arXiv subjects

Pedro Massey

Publications and source records attributed to Pedro Massey.

At least 19 recordsLinked to original sources

arXiv2026

Lambda admissible subspaces of self adjoint matrices

Given a self-adjoint matrix $A$ and an index $h$ such that $\lambda_h(A)$ lies in a cluster of eigenvalues of $A$, we introduce the novel class of $\Lambda$-admissible subspaces of $A$ of dimension $h$. First, we show that the low-rank approximation of the form $P_{\mathcal{T}} A P_{\mathcal{T}}$, for a subspace $\mathcal{T}$ that is close to any $\Lambda$-admissible subspace of $A$, has nice properties. Then, we prove that some well-known iterative algorithms (such as the Subspace Iteration Method, or the Krylov subspace method) produce subspaces that become arbitrarily close to $\Lambda$-admissible subspaces. We obtain upper bounds for the distance between subspaces obtained by the Rayleigh-Ritz method applied to $A$ and the class of $\Lambda$-admissible subspaces. We also find upper bounds for the condition number of the (set-valued) map computing the class of $\Lambda$-admissible subspaces of $A$. Finally, we include numerical examples that show the advantage of considering this new class of subspaces in the clustered eigenvalue setting.

math.NA↗

arXiv2024

Block subspace expansions for eigenvalues and eigenvectors approximation

Let $A\in\mathbb C^{n\times n}$ and let $\mathcal X\subset \mathbb C^n$ be an $A$-invariant subspace with $\dim \mathcal X=d\geq 1$, corresponding to exterior eigenvalues of $A$. Given an initial subspace $\mathcal V\subset \mathbb C^n$ with $\dim \mathcal V=r\geq d$, we search for expansions of $\mathcal V$ of the form $\mathcal V+A(\mathcal W_0)$, where $\mathcal W_0\subset \mathcal V$ is such that $\dim \mathcal W_0\leq d$ and such that the expanded subspace is closer to $\mathcal X$ than the initial $\mathcal V$. We show that there exist (theoretical) optimal choices of such $\mathcal W_0$, in the sense that $\theta_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq \theta_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $\theta_i(\mathcal X,\mathcal T)$ denotes the $i$-th principal angle between $\mathcal X$ and $\mathcal T$, for $1\leq i\leq d\leq \dim \mathcal T$. We relate these optimal expansions to block Krylov subspaces generated by $A$ and $\mathcal V$. We also show that the corresponding iterative sequence of subspaces constructed in this way approximate $\mathcal X$ arbitrarily well, when $A$ is Hermitian and $\mathcal X$ is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.

math.NA↗

arXiv2024

Dominant subspace and low-rank approximations from block Krylov subspaces without a prescribed gap

We develop a novel convergence analysis of the classical deterministic block Krylov methods for the approximation of $h$-dimensional dominant subspaces and low-rank approximations of matrices $ A\in\mathbb K^{m\times n}$ (where $\mathbb K=\mathbb R$ or $\mathbb C)$ in the case that there is no singular gap at the index $h$ i.e., if $σ_h=σ_{h+1}$ (where $σ_1\geq \ldots\geq σ_p\geq 0$ denote the singular values of $ A$, and $p=\min\{m,n\}$). Indeed, starting with a (deterministic) matrix $ X\in\mathbb K^{n\times r}$ with $r\geq h$ satisfying a compatibility assumption with some $h$-dimensional right dominant subspace of $A$, we show that block Krylov methods produce arbitrarily good approximations for both problems mentioned above. Our approach is based on recent work by Drineas, Ipsen, Kontopoulou and Magdon-Ismail on the approximation of structural left dominant subspaces. The main difference between our work and previous work on this topic is that instead of exploiting a singular gap at the prescribed index $h$ (which is zero in this case) we exploit the nearest existing singular gaps.

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arXiv2023

Geometric approach to the Moore-Penrose inverse and the polar decomposition of perturbations by operator ideals

We study the Moore-Penrose inverse of perturbations by a symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore-Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach-Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore-Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

math.FA↗

arXiv2023

Restricted orbits of closed range operators and equivalences between frames for subspaces

Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space and let $\mathcal{J}$ be a two-sided ideal of the algebra of bounded operators $\mathcal{B}(\mathcal{H})$. The groups $\mathcal{G} \ell_\mathcal{J}$ and $\mathcal{U}_{\mathcal{J}}$ consist of all the invertible operators and unitary operators of the form $I + \mathcal{J}$, respectively. We study the actions of these groups on the set of closed range operators. First, we find equivalent characterizations of the $\mathcal{G} \ell_\mathcal{J}$-orbits involving the essential codimension. These characterizations can be made more explicit in the case of arithmetic mean closed ideals. Second, we give characterizations of the $\mathcal{U}_{\mathcal{J}}$-orbits by using recent results on restricted diagonalization. Finally we introduce the notion of $\mathcal{J}$-equivalence and $\mathcal{J}$-unitary equivalence between frames for subspaces of a Hilbert space, and we apply our abstract results to obtain several results regarding duality and symmetric approximation of $\mathcal{J}$-equivalent frames.

