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

Dynamically enhanced magnetic incommensurability: Effects of local dynamics on non-local spin-correlations in a strongly correlated metal

We compute the spin susceptibility of the two-dimensional Hubbard model away from half-filling, and analyze the impact of frequency dependent vertex corrections as obtained from the dynamical mean field theory (DMFT). We find that the local dynamics captured by the DMFT vertex strongly affects non-local spin correlations, and thus the momentum dependence of the spin susceptibility. While the widely used random phase approximation yields commensurate N\'eel-type antiferromagnetism as the dominant instability over a wide doping range, the vertex corrections favor incommensurate ordering wave vectors away from $(\pi,\pi)$. Our results indicate that the connection between the magnetic ordering wave vector and the Fermi surface geometry, familiar for weakly interacting systems, can hold in a strongly correlated metal, too

Projections in operator ranges

If \H is a Hilbert space, $A$ is a positive bounded linear operator on \cH and \cS is a closed subspace of \cH, the relative position between \cS and A^{-1}(\cS \orto) establishes a notion of compatibility. We show that the compatibility of (A,\cS) is equivalent to the existence of a convenient orthogonal projection in the operator range $R(A^{1/2})$ with its canonical Hilbertian structure

Non-separable frequency dependence of two-particle vertex in interacting fermion systems

We derive functional flow equations for the two-particle vertex and the self-energy in interacting fermion systems which capture the full frequency dependence of both quantities. The equations are applied to the hole-doped two-dimensional Hubbard model as a prototype system with entangled magnetic, charge and pairing fluctuations. Each fluctuation channel acquires substantial dependencies on all three Matsubara frequencies, such that the frequency dependence of the vertex cannot be accurately represented by a channel sum with only one frequency variable in each term. At the temperatures we are able to access, the leading instabilities are mostly antiferromagnetic, with an incommensurate wave vector. However, at large doping, a divergence in the charge channel occurs at a finite frequency transfer, if the vertex flow is computed without self-energy feedback. This enigmatic instability was already observed in a calculation by Husemann et al. [Phys. Rev. B 85, 075121 (2012)], who used an approximate separable ansatz for the frequency dependence of the vertex. We identify a simple mechanism for this instability in terms of a random phase approximation for the charge channel with a frequency dependent effective magnetic interaction as input. In spite of the strong momentum and frequency dependence of the vertex, the self-energy has a Fermi liquid form. At the moderate interaction strength where our approach is applicable, we obtain a moderate reduction of the quasi-particle weight and a sizable decay rate with a pronounced momentum dependence. Nevertheless, the self-energy feedback into the vertex flow turns out to be crucial, as it suppresses the unphysical finite frequency charge instability

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.Comment: 23 page

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 $0<\alpha_1\leq \ldots\leq \alpha_n$ we characterize the finite sequences \cF\in\cW^n such that $\|f_i\|^2=\alpha_i$, 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_\varphi induced by \cW and an arbitrary convex function $\varphi:\R_+\rightarrow \R_+$.Comment: 31 pages. Accepted in the JFA. This revised version has several changes in the notation and the organization of the text. There exists text overlap with arXiv:1508.01739 in the preliminary section

Multiplicative Lidskii's inequalities and optimal perturbations of frames

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame \cF for \hil\cong\C^d we compute those dual frames \cG of \cF that are optimal perturbations of the canonical dual frame for \cF under certain restrictions on the norms of the elements of \cG. On the other hand, for a fixed finite frame \cF=\{f_j\}_{j\in\In} for \hil we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame V\cdot \cF=\{V\,f_j\}_{j\in\In} - which is equivalent to \cF - is optimal among such perturbations of \cF. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.Comment: 22 page