Some conditions implying normality of operators

Let $T\in\mathbb{B}(\mathscr{H})$ and $T=U|T|$ be its polar decomposition. We proved that (i) if $T$ is log-hyponormal or $p$-hyponormal and $U^n=U^\ast$ for some $n$, then $T$ is normal; (ii) if the spectrum of $U$ is contained in some open semicircle, then $T$ is normal if and only if so is its Aluthge transform $\widetilde{T}=|T|^{1\over2}U|T|^{1\over2}$

Evidence for entanglement at high temperatures in an engineered molecular magnet

The molecular compound [Fe$_{2}$($\mu_{2}$-oxo)(C$_{3}$H$_{4}$N$_{2}$)$_{6}$(C$_{2}$O$_{4}$)$_{2}$] was designed and synthesized for the first time and its structure was determined using single-crystal X-ray diffraction. The magnetic susceptibility of this compound was measured from 2 to 300 K. The analysis of the susceptibility data using protocols developed for other spin singlet ground-state systems indicates that the quantum entanglement would remain at temperatures up to 732 K, significantly above the highest entanglement temperature reported to date. The large gap between the ground state and the first-excited state (282 K) suggests that the spin system may be somewhat immune to decohering mechanisms. Our measurements strongly suggest that molecular magnets are promising candidate platforms for quantum information processing

Satellite Galaxies and Fossil Groups in the Millennium Simulation

We use a semianalytic galaxy catalogue constructed from the Millennium Simulation to study the satellites of isolated galaxies in the LCDM cosmogony. This sample (~80,000$ bright primaries, surrounded by ~178,000 satellites) allows the characterization, with minimal statistical uncertainty, of the dynamical properties of satellite/primary galaxy systems in a LCDM universe. We find that, overall, the satellite population traces the dark matter rather well: its spatial distribution and kinematics may be approximated by an NFW profile with a mildly anisotropic velocity distribution. Their spatial distribution is also mildly anisotropic, with a well-defined ``anti-Holmberg'' effect that reflects the misalignment between the major axis and angular momentum of the host halo. The isolation criteria for our primaries picks not only galaxies in sparse environments, but also a number of primaries at the centre of ''fossil'' groups. We find that the abundance and luminosity function of these unusual systems are in reasonable agreement with the few available observational constraints. We recover the expected L_{host} \sigma_{sat}^3 relation for LCDM models for truly-isolated primaries. Less strict primary selection, however, leads to substantial modification of the scaling relation. Our analysis also highlights a number of difficulties afflicting studies that rely on blind stacking of satellite systems to constrain the mean halo mass of the primary galaxies.Comment: 18 pages, 14 figures, MNRAS in press. Accepted version with minor changes. Version with high resolution figures available at: http://www.astro.uvic.ca/~lsales/SatPapers/SatPapers.htm