math.FA↗

arXiv2022

Norm inequalities for the spectral spread of Hermitian operators

In this work we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the Arithmetic-Geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).

math.FA↗

arXiv2021

On restricted diagonalization

Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space, $\mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators acting on $\mathcal{H}$ and $\mathcal{J}$ a proper two-sided ideal of $\mathcal{B}(\mathcal{H})$. Denote by $\mathcal{U}_\mathcal{J}(\mathcal{H})$ the group of all unitary operators of the form $I+\mathcal{J}$. Recall that an operator $A \in \mathcal{B}(\mathcal{H})$ is diagonalizable if there exists a unitary operator $U$ such that $UAU^*$ is diagonal with respect to some orthonormal basis. A more restrictive notion of diagonalization can be formulated with respect to a fixed orthonormal basis $\mathrm{e}=\{ e_n\}_{n\geq 1}$ and a proper operator ideal $\mathcal{J}$ as follows: $A \in \mathcal{B}(\mathcal{H})$ is called restricted diagonalizable if there exists $U\in \mathcal{U}_\mathcal{J}(\mathcal{H})$ such that $UAU^*$ is diagonal with respect to $\mathrm{e}$. In this work we give necessary and sufficient conditions for a diagonalizable operator to be restricted diagonalizable. Our conditions become a characterization of those diagonalizable operators which are restricted diagonalizable when the ideal is arithmetic mean closed. Then we obtain results on the structure of the set of all restricted diagonalizable operators. In this way we answer several open problems recently raised by Beltiţ$\breve{\text{a}}$, Patnaik and Weiss.

math.FA↗

arXiv2021

The spectral spread of Hermitian matrices

Let $A$ be a $n\times n$ complex Hermitian matrix and let $λ(A)=(λ_1,\ldots,λ_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$, given by $\text{Spr}^+(A) =(λ_1-λ_{n}\, , \, λ_2-λ_{n-1},\ldots, λ_{k}-λ_{n-k+1})\in \mathbb{R}^k$, where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of $A$, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.

math.FA↗

arXiv2021

Absolute variation of Ritz values, principal angles and spectral spread

Let $A$ be a $d\times d$ complex self-adjoint matrix, $\mathcal{X},\mathcal{Y}\subset \mathbb{C}^d$ be $k$-dimensional subspaces and let $X$ be a $d\times k$ complex matrix whose columns form an orthonormal basis of $\mathcal{X}$. We construct a $d\times k$ complex matrix $Y_r$ whose columns form an orthonormal basis of $\mathcal{Y}$ and obtain sharp upper bounds for the singular values $s(X^*AX-Y_r^*\,A\,Y_r)$ in terms of submajorization relations involving the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ and the spectral spread of $A$. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of $A$ associated with the subspaces $\mathcal{X}$ and $\mathcal{Y}$, that partially confirm conjectures by Knyazev and Argentati.

math.FA↗

arXiv2020

Optimal frame designs for multitasking devices with weight restrictions

Let $\mathbf d=(d_j)_{j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence (of dimensions) and $α=(α_i)_{i\in\mathbb I_n}$ be a sequence of positive numbers (of weights), where $\mathbb I_k=\{1,\ldots,k\}$ for $k\in\mathbb N$. We introduce the $(α\, , \,\mathbf d)$-designs i.e., $m$-tuples $Φ=(\mathcal F_j)_{j\in\mathbb I_m}$ such that $\mathcal F_j=\{f_{ij}\}_{i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{d_j}$, $j\in\mathbb I_m$, and such that the sequence of non-negative numbers $(\|f_{ij}\|^2)_{j\in\mathbb I_m}$ forms a partition of $α_i$, $i\in\mathbb I_n$. We characterize the existence of $(α\, , \, \mathbf d)$-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist $(α\, , \, \mathbf d)$-designs $Φ^{\rm op}=(\mathcal F_j^{\rm op})_{j\in\mathbb I_m}$ that are universally optimal; that is, for every convex function $φ:[0,\infty)\rightarrow [0,\infty)$ then $Φ^{\rm op}$ minimizes the joint convex potential induced by $φ$ among $(α\, , \, \mathbf d)$-designs, namely $$ \sum_{j\in\mathbb I_m}\text{P}_φ(\mathcal F_j^{\rm op})\leq \sum_{j\in \mathbb I_m}\text{P}_φ(\mathcal F_j) $$ for every $(α\, , \, \mathbf d)$-design $Φ=(\mathcal F_j)_{j\in\mathbb I_m}$, where $\text{P}_φ(\mathcal F)=tr(φ(S_{\mathcal F}))$; in particular, $Φ^{\rm op}$ minimizes both the joint frame potential and the joint mean square error among $(α\, , \, \mathbf d)$-designs. We show that in this case $\mathcal F_j^{\rm op}$ is a frame for $\mathbb C^{d_j}$, for $j\in\mathbb I_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.

