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Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then, the Aluthge transform is defined by ∆ (T) = |T | 1/2 U |T | 1/2. Let ∆ n (T) denote the n-times iterated Aluthge transform of T , i.e. ∆ 0 (T) = T and ∆ n (T) = ∆(∆ n−1 (T)), n ∈ N. We prove that the sequence {∆ n (T)} n∈N converges for every r × r matrix T. This result(More)
Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then, the Aluthge transform is defined by ∆ (T) = |T | 1/2 U |T | 1/2. Let ∆ n (T) denote the n-times iterated Aluthge transform of T , i.e. ∆ 0 (T) = T and ∆ n (T) = ∆(∆ n−1 (T)), n ∈ N. We prove that the sequence {∆ n (T)} n∈N converges for every r × r diagonalizable matrix(More)
In this paper we study shorted operators relative to two different subspaces, for bounded operators on infinite dimensional Hilbert spaces. We define two notions of " complementability " in the sense of Ando for operators, and study the properties of the shorted operators when they can be defined. We use these facts in order to define and study the notions(More)
Let H be a (separable) Hilbert space and {e k } k≥1 a fixed orthonormal basis of H. Motivated by many papers on scaled projections, angles of subspaces and oblique projections, we define and study the notion of compatibility between a subspace and the abelian algebra of diagonal operators in the given basis. This is used to refine previous work on scaled(More)
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allow us to give a simple necessary and sufficient condition for uniqueness. We(More)
We prove the Lorentz-Shimogaki and Boyd theorems for the spaces Λ p u (w). As a consequence, we give the complete characterization of the strong boundedness of H on these spaces in terms of some geometric conditions on the weights u and w, whenever p > 1. For these values of p, we also give the complete solution of the weak-type boundedness of the(More)
Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a, b] ⊂ R, we study the action defined in the Lie group of n × n unitary matrices U(n) by S(α) = b a L(˙ α(t)) dt , where α : [a, b] → U(n) is a rectifiable curve. We prove that the one-parameter subgroups of U(n) are the optimal paths,(More)
Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then the Aluthge transform is defined by ∆ (T) = |T | 1/2 U |T | 1/2. Let ∆ n (T) denote the n-times iterated Aluthge transform of T , i.e. ∆ 0 (T) = T and ∆ n (T) = ∆(∆ n−1 (T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of the(More)