An Assessment of Aluthge and Duggal Transformations: a Brief Survey

Letbe a separable Hilbert space withandbe the C*- algebra of all bounded linear operators on. Let. Let be the unique polar decomposition of T, where U is a partial isornetry such that and. Obviouslyis a positive operator. Alsofor all, and ifthen E is the initial projection of U (ie.,where ) and E is the support of T as well as the support of. The following definition is due to Aluthge. Definition 3.1 (Aluthge transformation). Ifandis the polar decomposition of T, then is called the Aluthge transformation of T. Definition 3.2 ( Duggal transformation). Ifandis the polar decomposition of T, then is called the Duggal transformation of T. Definition 3.3 (—Aluthge transformation). If,the polar decomposition of T, and, thenis called the —Aluthge transformation of T. When, the—Aluthge transformation is the Aluthge transformation. The notion of Aluthge transformation was first studied in relation with the p—hyponormal and log — hyponormal operators. Roughly speaking, the Aluthge transformation of an operator is closer to being normal. Aluthge transformation has received much attention in recent years. One reason is the connection of Aluthge transformation with the invariant subspace problem. Jung, Ko and Pearcy proved in that T has a nontrivial invariant subspace if iterated Aluthge transformation. Foias, Jung, Ko and Pearcy introduced the the concept of Dug- gal transformations, and proved several analogous results for Aluthge transformations and Duggal transformations. Yamazaki proved that for every , the sequence of the norms of the Aluthge iterates of T converges to the spectral radius. Denning Wang gave another proof of this result. We started studying Aluthge and Duggal transformations hoping to prove, the analogue of the result of Yamazaki, that the sequence of the norms of the Duggal iterates of T converges to the spectral radius, for every Several finite dimensional examples suggested that the result is true for Duggal transformations. We succeeded in proving that the sequence of the norms of the Duggal iterates converges to the spectral radius, for certain classes of operators. Further investigation led to an example of a finite dimensional operator showing that there exist operators such that the sequence of the norms of the Duggal iterates does not converge to the spectral radius.

Letbe a Hilbert space and T a bounded operator defined onwhose (left) polar decomposition is. The Aluthge transform of T is the operator defined by

(3.3)

This transform was introduced in 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 that T has a nontrivial invariant subspace if an only ifdoes. On the other hand, Dykema and Schultz proved that the Brown measure is preserved by the Aluthge transform. Another reason is related with the iterated Aluthge transform. Letand) for every Jung, Ko and Peacv raised the following conjecture: Conjecture 1. The sequence of iterates converges, for every matrix T. A 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 operatorsuch that the sequence does not converge even in the weak operator topology. 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 3.2 (Aluthge). Letbe p-hvponormal. Then

• If, thenis hn,

• If, thenis-hn,

• It holds thatis hn. □

Later on, Jung, Ko and Pearcy proved the next result that allowed to extend Brown's result to p-hyponormal operators: Theorem 3.3 (Jung-Ko-Pearcv). If Lat(T) denotes the lattice of invariant subspaces of a given operator, then. □ This result led to the first version of Jung-Ko-Pearcv conjecture on the iterated Aluthge transform sequence: The sequence of iteratesconverges 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 was one of the most important:

. However, after several positive partial results, some counterexamples appeared. One of the most interesting was found by Yanahida's. 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: letbe the canonical basis of, and the weighted shift operator defined by where Straightforward computations show thatis also a weighted shift with weights: ,. Then, using some tricky estimates, it can be proved that the sequencedoes not converge, which implies that the sequence of iterates does not converge in the weak operator topology.

Letbe a complex Hilbert space, and letbe the algebra of bounded linear operators on. Given, consider its (left) polar decomposition. In order to study the relationship among p-hyponormal operators, Aluthge introduced the transformation defined by . Later on, this transformation, now called Aluthge transform, was also studied in other contexts by several authors, such as Jung, Ko and Pearcy, Ando, Ando and Yamazaki (3|, Yamazaki, Okubo pEj and Wu pRJf among others. In this study, givenand, we study the so-called-Aluthge transform of T defined by transform of T, i.e. and

. (1.1)

In a previous study, we show that the iterates of usual Aluthge transformconverge to a normal matrixfor every diagonalizable matrix(of any size). We also proved the smoothness of the mapwhen it is restricted to a similarity orbit, or to the (open and dense) setof invertible rxr matrices with r different eigenvalues. The key idea was to use a dynamical systems approach to the Aluthge transform, thought as acting on the similarity orbit of a diagonal invertible matrix. Recently, Huajun Huang and Tin-Yau Tam showed, with other approach, that the iterates of every-Aluthge transform converge, for every matrixwith all its eigenvalues of different, moduli.

We shall prove thatfor operators T belonging to certain classes of operators in C(H). By the inequality,for all. Moreover, and hencefor all Thusis a decreasing sequence which is bounded below by The following lemma is an easy consequence. Lemma 3.2. There is anfor which Remark 3.1. We notice one more analogy between Aluthge and Duggal transformations. The sequenceis decreasing such that, sequenceis decreasing such that, andfor all n. Theorem 3.6 (Mc Intosh inequality ). For bounded linear operators A,B and x, Theorem 3.7 (Heinz inequality ). For positive linear operators A and B, and bounded linear operator X. for all Using these inequalities we prove the following results. Lemma 3.3. For any positive integer k, for all. Consequently, the decreasing sequenceis convergent. Proof. Leta neighborhood of, and note that We have. Lemma 3.4. Ifis the polar decomposition of, then for any positive integer k, Proof. We haveand therefore Hence by theorem 3.6,

Some of the problems that were thought about and where further research is possible. It is proved in this paper that if T is a bounded linear operator on a r(T'), if T satisfies certain conditions. In general, unlike Aluthge transformations, the sequence need not converge to r(T). One can attempt to characterize operators T on a Hilbert space, having the property that the sequence of the norms of Duggal iterates of T converges to the spectral radius r(T).

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• Ciprian Foias, Il Bong Jung, Eungil Ko, and Carl Pearcy. Complete contractivity of maps associated with the Aluthge and Duggal transforms. Pacific J. Math., 209(2):249–259, 2003.

• Il Bong Jung, Eungil Ko, and Carl Pearcy. Aluthge transforms of operators. Integral Equations Operator Theory, 37:437–448, 2000.

• Jung, E. Ko, and C. Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory 37(2000), 437-448.

 M. Cho, I. B. Jung, and W. Y. Lee. On Aluthge transform of p hyponormal operators. Integral Equations Operator Theory, 53:321–329, 2005.