistas/ruma/v49n1/1a04147x.png)
If the Aluthge transform is studied restricted to the similarity orbit, the diagonalizable case has some advantages. Note that, if , then
contains a compact submanifold of fixed points, and the sequence
goes to this submanifold as
. In fact, since
, then
for some diagonal matrix
which has the same characteristic polynomial as
. The unitary orbit
of
, is a compact submanifold of
that consists of all normal matrices in
. By Proposition 2.4,
is fixed by the Aluthge transform and every limit points of the sequence
belongs to
. In contrast, if
, then
does not have fixed points, and the sequence of iterated Aluthge transforms still goes to
, which is not contained in
, but in its boundary. The key result in order to perform the dynamical approach to this problem is the following:
Theorem 3.1. Let be an invertible diagonal matrix. For every
, there exists a subspace
of the tangent space
such that
;
- Both,
and
, are
-invariant;
and
, where
- If
satisfies
, then
.
In particular, the map is smooth. This fact can be formulated in terms of the projections
onto
parallel to
,
.
The basic idea of the proof is that has an easy description in terms of coordinates (see Eq (4) in Appendix B). By a sequence of steps, one can describe
, for
, as a Hadamard multiplication
, for a matrix
. These facts allow to find the subspace
as well as bounds for
. The general case (
) follows by unitary conjugations.
The idea behind Theorem 3.1 is the following: when we iterate the derivative of the Aluthge transform on an element of the tangent space of , for some
, the sequence of iterates converge exponentially to
. This is the behavior that one expects the Aluthge transform (instead of its derivative) to have. In order to extrapolate this result to the non-linear setting, we used the stable manifold theorem, which is a well known result of dynamical systems introduced independently by Hadamard and Perron (see Appendix). Under the conditions which Theorem 3.1 assures for the Aluthge transform, this theorem states that there exists a local submanifold
through each
such that:
, in particular
is transversal to
.
- The submanifold
is characterized as the set of matrices near
that converge exponentially to
by the iteration of the Aluthge transform.
Since the problem of the convergence of the iterates of the Aluthge transform has a symmetry due to the invariance by unitary conjugation, the size of the submanifolds as well as the exponential rate of convergence is uniform along the unitary orbit
. These fact allow to prove, using arguments that involve the inverse mapping theorem, that the union of the submanifolds
form an open neighborhood of
(see Fig. 1). Thus, as the sequence
goes toward
, for some
large enough the sequence of iterated Aluthge transforms enters into this open neighborhood, and, from that
on, the sequence converges exponentially. Moreover, standard computations also show that the functional sequence
converges uniformly on
to a limit function, denoted by
, which is a strong (continuous) retraction from
onto
.
Figure 1: Union of stable manifolds
From the above mentioned facts, we can only deduce that is continuous. However, better regularity properties can be proved. Let
be an open neighborhood of
contained in the union of the submanifolds
. As
by the uniqueness of the limit, we can define a projection
by:
It is not difficult to prove that this map is of class . Moreover, the limit function
can be locally written as the composition of
with
. Since both functions are
on
,
is also of class
on
.
Similar arguments, which involve a more specific version of the stable manifold theorem, allow to prove that the limit function is also
when it is restricted to the open dense set consisting of those matrices have all their eigenvalues different.
3.2. The nondiagonalizable case. The non-diagonalizable case is different, since the geometry context of the problem is more complicated. Let be non diagonalizable and
such that
. Then
is contained in the boundary of
, which also contains the orbits of matrices with smaller Jordan forms than the Jordan form of
. The boundary of
can be thought as a sort of lattice of boundaries. Therefore, in order to prove Conjecture 1, the problem is set in a more appropriate context so that both cases, diagonalizable and non-diagonalizable can be analyzed together. In this new approach, the Aluthge transform is thought as an endomorphism of the open set
, and all the orbits mentioned before are considered, not as a manifold, but as the basin of attraction
of
. By definition, in this case, the basin of attraction consists of those matrices
such that the sequence
goes to
as
. Note that, by Proposition 2.4, the basin
can also be characterized as the set of those matrices that have the same characteristic polynomial as
.
Since the Aluthge transform is thought as an endomorphism on the open set , the descomposition of Theorem 3.1 has to be extended to a decomposition of
in
-invariant subspaces, for each
. This extension follows using that,
![]() | (2) |
This fact can be proved by the standard properties of , since
can be characterized as the commutant of
. Hence, just take the decomposition
. The stable manifold theorem used in this context is a standard extension to
(see Remark A.4), and no differential structure is required in the basin. This theorem implies the existence of
-invariant manifolds
through each
in the basin close enough to
. One of the most important properties of these manifolds is that
![]() | (3) |
where and
are constants that only depend on the distance among different eigenvalues of
. Hence, if the sequence
converges for some
, then the same must happen for
(with the same limit).
