Banach space projections and Petrov-Galerkin estimates

We sharpen the classic a priori error estimate of Babuska for Petrov-Galerkin methods on a Banach space. In particular, we do so by (i) introducing a new constant, called the Banach-Mazur constant, to describe the geometry of a normed vector space; (ii) showing that, for a nontrivial projection $P$, it is possible to use the Banach-Mazur constant to improve upon the naive estimate $\| I - P \| \leq 1 + \| P \|$; and (iii) applying that improved estimate to the Petrov-Galerkin projection operator. This generalizes and extends a 2003 result of Xu and Zikatanov for the special case of Hilbert spaces.Comment: 9 pages; v2: added new section on application to Lp and Sobolev space

Radiation Pressure Confinement - I. Ionized Gas in the ISM of AGN Hosts

We analyze the hydrostatic effect of AGN radiation pressure on optically thick gas in the host galaxy. We show that in luminous AGN, the radiation pressure likely confines the ionized layer of the illuminated gas. Radiation pressure confinement (RPC) has two main implications. First, the gas density near the ionization front is 7x10^4 L_{i,45} r_{50}^{-2} cm^{-3}, where L_{i,45} is the ionizing luminosity in units of 10^45 erg/s and r_{50} is the distance of the gas from the nucleus in units of 50 pc. Second, as shown by Dopita et al., the solution of the ionization structure within each slab is unique, independent of the ambient pressure. We show that the RPC density vs. distance relation is observed over a dynamical range of ~10^4 in distance, from sub-pc to kpc from the nucleus, and a range of ~10^8 in gas density, from 10^3 to 10^11 cm^{-3}. This relation implies that the radiative force of luminous AGN can compress giant molecular clouds in the host galaxy, and possibly affect the star formation rate. The unique ionization structure in RPC includes a highly ionized X-ray emitting surface, an intermediate layer which emits coronal lines, and a lower ionization inner layer which emits optical lines. This structure can explain the observed overlap of the extended X-ray and optical narrow line emission in nearby AGN. We further support RPC by comparing the predicted ratios of the narrow lines strength and narrow line widths with available observations. We suggest a new method, based on the narrow line widths, to estimate the black hole mass of low luminosity AGN.Comment: Accepted for publication in MNRAS. The first author will be on regular email contact from December 201