On the Convergence of Aluthge Sequence: a Review

Given X ∈ Cn×n, the polar decomposition [9] asserts that X = UP, where U is unitary and P is positive semi definite, and the decomposition is unique if X is nonsingular[20]. Though the polar decomposition may not be unique, the Althuge transform [1] of X: ∆(X) := P 1/2UP1/2 (P 1/2XP −1/2 if X is nonsingular) is well defined [17, Lemma 2]. Aluthge transform has been studied extensively, for example, [1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 14, 16, 17]. Recently Yamazaki [16] established the following interesting result (1.1) lim m→∞ k∆m(X)k = r(X), where r(X) is the spectral radius of X and kXk := max kvk2=1 kXvk2 is the spectral norm of X. Suppose that the singular values s1(X), . . . , sn(X) and the eigenvalues λ1(X), . . . , λn(X) of X are arranged in nonincreasing order s1(X) ≥ s2(X) ≥ · · · ≥ sn(X), |λ1(X)| ≥ |λ2(X)| ≥ · · · ≥ |λn(X)|. Since kXk = s1(X) and r(X) := |λ1(X)|, the following result of Ando [3] is an extension of (1.1). Theorem 1.1. (Yamazaki-Ando) Let X ∈ Cn×n. Then (1.2) lim m→∞ si(∆m(X)) = |λi(X)|, i = 1, . . . , n[20].

Aluthge transform ∆(T) is also defined for Hilbert space bounded linear operator T [17] and (1.1) remains true [16]. Yamazaki’s result (1.1) provides support for the following conjecture of Jung et al [11, Conjecture 1.11] for any T ∈ B(H) where B(H) denotes the algebra of bounded linear operators on the Hilbert space H. Conjecture 1.2. Let T ∈ B(H). The Aluthge sequence {∆m(T)}m∈N is norm convergent to a quasinormal Q ∈ B(H), that is, k∆m(T) − Qk → 0 as m → ∞, where k · k is the spectral norm. It is known [11, Propositioin 1.10] that if the Aluthge sequence of T ∈ B(H) converges, its limit L is quasinormal, that is, L commutes with L ∗L, or equivalently, UP = P U where L = UP is a polar decomposition of L [9]. However very recently it is known [7] that Conjecture 1.2 is not true for infinite dimensional Hilbert space. Ch¯o, Jung and Lee [7, Corollary 3.3] constructed a unilateral weighted shift operator T : `2(N) → `2(N) such that the sequence {∆m(T)}m∈N does not converge in weak operator topology[20]. They also constructed [7, Example 3.5] a hyponormal bilateral weighted shift B : `2(Z) → `2(Z) such that {∆m(B)}m∈N converges in the strong operator topology, that is, for some L : `2(Z) → `2(Z), k∆m(B)x−Lxk → 0 as m → ∞ for all x ∈ `2(Z), where kxk is the norm induced by the inner product. However {∆m(B)}m∈N does not converge in the norm topology. So the study of Conjecture 1.2 is reduced to the finite dimensional case Cn×n. Since the three (weak, strong, norm) topologies coincide and quasinormal and normal coincide [9] in the finite dimensional case, the limit points of the Aluthge sequence are normal [13, Proposition 3.1], [3, Theorem 1]. Also see [11, Proposition 1.14]. Moreover the eigenvalues of ∆(X) and the eigenvalues of X are identical, counting multiplicities[20].

The iterates of usual Aluthge transform ∆n 1/2 (T) converge to a normal matrix ∆∞ 1/2 (T) for every diagonalizable matrix T ∈ Mr(C) (of any size). We also proved in [21] the smoothness of the map T 7→ ∆∞ 1/2 (T) when it is restricted to a similarity orbit, or to the (open and dense) set D∗ r (C) of invertible r×r 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 [22] showed, with other approach, that the iterates of every λ-Aluthge transform ∆n

Available online at www.ignited.in Page 2

λ (T) converge, for every matrix T ∈ Mr(C) with all its eigenvalues of different moduli.

In this paper, we study the general case of λ-Aluthge transforms by means of a dynamical systems approach. This allows us to generalize Huajun Huang and Tin-Yau Tam result for every diagonalizable matrix T ∈ Mr(C), as well as to show regularity results for the two parameter map (λ, T) 7→ ∆∞ λ (T) = limn∈N ∆n λ (T).

[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory, 13 (1990), 307–315. [2] A. Aluthge, Some generalized theorems on p-hyponormal operators, Integral Equations Operator Theory, 24 (1996), 497–501. [3] T. Ando, Aluthge transforms and the convex hull of the eigenvalues of a matrix, Linear Multilinear Algebra, 52 (2004), 281–292. [4] T. Ando and T. Yamazaki, The iterated Aluthge transforms of a 2-by-2 matrix converge, Linear Algebra Appl., 375 (2003), 299–309. [5] J. Antezana, P. Massey and D. Stojanoff, λ-Aluthge transforms and Schatten ideals, Linear Algebra Appl., 405 (2005) 177–199. [6] R. Bhatia and F. Kittaneh, Some inequalities for norms of commutators, SIAM J. Matrix Anal. Appl., 18 (1997) 258–263. [7] M. Ch¯o, I.B. Jung and W.Y. Lee, On Aluthge transform of p-hyponormal operators, Integral Equations Operator Theory, 53 (2005), 321–329. [8] C. Foia¸s, I.B. Jung, E. Ko and C. Pearcy, Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math., 209 (2003), 249–259. [9] P.R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, 1974. [10] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [11] I.B. Jung, E. Ko and C. Pearcy, Aluthge transforms of operators, Integral Equations Operator Theory, 37 (2000), 437–448. [12] I.B. Jung, E. Ko and C. Pearcy, Spectral pictures of Aluthge transforms of operators, Integral Equations Operator Theory, 40 (2001), 52–60. [13] I.B. Jung, E. Ko and C. Pearcy, The iterated Aluthge transform of an operator, Integral Equations Operator Theory, 45 (2003), 375–387. [14] K. Okubo, On weakly unitarily invariant norm and the Aluthge transformation, Linear Algebra Appl., 371 (2003), 369–375. [15] A.L. Onishchik and E.B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin, 1990. [16] T. Yamazaki, An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., 130 (2002), 1131–1137. [17] T. Yamazaki, On numerical range of the Aluthge transformation, Linear Algebra Appl., 341 (2002) 111–117. [18] http://www.auburn.edu/~tamtiny/lambda13.pdf [19] http://arxiv.org/pdf/0706.1234.pdf [20] UAJUN HUANG AND TIN-YAU TAM, On the convergence of aluthge sequence,available at: http://www.auburn.edu/~tamtiny/lambda13.pdf [21] J. Antezana, E. Pujals and D. Stojanoff, Convergence of iterated Aluthge transform sequence for diagonalizable matrices, Advances in Math., to appear. Los Alamos preprint version in www.arxiv.org/abs/math.FA/0604283. [22] Huajun Huang and Tin-Yau Tam, On the Convergence of the Aluthge sequence, Math Inequalities and Appl. to appear