In the diagonalizable case, we have considered only the stable manifolds for points
and it was proved that the union of these manifolds contains an open neighborhood of
in
.
That approach fails in the general case, because the basin in not a manifold. So, a different idea is used. Inside the basin, there is a distinguished subset of matrices which satisfy Conjecture 1. This set, denoted by consists of those matrices, in the basin, with orthogonal spectral projections. Indeed, given
, let
be the spectrum of
,
the spectral projection of
associated to each
, and
. Observe that
is uniquely determined (as a normal matrix) by its vector
and the spectral projections
. But Proposition 2.4 assures that all the limit points of the sequence
must be normal matrices with this vector, and item 6 of Proposition 2.6 assures that they must have the same spectral projections as
(because they are orthogonal). Hence
is the unique possible limit point, so that
.
Having identified this set in the basin, the strategy is to prove that, for every in the basin near
, the stable manifolds
intersect the set
. This fact would be enough by the remark which follows Eq. (3), and the previous study about
.
However, does not have a differential structure, which is an important technical obstacle. To avoid this problem, the stable manifolds
as well as
are projected onto the orbit
, using the above mentioned function
, which is smooth (see Kato's book [17]). Observe that
if and only if
. On the other hand, the derivative of
at
is an orthogonal projection with range equal to
. By a continuity argument, this implies that, for every
close enough to
, the nullspace of the derivative of
at the different points of
is transversal to the corresponding tangent spaces of
. This implies that the projection onto
of the manifolds
are submanifolds of
. Moreover, it can be proved that
is "close" in some sense to
, where
is certain normal operator close to
. Observe that
is one of the stable manifolds studied in the diagonalizable case. Therefore
intersects transversally
. These facts imply, by some well known results about transversal intersections (see [11]), that
also intersects
. Finnaly, if
, then
is the projection of a matrix
(see Fig. 2).
Figure 2: The projection argument
Similar (but slightly more complicated) arguments show that the limit map is continuous on
. Nevertheless, the previous techniques are not useful to study continuity outside of
because the map
fails to be differentiable there.
Rate of convergence. In [6] it was proved that, if is diagonalizable, then after some iterations the rate of convergence of the sequence
becomes exponential. More precisely, for some
and every
, there exist
and
such that
. This exponential rate depends on the spectrum of
. Actually, if
for some diagonal matrix
, then
, the constant which appears in Theorem 3.1. Using the formula for
, one can see that it is closer to
(so that the rate of convergence becomes slower) if the different eigenvalues are closer one to each other.
These facts are no longer true if is not diagonalizable, since the rate of convergence for such a
depends on the rate of convergence for some
, which can be much slower (and not exponential). Observe that the proof of the convergence of the sequence
, does not study the rate of convergence. It only shows that there exists an unique possible limit point for the sequence.
Nevertheless, if one denotes by the system of spectral projections of a matrix
associated to its different eigenvalues, the previous approach shows that
converges to
exponentially, because
. As in the case of diagonalizable matrices the rate of convergence of the spectral projections depends on the spectrum of
, which agree with the spectrum of
. Note that the spectrum of
and the spectral projections of
completely characterize the limit
. Indeed, if
, then
-Aluthge transform. Given
and a matrix
whose polar decomposition is
, the
-Aluthge transform of
is defined by
![λ](/img/revistas/ruma/v49n1/1a04370x.png)
![λ ∈ (0,1)](/img/revistas/ruma/v49n1/1a04371x.png)
![λ](/img/revistas/ruma/v49n1/1a04372x.png)
![k D](/img/revistas/ruma/v49n1/1a04373x.png)
![λ](/img/revistas/ruma/v49n1/1a04374x.png)
![λ ∈ (0,1)](/img/revistas/ruma/v49n1/1a04375x.png)
![(0,1)](/img/revistas/ruma/v49n1/1a04376x.png)
Another result which depends particularly on the Aluthge transform is Eq. (2), and the extended decomposition . Nevertheless, it is easy to see that both results are still true for every
. On the other hand, the proof of the continuity of the map
on
uses the same facts about the Aluthge transform. So that, it also remains true for
, for every
. We resume all these remarks in the following statement:
Theorem 3.2. For every and
, the sequence
converges to a normal matrix
. The map
is continuous on
.
