arXiv subjects

Jorge Antezana

Publications and source records attributed to Jorge Antezana.

28 records · Page 2Linked to original sources

arXiv2011

Short paths for symmetric norms in the unitary group

For a given symmetrically normed ideal I on an infinite dimensional Hilbert space H, we study the rectifiable distance in the classical Banach-Lie unitary group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that one-parameter subgroups of U_I are short paths, provided the spectrum of the exponent is bounded by $π$, and that any two elements of U_I can be joined with a short path, thus obtaining a Hopf-Rinow theorem in this infinite dimensional setting, for a wide and relevant class of (non necessarily smooth) metrics. Then we prove that the one-parameter groups are the unique short paths joining given endpoints, provided the symmetric norm considered is strictly convex.

math.MG↗

arXiv2011

Optimal paths for symmetric actions in the unitary group

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval $[a,b]\subset\mathbb R$, we study the action defined in the Lie group of $n\times n$ unitary matrices $\mathcal{U}(n)$ by $$ S(α)=\int_a^b L(\dotα(t))\,dt\,, $$ where $α:[a,b]\to\mathcal{U}(n)$ is a rectifiable curve. We prove that the one-parameter subgroups of $\mathcal{U}(n)$ are the optimal paths, provided the spectrum of the exponent is bounded by $π$. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in $\mathcal{U}(n)$ as well as angular metrics in the Grassmann manifold

math.DG↗

arXiv2011

Thompson-type formulae

Let X and Y be two nxn Hermitian matrices. In the article "Proof of a conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986, 187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and V such that $$ e^{i X}e^{i Y}=e^{i (UXU^*+VBV^*)}. $$ In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II$_1$ factor.

math.FA↗

arXiv2007

The iterated Aluthge transforms of a matrix converge

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ Δ(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $Δ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $Δ^{0}(T)=T$ and $Δ^{n}(T)=Δ(Δ^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ matrix $T$. This result was conjecturated by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.

math.FA↗

arXiv2007

Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $λ$-Aluthge transform

Let $λ\in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ Δ_λ(T )= |T|^λ U |T |^{1-λ}. $$ Let $Δ_λ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ_λ^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map $(\la, T) \mapsto \alulit{\infty}{T}$, and we study for which matrices the map $(0,1)\ni λ\mapsto Δ_λ^{\infty}(T)$ is constant.

math.FA↗

arXiv2005

Bilateral Shorted Operators and Parallel Sums

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 of parallel sum and substraction, in this Hilbertian context.

math.FA↗

arXiv2004

Jensen's Inequality and majorization

Let $\mathcal{A}$ be a $C^*$-algebra and $ϕ:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we obtain a Jensen's-type inequality $f(ϕ(a)) \leq ϕ(f(a))$ where $\le$ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered i.e., operator convex, monotone convex and arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.

math.FA↗

arXiv2004

Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.

math.FA↗

arXiv2004

Weighted projections and Riesz frames

Let $\mathcal{H}$ be a (separable) Hilbert space and $\{e_k\}_{k\geq 1}$ a fixed orthonormal basis of $\mathcal{H}$. 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 projections, and to obtain a new characterization of Riesz frames.

math.FA↗

arXiv2004

Spectral shorted operators

If $\mathcal H$ is a Hilbert space, $\mathcal S \subseteq \mathcal H$ is a closed subspace of $\mathcal H$, and $A $ is a positive bounded linear operator on $\mathcal H$, the spectral shorted operator $ρ(\mathcal S, A)$ is defined as the infimum of the sequence $Σ(\mathcal S, A^n)^{1/n}$, where $Σ(\mathcal S, B)$ denotes the shorted operator of $B$ to $\mathcal S$. We characterize the left spectral resolution of $ρ(\mathcal S, A)$ and show several properties of this operator, particularly in the case that $\dim \mathcal S = 1$. We use these results to generalize the concept of Kolmogorov complexity for the infinite dimesional case and for non invertible operators.

math.FA↗