Interated Aluthge Transforms: a Review

Let H be a separable, infinite dimensional, complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. An arbitrary operator T in L(H) has a unique polar decomposition T = U|T|, where |T| = (T ∗T) and U is the appropriate partial isometry (with ker U = ker T and ker U ∗ = ker T ∗ ). An operator T ∈ L(H) is said to be p-hyponormal if (T ∗T) p − (T T ∗ ) p ≥ 0, p ∈ (0, ∞) ([6]). If p = 1, T is hyponormal and if p = 1 2 , T is semi-hyponormal ([8]). The L˝owner-Heinz inequality ([7]) implies that p-hyponormal operators are qhyponormal operators for q ≤ p. In particular, T is said to be ∞-hyponormal if T is p-hyponormal for every p > 0 ([9]). It is well known that every qusinormal operators are ∞-hyponormal. Associated with T there is a very useful related operator Ter,t = |T| tU|T| r−t for r ≥ t ≥ 0, called the generalized Aluthge transform of T ([16]). This transform Ter,t is said to be (r, t)-Aluthge transform. Then (1, 1 2 )- Aluthge transform is refered as the Aluthge transform which is denoted by Te ([10]). The (1, 1)-Aluthge transform is refered as Duggal transform of T, which is denoted by Tb (i.e., Tb := |T|U) ([11]). In many cases Aluthge and Duggal transforms are useful, and ones concentrate to discuss here their transforms. An operator T is (r, t)-weakly hyponormal if |Ter,t| ≥ |T| ≥ |Te∗ r,t| ([12]). The (1, 1 2 )-weakly hyponormal operator is refered as w-hyponormal operator ([13], [14]). Given an r × r complex matrix T, if is the polar decomposition ofT, then the Aluthge transform is defined by Let denote the n-times iterated Aluthge transform of T, i.e. and , . In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the fact that the sequence converges for every r × r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003.

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 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 [2] that has a nontrivial invariant subspace if an only if does. On the other hand, Dykema and Schultz proved in [3] that the Brown measure is preserved by the Aluthge transform. Another reason is related with the iterated Aluthge transform. Let and for every . In [4] Jung, Ko and Peacy raised the following conjecture: Theorem 1 (Aluthge [1]). Let be p-hyponormal. Then

• 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 (Jung-Ko-Pearcy [5]). 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 [5]) was one of the most important.

We summarize two works ([16] and [17]) which contain a proof of a positive answer to Conjecture [18] 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

[1] A. Aluthge, On p-hyponormal operators for , Integral Equations Operator Theory 13 (1990), 307-315. [2] K. Dykema and H. Schultz, On Aluthge Transforms: continuity properties and Brown measure, preprint available in arXiv. [3] I. Jung, E. Ko, and C. Pearcy, Aluthge transform of operators, Integral Equations Operator Theory 37 (2000), 437-448. [4] I. Jung, E. Ko, and C. Pearcy, The Iterated Aluthge Transform of an operator, Integral Equations Operator Theory 45 (2003), 375-387. [5] D. Wang, Heinz and McIntosh inequalities, Aluthge Transformation and the spectral radius, Mathematical Inequalities and Applications Vol.6 No.1 (2003), 121-124.

51(1975), 243-257.

[7] E. Heinz, Beitr¨ae zur St¨orungstheorie der Spektralzerlegung, Math. Ann. 123(1951), 415-438 [8] D. Xia, Spectral theory of hyponormal operators, Birkh¨auuser Verlag, 1983. [9] S. Miyajima and I. Saito, ∞-hyponormal operators and their spectral properties, Acta Sci. Math. (Szeged), 67 (2001), 357-371. [10] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory 13(1990), 307-315. [11] C. Foias, I. Jung, E. Ko and C. Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math. 209(2003), 249-259. [12] I. Jung, On (r, t)-weakly hyponormal, in preparation. [13] A. Aluthge and D. Wang, w-hyponormal operators, Integral Equations Operator Theory, 36(2000), 1-10. [14] w-hyponormal operators, II, Integral Equations Operator Theory, 37(2000), 324-331 [15] http://www.scielo.org.ar/scielo.php? script=sci_ arttext & pid=S0041-69322008000100004 [16] J. Antezana, E. Pujals and D. Stojanoff, Convergence of iterated Aluthge transform sequence for diagonalizable matrices, Advances in Mathematics 216 (2007) 255-278. [17] J. Antezana, E. Pujals and D. Stojanoff, The iterated Aluthge transforms of a matrix converge, preprint available in arXiv. [18] A. Aluthge, On p-hyponormal operators for , Integral Equations Operator Theory 13 (1990), 307-315. [19] Jorge Antezana, Enrique R. Pujals and Demetrio Stojanoff,] Iterated Aluthge transforms: a brief survey.