Frames of subspaces and operators

We study the relationship between operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space $\mathcal{H}$. We get sufficient conditions on an orthonormal basis of subspaces $\mathcal{E} = \{E_i \}_{i\in I}$ of a Hilbert space $\mathcal{K}$ and a surjective $T\in L(\mathcal{K}, \mathcal{H})$ in order that $\{T(E_i)\}_{i\in I}$ is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J. A. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames. Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinament of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.Comment: 21 pages, LaTeX; added references and comments about fusion frame

Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: λ\lambda-Aluthge transform

Let $\lambda \in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the \la- Aluthge transform is defined by $$\Delta_\lambda (T )= |T|^{\lambda} U |T |^{1-\lambda}.$$ Let $\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map (\la, T) \mapsto \alulit{\infty}{T}, and we study for which matrices the map $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}(T)$ is constant.Comment: 24 page

Bilateral Shorted Operators and Parallel Sums

In this paper we study shorted operators relative to two different subspaces, for bounded operators on infinite dimensional Hilbert spaces. We define two notions of complementability in the sense of Ando for operators, and study the properties of the shorted operators when they can be defined. We use these facts in order to define and study the notions of parallel sum and substraction, in this Hilbertian context