Pedro Massey

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.

Multivariable Schur–Horn theorems

We prove a variety of results describing the possible diagonals of tuples of commuting hermitian operators in type

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

II_1

Non-commutative Schur–Horn theorems and extended majorization for Hermitian matrices

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

Schur-Horn theorems in II

\mathcal{B}(\mathcal{H})

_\infty

and give approximate characterizations in this case as well.

-factors

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

THE SCHUR-HORN THEOREM FOR OPERATORS AND FRAMES WITH PRESCRIBED NORMS AND FRAME OPERATOR.

Let 𝒜 ⊆ ℳ n (ℂ) be a unital *-subalgebra of the algebra ℳ n (ℂ) of all n × n complex matrices and let B be an Hermitian matrix. Let 𝒰 n (B) denote the unitary orbit of B in ℳ n (ℂ) and let ℰ𝒜 denote the trace preserving conditional expectation onto 𝒜. We give a spectral characterization of the set 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 spectal relations that come together with extended (non-commutative) Schur–Horn type theorems.

The Structure of Minimizers of the Frame Potential on Fusion Frames

We describe majorization between selfadjoint operators in a

λ-Aluthge transforms and Schatten ideals

\sigma