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Let $(G,+)$ be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism $\tau _g$ is ergodic. The main result of this paper is a new ergodic theorem for functions in $L^1(G,M)$ , where M is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit $\{\tau _g^n(h)\}_{n\in \mathbb {N}}$ . The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to $L^1$ functions. The extension is based on a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and does not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast to those results that need to use the tangent space or some chart to define the mollifier.

Introducción: Diversos estudios reportan que la depuración del lactato está asociado a la mortalidad en los pacientes críticos. Se describe la relación entre la depuración del lactato y la normalización del lactato con la mortalidad en los pacientes residentes de la gran altitud con trauma grave. Por lo cual es importante conocer la mortalidad de los pacientes con trauma grave en la altitud y la distribución lesional del trauma grave en la altitud. Material y Métodos: Estudio de cohorte, retrospectivo, realizado en una unidad de cuidados intensivos a 4,150 “msnm” en pacientes que ingresaron por trauma grave. Los criterios de inclusión fueron: a) Diagnóstico de trauma grave. b) Injury severity score mayor a 16. c) Presencia del resultado del análisis del lactato al ingreso y a las 6 horas en UCI y d) Residente de la altitud desde los 12 años. Se excluyeron los pacientes con historias clínicas incompletas y no legibles. Resultados: Se incluyeron 160 pacientes, Mortalidad del 15%, 65% de las lesiones fueron por trauma encefálico grave. En el grupo de supervivientes (n=136), la depuración de lactato fue del 52,27% y en el grupo de fallecidos fue 21,38%. En relación a la normalización (lactato< 2 mmol/L), el grupo de supervivientes tuvo 47% frente a 1% de los fallecidos. Conclusiones: La depuración de lactato y la normalización del lactato a las 6 seis horas son factores de protección para disminuir la mortalidad por trauma grave. La mortalidad por trauma grave es 15%. El trauma encefálico grave fue la lesión más frecuente.

We study the existence of Gabor orthonormal bases with window the characteristic function of the set Ω = [ 0 , α ] ∪ [ β + α , β + 1 ] of measure 1, with α , β > 0 . By the symmetries of the problem, we can restrict our attention to the case α ≤ 1 / 2 . We prove that either if α 1 / 2 or ( α = 1 / 2 and β ≥ 1 / 2 ) there exist such Gabor orthonormal bases, with window the characteristic function of the set Ω, if and only if Ω tiles the line. Furthermore, in both cases, we completely describe the structure of the set of time–frequency shifts associated to these bases., Departamento de Matemática

Let Ω be a smooth, bounded, convex domain in Rn and let Λk be a finite subset of Ω. We find necessary geometric conditions for Λk to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k., Facultad de Ciencias Exactas

Let $\mu$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},\mu)$ using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials $\{z^n\}_{n\in \mathbb{N}}$ is effective in $L^2(\mathbb{T},\mu)$, it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame $P_\varphi(z^n)$ for the model subspace $\mathcal{H}(\varphi)= H^2 \ominus \varphi H^2$, where $P_\varphi$ is the orthogonal projection from the Hardy space $H^2$ onto $\mathcal{H}(\varphi)$. The study of Fourier expansions in $L^2(\mathbb{T},\mu)$ also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures $\mu$ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization., Comment: 19 pages

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function φ ∈ L 2 ( 0 , 1 ) . More precisely, we study when the system { φ ( n x ) } n is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space H 2 of Dirichlet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.

Fil: Antezana, Jorge Abel. Universidad Nacional de La Plata; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica Alberto Calderon; Argentina

Given two subsets A and B of the algebra of bounded linear operators on a Hilbert space H we denote by A B : = { AB : A ∈ A , B ∈ B } . The goal of this article is to describe A B if A and B denote classes of projections, partial isometries, positive (semidefinite) operators, etc. Moreover, fixed T ∈ A B we shall describe ( A B ) T : = { ( A , B ) ∈ A × B : AB = T } , p 1 ( ( A B ) T ) : = { A ∈ A : T = AB for some B ∈ B } and p 2 ( ( A B ) T ) : = { B ∈ B : T = AB for some A ∈ A } .

