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Alejandro Varela

Publications and source records attributed to Alejandro Varela.

At least 19 recordsLinked to original sources
arXiv2026

Graphs of operators as points in the Grassmann manifold

We study the set $\Gamma$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\times H)$ of orthogonal projections in $H\times H$. We show that the subset $\Gamma^b$ of graphs of bounded operators is the open unit ball of $P(H\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\times\{0\}$). This ball is diffeomorphic to $B(H)$ via the map $T\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $\Gamma$. We also study the existence and characteristics of minimal geodesics of $P(H\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $\Gamma$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.

math.FA↗
arXiv2026

Best approximants relative to a C$^*$-subalgebra, joint numerical range and subdifferentials

We study the minimality of $n\times n$ Hermitian matrices $A$ respect to a $C^*$-subalgebra $\mathcal{B}$ of $M_n(\mathbb{C})$ in the spectral norm, that is \[\|A\|\leq \|A+B\|,\ \text{ for every } B\in \mathcal{B}.\] We generalize the notion of the moment of a subspace and relate it to the joint numerical range and the subdifferentials of the maximum eigenvalue. We extend results previously known for the subalgebra of diagonal operators and describe the subdifferential of the maximum eigenvalue in terms of the moment of the corresponding eigenspace. We also characterize $\mathcal{B}$-minimality via moments and subdifferentials, and provide examples.

math.FA↗
arXiv2025

The Riemann sphere of a C*-algebra

Given the unital C$^*$-algebra $A$, the unitary orbit of the projector $p_0=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ in the C$^*$-algebra $M_2(A)$ of $2\times 2$ matrices with coefficients in $A$ is called in this paper, the Riemann sphere $R$ of $A$. We show that $R$ is a homogeneous reductive C$^\infty$ manifold of the unitary group $U_2(A)\subset M_2(A)$ and carries the differential geometry deduced from this structure (including an invariant Finsler metric). Special attention is paid to the properties of geodesics and the exponential map. If the algebra $A$ is represented in a Hilbert space $H$, in terms of local charts of $R$, elements of the Riemann sphere may be identified with (graphs of) closed operators on $H$ (bounded or unbounded). In the first part of the paper, we develop several geometric aspects of $R$ including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of $H\times H$. In the last section we show some applications of the geometry of $R$, to the geometry of operators on a Hilbert space. In particular, we define the notion of bounded deformation of an unbounded operator and give some relevant examples.

math.OA↗
arXiv2024

Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems. Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization. [1] On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl.9 (1988), no 4, 905-918

math.FA↗
arXiv2023

Machine Learning Small Molecule Properties in Drug Discovery

Machine learning (ML) is a promising approach for predicting small molecule properties in drug discovery. Here, we provide a comprehensive overview of various ML methods introduced for this purpose in recent years. We review a wide range of properties, including binding affinities, solubility, and ADMET (Absorption, Distribution, Metabolism, Excretion, and Toxicity). We discuss existing popular datasets and molecular descriptors and embeddings, such as chemical fingerprints and graph-based neural networks. We highlight also challenges of predicting and optimizing multiple properties during hit-to-lead and lead optimization stages of drug discovery and explore briefly possible multi-objective optimization techniques that can be used to balance diverse properties while optimizing lead candidates. Finally, techniques to provide an understanding of model predictions, especially for critical decision-making in drug discovery are assessed. Overall, this review provides insights into the landscape of ML models for small molecule property predictions in drug discovery. So far, there are multiple diverse approaches, but their performances are often comparable. Neural networks, while more flexible, do not always outperform simpler models. This shows that the availability of high-quality training data remains crucial for training accurate models and there is a need for standardized benchmarks, additional performance metrics, and best practices to enable richer comparisons between the different techniques and models that can shed a better light on the differences between the many techniques.

q-bio.BM↗
arXiv2023

Minimal self-adjoint compact operators, moment of a subspace and joint numerical range

We define the (convex) joint numerical range for an infinite family of compact operators in a Hilbert space H. We use this set to determine whether a self-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimal respect to the set of diagonals in a fixed basis E of H in the operator norm, that is ||A|| <= ||A+D||, for all diagonal D. We also describe the moment set m_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms of joint numerical ranges and obtain equivalences between the intersection of moments of two subspaces and of its two related joint numerical ranges. Moreover, we relate the condition of minimality of A or the intersection of the moments of the eigenspaces of ||A|| and -||A|| to the intersection of the joint numerical ranges of two finite families of certain finite hermitian matrices. We also study geometric properties of the set m_S such as extremal curves related with the basis E. All these conditions are directly related with the description of minimal self-adjoint compact operators.