math.FA↗

arXiv2020

Majorization bounds for Ritz values of self-adjoint matrices

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan Θ$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.

math.FA↗

arXiv2018

Generalized frame operator distance problems

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ be a positive semidefinite $d\times d$ complex matrix and let $\mathbf a=(a_i)_{i\in\mathbb{I}_k}\in \mathbb{R}_{>0}^k$, indexed by $\mathbb{I}_k=\{1,\ldots,k\}$, be a $k$-tuple of positive numbers. Let $\mathbb T_{d}(\mathbf a )$ denote the set of families $\mathcal G=\{g_i\}_{i\in\mathbb{I}_k}\in (\mathbb{C}^d)^k$ such that $\|g_i\|^2=a_i$, for $i\in\mathbb{I}_k$; thus, $\mathbb T_{d}(\mathbf a )$ is the product of spheres in $\mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $\mathcal{M}_d(\mathbb{C})$, we consider the generalized frame operator distance function $Θ_{( N \, , \, S\, , \, \mathbf a)}$ defined on $\mathbb T_{d}(\mathbf a )$, given by $$ Θ_{( N \, , \, S\, , \, \mathbf a)}(\mathcal G) =N(S-S_{\mathcal G }) \quad \text{where} \quad S_{\mathcal G}=\sum_{i\in\mathbb{I}_k} g_i\,g_i^*\in\mathcal{M}_d(\mathbb{C})^+\,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $\mathcal G_0\in\mathbb T_{d}(\mathbf a )$ of $Θ_{( N \, , \, S\, , \, \mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.

math.FA↗

arXiv2018

Local Lidskii's theorems for unitarily invariant norms

Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.

math.FA↗

arXiv2016

Convex potentials and optimal shift generated oblique duals in shift invariant spaces

We introduce an extension of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in $L^2(\R^k)$. We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.

math.FA↗

arXiv2016

Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace $\cW\subset L^2(\R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(\cF)$ induced by finite sequences of vectors $\cF\in \cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $\cF$ and the spectra of $S_{E(\cF)}$ is prescribed in each fiber of $\text{Spec}(\cW)\subset \T^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<α_1\leq \ldots\leq α_n$ we characterize the finite sequences $\cF\in\cW^n$ such that $\|f_i\|^2=α_i$, for $1\leq i\leq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(\cF)$ have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential $P^\cW_φ$ induced by $\cW$ and an arbitrary convex function $φ:\R_+\rightarrow \R_+$.

math.FA↗

arXiv2014

Multivariable Schur-Horn theorems

We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type $II_1$ factors. These results are generalisations of the classical Schur-Horn theorem to the infinite dimensional, multivariable setting. Our description of these possible diagonals uses a natural generalisation of the classical notion of majorization to the multivariable setting. In the special case when both the given tuple and the desired diagonal have finite joint spectrum, our results are complete. When the tuples do not have finite joint spectrum, we are able to prove strong approximate results. Unlike the single variable case, the multivariable case presents several surprises and we point out obstructions to extending our complete description in the finite spectrum case to the general case. We also discuss the problem of characterizing diagonals of commuting tuples in $\mathcal{B}(\mathcal{H})$ and give approximate characterizations in this case as well.

math.OA↗

arXiv2014

Aliasing and oblique dual pair designs for consistent sampling

In this paper we study some aspects of oblique duality between finite sequences of vectors $\cF$ and $\cG$ lying in finite dimensional subspaces $\cW$ and $\cV$, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to $\cF$ lying in $\cV$; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for $\cF$ under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces $\cV$ and $\cW$ has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual of $U\cdot \cF$ minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual pair associated to $U\cdot \cF$ minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.

math.FA↗

arXiv2013

Optimal frame completions with prescribed norms for majorization

Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we characterize in a complete and explicit way the optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $\cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $\cF_0$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.

math.FA↗