arXiv subjects

Guillermina Fongi

Publications and source records attributed to Guillermina Fongi.

9 recordsLinked to original sources

arXiv2026

Proper splittings of Hermitian operators

In this article we deepen the study of proper splittings of Hilbert space operators, with special emphasis on proper splittings of Hermitian operators. On the one hand, we improve characterizations given in [Fongi $\&$ Gonzalez, J. Math. Anal. Appl., 545 (2025) 129093] of the convergence of both the polar proper and the Moore Penrose proper splittings. On the other hand, we introduce new proper splittings and we compare their convergence with those of the polar and the Moore Penrose proper splittings.

math.FA↗

arXiv2026

Antitonicity property of the Moore-Penrose inverse for selfadjoint operators

In this article we study the antitonicity property of the Moore-Penrose inverse in the class of selfadjoint operators, with respect to the Löwner order. For this purpose, we employ different positive decompositions that selfadjoint operators admit. In addition, we relate a weak version of the antitonicity property with Thompson components.

math.FA↗

arXiv2024

Proper splittings of Hilbert space operators

Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear operators defined on Hilbert spaces. We study the convergence of general proper splittings of operators in the infinite dimensional context. We also propose some particular splittings for special classes of operators and we study different criteria of convergence and comparison for them. In some cases, these criteria are given under hypothesis of operator order relations. In addition, we relate these results with the concept of the symmetric approximation of a frame in a Hilbert space.

math.FA↗

arXiv2022

Moore-Penrose inverse and partial orders on Hilbert space operators

In this article we explore several aspects concerning to the Moore-Penrose inverse of a bounded linear operator. On the one hand, we study monotonicity properties of the Moore-Penrose inverse with respect to the Löwner, star, minus, sharp and diamond orders. On the other hand, we analyze the validity of the reverse order law, $B^\dagger A^\dagger=(AB)^\dagger$, under hypothesis of operator ranges and also under hypothesis of order operators. Finally, we study the operator $B^\dagger A^\dagger$ as different weighted inverses of $AB$.

math.FA↗

arXiv2022

Polyak's theorem on Hilbert spaces

We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We give sufficient conditions which assure that the joint image is also closed. However, we show that, in general, the closedness part of Polyak's theorem does not hold in the infinite dimensional setting, even for quadratic functions generated by compact operators. We give some applications to S-lemma type results.

math.FA↗

arXiv2020

Total least squares problems on infinite dimensional spaces

In this work we study weighted total least squares problems on infinite dimensional spaces. We show that in most cases this problem does not admit a solution (except in the trivial case) and then, we consider a regularization on the problem. We present necessary conditions for the regularized problem to have a solution. We also show that, by restricting the regularized minimization problem to special subsets, the existence of a solution may be assured.

math.FA↗

arXiv2019

Global solutions of approximation problems in Hilbert spaces

We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way and the solvability of the associated operator problem in a fixed p-Schatten norm.

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arXiv2017

The minus order and range additivity

We study the minus order on the algebra of bounded linear operators on a Hilbert space. By giving a characterization in terms of range additivity, we show that the intrinsic nature of the minus order is algebraic. Applications to generalized inverses of the sum of two operators, to systems of operator equations and to optimization problems are also presented.

math.FA↗

arXiv2013

Weighted projections into closed subspaces

In this paper we study $A$-projections, i.e. operators of a Hilbert space $\HH$ which act as projections when a seminorm is considered in $\HH$. $A$-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite dimensional spaces. We relate this concept to the theory of compatibility between positive operators and closed subspaces of $\HH$. We also study the relationship between weighted least squares problems and compatibility.

math.FA↗