math.FA↗
arXiv2021

Moment of a subspace and joint numerical range

For a given complex finite dimensional subspace $S$ of $\mathbb{C}^n$ and a fixed basis, we study the compact and convex subset of $\left(\mathbb{R}_{\geq 0}\right)^n$ that we call the moment of $S$ $m_S=$ convex hull ($\{|s|^2\in\mathbb{R}^n_{\geq 0}: s\in S \wedge \|s\|=1\} )$ $\simeq \{ Diag(Y) \in M_n^h(\mathbb{C}):Y\geq 0, tr(Y)=1, P_S Y P_S=Y\}$ where $|s|^2=(|s_1|^2,|s_2|^2,\dots,|s_n|^2)$. This set is relevant in the determination of minimal hermitian matrices ($M\in M^h_n$ such that $\|M+D\|\leq D$ for every diagonal $D$ and $\| \|$ the spectral norm). We describe extremal points and curves of $m_S$ in terms of principal vectors that minimize the angle between $S$ and the coordinate axes. We also relate $m_S$ to the joint numerical range $W$ of $n$ rank one $n\times n$ matrices constructed with the orthogonal projection $P_S$ and the fixed basis used. This connection provides a new approach to the description of $m_S$ and to minimal matrices. As a consequence the intersection of two of these joint numerical ranges allow the construction or detection of a minimal matrix, a fact that is easier to corroborate than the equivalent condition for moments. It is also proved that $m_S$ is a semi-algebraic set equal to the intersection of the mentioned $W$ with a hyperplane and whose generated positive cone coincides with that of $W$.

math.FA↗
arXiv2020

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

math.FA↗
arXiv2019

Supports for minimal hermitian matrices

We study certain pairs of subspaces $V$ and $W$ of $\mathbb{C}^n$ we call supports that consist of eigenspaces of the eigenvalues $\pm\|M\|$ of a minimal hermitian matrix $M$ ($\|M\|\leq \|M+D\|$ for all real diagonals $D$). For any pair of orthogonal subspaces we define a non negative invariant $δ$ called the adequacy to measure how close they are to form a support and to detect one. This function $δ$ is the minimum of another map $F$ defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of $F$ in order to approximate $δ$. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

math.FA↗
arXiv2019

Geodesic neighborhoods in unitary orbits of self-adjoint operators of K+C

We study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group U_(K+C) of the unitization of the compact operators K(H)+C, or equivalently, the quotient U_(K+C)/ U_Diag(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

math.OA↗
arXiv2016

Unitary subgroups and orbits of compact self-adjoint operators

Let H be a separable Hilbert space, and D(B(H))^ah the anti-Hermitian bounded diagonals in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators U_kd = {u in U(H): such that u-e^D in K(H) for D in D(B(H))^ah} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob={ e^K b e^{-K} : K in K(H)^ah } of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on Ob defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space K(H)^ah / D(K(H)^ah). Then for every c in Ob and x in T(\ob)_c there exist a minimal lifting Z_0 in B(H)^ah (in the quotient norm, not necessarily compact) such that g(t)=e^{t Z_0} c e^{-t Z_0} is a short curve on Ob in a certain interval.

math.FA↗
arXiv2015

Minimal length curves in unitary orbits of a Hermitian compact operator

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

math.FA↗
arXiv2014

Inequalities related to Bourin and Heinz means with a complex parameter

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\{z \in {\mathbb C}: Re(z) \in [1/4;3/4]\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by $s_j(A^tB^{1-t}+B^tA^{1-t})\le s_j(A+B)$, answering in the negative a question made by K. Audenaert and F. Kittaneh.

math.FA↗
arXiv2013

Best approximation by diagonal compact operators

We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such that ||C|| is less or equal than || C + D ||, for all D in D(K(H)). This property is equivalent to || C || = min{||C+D||: D in D(K(H))} = dist (C,D(K(H))), where D(K(H)) denotes the space of compact diagonal operators in a fixed base of H and ||.|| is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.

math.FA↗
arXiv2011

The left invariant metric in the general linear group

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.

math.DG↗
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↗