We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group G admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the n-dimensional Euclidean space Rⁿ can be extended to G. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in generalized Paley-Wiener spaces., Facultad de Ciencias Exactas

Consider the Lie group of n × n complex unitary matrices U ( n ) endowed with the bi-invariant Finsler metric given by the spectral norm, ‖ X ‖ U = ‖ U ⁎ X ‖ ∞ = ‖ X ‖ ∞ for any X tangent to a unitary operator U. Given two points in U ( n ) , in general there exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness. Similar studies are done for the Grassmann manifolds. On the other hand, consider the cone of n × n positive invertible matrices G l ( n ) + endowed with the bi-invariant Finsler metric given by the trace norm, ‖ X ‖ 1 , A = ‖ A − 1 / 2 X A − 1 / 2 ‖ 1 for any X tangent to A ∈ G l ( n ) + . In this context, also exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves proving first a characterization of the minimal curves joining two Hermitian matrices X , Y ∈ H ( n ) . The last description is also used to construct minimal paths in the group of unitary matrices U ( n ) endowed with the bi-invariant Finsler metric ‖ X ‖ 1 , U = ‖ U ⁎ X ‖ 1 = ‖ X ‖ 1 for any X tangent to U ∈ U ( n ) . We also study the set of intermediate points in all the previous contexts.

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 then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in M. Frank, V. Paulsen, T. Tiballi, Symmetric Approximation of frames and bases in Hilbert Spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. In addition, we characterize symmetric approximations of frames belonging to a prescribed subspace., 21 pages

We study the point process given by the set of real zeros of random series generated with orthonormal bases of reproducing kernels of de Branges spaces. We find an explicit formula for the intensity function in terms of the phase of the Hermite-Biehler function generating the de Branges space. We prove that the intensity of the point process completely characterizes the underlying de Branges space., Facultad de Ciencias Exactas

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 $$ S(\alpha)=\int_a^b L(\dot\alpha(t))\,dt\,, $$ where $\alpha:[a,b]\to\mathcal{U}(n)$ 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 $\pi$. 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, Comment: 20 pages

The main goal of this paper is to provide a characterization of the weak-type boundedness of the Hardy–Littlewood maximal operator, M, on weighted Lorentz spaces Λpᵤ(w), whenever p > 1. This solves a problem left open in (Carro et al., Mem Am Math Soc. 2007). Moreover, with this result, we complete the program of unifying the study of the boundedness of M on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph., Facultad de Ciencias Exactas

Let X and Y be two n×n Hermitian matrices. In the article Proof of a conjectured exponential formula (Linear Multilinear Algebra 19 (1986) 187-197) R.C. Thompson proved that there exist two n×n unitary matrices U and V such that eiXeiY=eiUXU*+VYV*. In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II1 factor., Facultad de Ciencias Exactas

Given a closed subspace \({\mathcal{S}}\) of a Hilbert space \({\mathcal{H}}\), we study the sets \({\mathcal{F}_\mathcal{S}}\) of pseudo-frames, \({\mathcal{C}\mathcal{F}_\mathcal{S}}\) of commutative pseudo-frames and \({\tiny{\mathfrak{X}}_{\mathcal{S}}}\) of dual frames for \({\mathcal{S}}\), via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair \({(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})}\), $$F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,$$ and $$H:\ell^2 \to\,\mathcal{H}, \quad H=(\{c_n\}_{n\in\mathbb{N}})=\sum_n c_n h_n.$$ We prove that, with this identification, the sets \({\mathcal{F}_\mathcal{S}}\), \({\mathcal{C}\mathcal{F}_\mathcal{S}}\) and \({\tiny{\mathfrak{X}}_{\mathcal{S}}}\) are complemented submanifolds of \({\mathcal{B}(\ell^2,\mathcal{H})\times \mathcal{B}(\ell^2,\mathcal{H})}\). We examine in more detail \({\tiny{\mathfrak{X}}_{\mathcal{S}}}\), which carries a locally transitive action from the general linear group GL(l2). For instance, we characterize the homotopy theory of \({\tiny{\mathfrak{X}}_{\mathcal{S}}}\) and we prove that \({\tiny{\mathfrak{X}}_{\mathcal{S}}}\) is a strong deformation retract both of \({\mathcal{F}_\mathcal{S}}\) and \({\mathcal{C}\mathcal{F}_\mathcal{S}}\) ; therefore these sets share many of their topological properties.

