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Pedro Massey

Publications and source records attributed to Pedro Massey.

29 records · Page 2Linked to original sources
arXiv2011

Robust dual reconstruction systems and fusion frames

We study the duality of reconstruction systems, which are $g$-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.

math.FA↗
arXiv2011

Schur-Horn theorems in II$_\infty$-factors

We describe majorization between selfadjoint operators in a $σ$-finite II$_\infty$ factor $(\mathcal{M},τ)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and for the unitary and contractive orbits of $τ$-integrable operators in $\mathcal{M}$.

math.OA↗
arXiv2010

Injective Envelopes and Local Multiplier Algebras of Some Spatial Continuous Trace C*-Algebras

A precise description of the injective envelope of a spatial continuous trace C*-algebra A over a Stonean space Delta is given. The description is based on the notion of a weakly continuous Hilbert bundle, which we show to be a Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use the description of the injective envelope of A to study the first- and second-order local multiplier algebras of A. In particular, we show that the second-order local multiplier algebra of A is precisely the injective envelope of A.

math.OA↗
arXiv2010

Duality in reconstruction systems

We consider the notion of finite dimensional reconstructions systems (RS's), which includes the fusion frames as projective RS's. We study erasures, some geometrical properties of these spaces, the spectral picture of the set of all dual systems of a fixed RS, the spectral picture of the set of RS operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting. We give several examples.

math.FA↗
arXiv2008

The structure of minimizers of the frame potential on fusion frames

In this paper we study the fusion frame potential, that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. The structure of local and global minimizers of this potential is studied, when we restrict the frame potential to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. Thus, we exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko's compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, varying the sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima.

math.FA↗
arXiv2008

Towards the Carpenter's Theorem

Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every discrete a in the positive part of the unit ball of A it is possible to find a projection p in M such that E(p)=a$. We also show an example of a diffuse operator x in A such that there exists a projection q in M with E(q)=x. These results show a new family of instances of a conjecture by Kadison, the so-called "Carpenter's Theorem".

math.OA↗
arXiv2007

Non-commutative Schur-Horn theorems and extended majorization for hermitian matrices

Let $\mathcal A\subseteq \mat$ be a unital $*$-subalgebra of the algebra $\mat$ of all $n\times n$ complex matrices and let $B$ be an hermitian matrix. Let $\U_n(B)$ denote the unitary orbit of $B$ in $\mat$ and let $\mathcal E_\mathcal A$ denote the trace preserving conditional expectation onto $\mathcal A$. We give an spectral characterization of the set $$ \mathcal E_\mathcal A(\U_n(B))=\{\mathcal E_\mathcal A(U^* B U): U\in \mat,\ \text{unitary matrix}\}.$$ We obtain a similar result for the contractive orbit of a positive semi-definite matrix $B$. We then use these results to extend the notions of majorization and submajorization between self-adjoint matrices to spectral relations that come together with extended (non-commutative) Schur-Horn type theorems.

math.OA↗
arXiv2007

Minimization of convex functionals over frame operators

We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.

math.FA↗
arXiv2006

The local form of doubly stochastic maps and joint majorization in II$_1$ factors

We find a description of the restriction of doubly stochastic maps to separable abelian $C^*$-subalgebras of a II$_1$ factor $\cM$. We use this local form of doubly stochastic maps to develop a notion of joint majorization between $n$-tuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II$_1$ factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian $C^*$-subalgebra of $\cM$ can be embedded into a separable abelian $C^*$-subalgebra of $\cM$ with diffuse spectral measure.

math.OA↗
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↗