3.3. Some open problems. Concerning the convergence of iterated Aluthge sequences, the following problems are of great interest, and they still remain unsolved:
- The continuity of the map
. Using the techniques mentioned in this survey, it can be proved that this map is continuous in
. But, as
is not globally
outside
, new methods should be developed in order to prove the continuity of
in
. We remark that this fact is supported by computational evidence.
- If
is a separable Hilbert space, to get a characterization of those
such that the sequence
converges. The first step might be to study compact operators, using the convergence for matrices.
- To prove that, if
is a II
factor (i.e. an infinite dimensional finite von Neumann algebra with trivial center), then the sequence
converges to a normal element of
, for every
. This fact might be very important in order to get an affirmative answer of the ISP for these algebras, a problem which has great interest in operator theory and remains open. As in the case of compact operators, the finite dimensional case could be useful to prove Conjecture 1 in this setting, because there exist good finite dimensional methods of approximation for these particular class of von Neumann algebras.
Appendix A. The stable manifold theorem
As a general reference of this theory, we refer to the books [12] and [18]. Let be a smooth Riemann manifold and
a submanifold (not necessarily compact). Throughout this subsection
denotes the tangent bundle of
restricted to
.
Definition A.1. A pre-lamination indexed by
is a continuous choice of a
embedded disc
through each
. Continuity means that
is covered by open sets
in which
is given by
where
is a continuous section. Note that
is a
fiber bundle over
whose projection is
. Thus
. If the sections mentioned above are
,
, we say that the
pre-lamination is of class
. A pre-lamination is called self coherent if the interiors of each pair of its discs meet in a relatively open subset of each one.
Theorem A.2 (Stable manifold theorem). Let be a
endomorphism of
and let
be a compact subset of
consisting of fixed points of
. Assume that there exist two continuous subbundles of
, denoted by
and
, such that, for every
,
.
is
-invariant.
- There exists
such that
.
Then, there is a continuous, -invariant and self coherent
-pre-lamination
(endowed with the
-topology) such that, for every
,
,
is tangent to
,
.
Remark A.3. As is compact and
, using the so-called
-prelamination theorem, it can be proved that the prelamination
is of class
.
Remark A.4. Under the hypothesis of Theorem A.2, recall that the basin of attraction of is the set
. Also recall that, for every
, a local basin of
is the set
![s E x](/img/revistas/ruma/v49n1/1a04466x.png)
![Fx](/img/revistas/ruma/v49n1/1a04467x.png)
![Bf (N )ɛ](/img/revistas/ruma/v49n1/1a04468x.png)
![ɛ > 0](/img/revistas/ruma/v49n1/1a04469x.png)
![^Esx](/img/revistas/ruma/v49n1/1a04470x.png)
![^Fx](/img/revistas/ruma/v49n1/1a04471x.png)
.
is
-invariant in the sense that
.
- There exists
such that
restricted to
expand it by a factor greater than
, and
has norm lower than
.
In this case, an extended version of the stable manifold theorem assures that there is a -pre-lamination
which is continuous,
-invariant, self coherent and satisfies for every
(1) and (2) of Theorem A.2 and the following modified version of (3):
.
Appendix B. Similarity orbit of a diagonal matrix
Let be diagonal, with
,
. By means of
we denote the similarity orbit of
, i.e.,
. On the other hand,
denotes the unitary orbit of
. We denote by
the
map defined by
. With the same name we note its restriction to the unitary group:
.
Proposition B.1 (See [9] or [4]). The similarity orbit is a
submanifold of
, and the projection
becomes a submersion. Moreover,
is a compact submanifold of
, which consists of the normal elements of
, and
is a submersion.
As a consequence of this result, it is not difficult to see that:
TD ![]() ![]() | (4) |
Straightforward computations also show that, provided that
. We consider on
(and on
) the Riemannian structure inherited from
(using the usual inner product on their tangent spaces). For
, we denote by
the Riemannian distance between
and
(in
). Observe that, for every
, one has that
and the map
is isometric, on
, with respect to the Riemannian metric as well as with respect to the
metric of
.
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Jorge Antezana
Departamento de Matemática,
Facultad de Ciencias Exactas,
Universidad Nacional de La Plata,
Calle 50 y 115, La Plata, Argentina
and IAM-CONICET
antezana@mate.unlp.edu.ar
Enrique R. Pujals
Instituto Nacional de Matemática Pura y Aplicada (IMPA),
Rio de Janeiro, Brasil
enrique@impa.br
Demetrio Stojanoff
Departamento de Matemática,
Facultad de Ciencias Exactas,
Universidad Nacional de La Plata,
Calle 50 y 115, La Plata, Argentina
and IAM-CONICET
demetrio@mate.unlp.edu.ar
Recibido: 10 de abril de 2008
Aceptado: 22 de abril de 2008