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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 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)
If H is a Hilbert space, S ⊆ H is a closed subspace of H, and A is a positive bounded linear operator on H, the spectral shorted operator ρ(S, A) is defined as the infimum of the sequence Σ(S, A n) 1/n , where Σ(S, B) denotes the shorted operator of B to S. We characterize the left spectral resolution of ρ(S, A) and show several properties of this operator,(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)