Let W and M be two finite dimensional subspaces of a Hilbert space H such that H = W ⊕ M>SUP>⊥, and let PW ‖ M⊥ denote the oblique projection with range W and nullspace M⊥. In this article we get the following formula for the singular values of P W‖M⊥: 2 (sk (P W ‖ M⊥) - 1) = min, (F, H) ∈ X (W, M) Sk (F - H)2,where the minimum is taken over the set of all operator pairs (F, H) on H such that R (F) = W, R (H) = M and FH* = P W ‖ M⊥, and k ∈ {1, ..., dim W}. We also characterize all the pairs where the minimum is attained., Facultad de Ciencias Exactas

Let H be a Hilbert space, W a closed subspace of H, and Q a (linear bounded) projection from H onto W with null space M⊥. We study decompositions like Qf = ∑n∊ℕ 〈 f, hn〉 fn, where {fn}n∊ℕ and {hn}n∊ℕ are frames for the subspaces W and M, respectively. This type of decompositions corresponds to sampling formulae. By considering the synthesis operator F (resp. H) of the sequence {fn}n∊ℕ (resp. {hn}n∊ℕ), the formula above can be expressed as the factorization Q = FH*. We study different properties of these factorizations and decompositions of oblique and orthogonal projections. Several characterizations of these decompositions are presented. By means of an operator inequality for positive operators, we get a result which minimizes the norm of F — H., Facultad de Ciencias Exactas, Facultad de Ingeniería

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 $\rho(\mathcal S, A)$ is defined as the infimum of the sequence $\Sigma (\mathcal S, A^n)^{1/n}$, where $\Sigma (\mathcal S, B)$ denotes the shorted operator of $B$ to $\mathcal S$. We characterize the left spectral resolution of $\rho(\mathcal S, A)$ 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., Comment: 19 pages

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., Comment: 16 pages

We study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length., Facultad de Ciencias Exactas

We study the star order on the algebra L(H) of bounded operators on a Hilbert space H. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finitedimensional setting. We also study the existence of strong limits of starmonotone sequences and nets. Fil: Antezana, Jorge Abel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina Fil: Cano, Cristina. Universidad Nacional del Comahue; Argentina Fil: Mosconi, Irene. Universidad Nacional del Comahue; Argentina Fil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

Let $\lambda \in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ \Delta_\lambda (T )= |T|^{\lambda} U |T |^{1-\lambda}. $$ Let $\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ 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 $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}(T)$ is constant., Comment: 24 pages

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 $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show that the limit $\Delta^{\infty}(\cdot)$ is a map of class $C^\infty$ on the similarity orbit of a diagonalizable matrix, and %of class $C^\infty$ on the (open and dense) set of $r\times r$ matrices with $r$ different eigenvalues., Comment: 25 pages

In a recent article, Smale and Zhou define a notion of rich data for sampling problems and reconstruction of signals from a discrete set of samples and study different least-square problems related with the minimization of the error. They obtain different error estimations assuming that the original signal belong to the reconstruction subspace and they propose to find error estimations if this assumption does not hold. In this paper, using projection methods, we find such estimates and we extend from reproducing kernel Hilbert spaces to abstract Hilbert spaces some of their results on function reconstruction from point values. Fil: Antezana, Jorge Abel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina

We characterize those frames on a Hilbert space H which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension K of H. We show that all frames with infinite excess and frame bounds 1 ≤ A ≤ B are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames., Facultad de Ciencias Exactas

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}} $ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$, to be frame admissible., Comment: To appear in Illinois Journal of Math

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 subtraction, in this Hilbertian context., Facultad de Ciencias Exactas

Let T∈L(H), and let T = U|T| = |T*|U be the polar decomposition of T. Then, for every λ ∈ [0, 1] the λ-Aluthge transform is defined by Δλ(T) = |T|λU|T| 1-λ. We show that several properties which are known for the usual Aluthge transform (i.e. the case λ = 1/2) also hold for λ-Aluthge transforms with λ ∈ (0, 1). Moreover, we get several results which are new, even for the usual Aluthge transform., Facultad de Ciencias Exactas

Given a n × n positive semidefinite matrix A and a subspace S of ℂn, ∑(S, A) denotes the shorted matrix of A to S. We consider the notion of spectral shorted matrix ρ(S, A) = limm→∞ ∑(S, Am)1/m. We completely characterize this martix in terms of script S sign and the spectrum and the eigenspaces of A. We show the relation of this notion with the spectral order of matrices and the Kolmogorov's complexity of A to a vector ξ ℂn., Facultad de Ciencias Exactas

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., Comment: 23 pages, to appear in Linear Algebra and its Applications

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

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 $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta^{n}(T)\}_{n\in\mathbb{N}}$ 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., Comment: 23 pages