On excesses of frames

We show that any two frames in a separable Hilbert space that are dual to each other have the same excess. Some new relations for the analysis resp. synthesis operators of dual frames are also derived. We then prove that pseudo-dual frames and, in particular, approximately dual frames have the same excess. We also discuss various results on frames in which excesses of frames play an important role.Comment: 12 page

Oblique projections and frames

We characterize those frames on a Hilbert space H which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension K of H. We show that all frames with infinite excess, and frame bounds 1 ≤ A ≤ B,are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames.Fil: Antezana, Jorge Abel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin

The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}}$ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$, to be frame admissible.Comment: To appear in Illinois Journal of Mat

Identification of Faint Chandra X-ray Sources in the Core-Collapsed Globular Cluster NGC 6752

We have searched for optical identifications for 39 Chandra X-ray sources that lie within the 1.9 arcmin half-mass radius of the nearby (d = 4.0 kpc), core-collapsed globular cluster, NGC 6752, using deep Hubble Space Telescope ACS/WFC imaging in B435, R625, and H alpha. Photometry of these images allows us to classify candidate counterparts based primarily on color-magnitude and color-color diagram location. The color-color diagram is particularly useful for quantifying the H alpha line equivalent width. In addition to recovering 11 previously detected optical counterparts, we propose 20 new optical IDs. In total, there are 16 likely or less certain cataclysmic variables (CVs), nine likely or less certain chromospherically active binaries, three galaxies, and three active galactic nuclei (AGNs). The latter three sources, which had been identified as likely CVs by previous investigations, now appear to be extragalactic objects based on their proper motions. As we previously found for NGC 6397, the CV candidates in NGC 6752 fall into a bright group that is centrally concentrated relative to the turnoff-mass stars and a faint group that has a spatial distribution that is more similar to that of the turnoff-mass stars. This is consistent with an evolutionary scenario in which CVs are produced by dynamical interactions near the cluster center and diffuse to larger radius orbits as they age.Comment: 26 pages, 18 figure