Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.

Ultraluminous X-ray sources with the SKA

The discovery almost three decades ago of non-nuclear, point-like X-ray sources with X-ray luminosities LX ≥ 3 × 1039 erg s−1 revolutionized the physics of black hole accretion. If of stellar origin, such Ultraluminous X-ray sources (ULXs) would have to accrete at super-Eddington rates in order to reach the observed high X-ray luminosities. Alternatively, ULXs could host sub-Eddington accreting intermediate-mass black holes, which are the long-time sought missing link between stellar and supermassive black holes and the possible seeds of the supermassive black holes that formed in the early Universe. The nature of ULXs can be better investigated in those cases for which a radio counterpart is detected. Radio observations of ULXs have revealed a wide variety of morphologies and source types, from compact and extended jets to radio nebulae and transient behaviours, providing the best observational evidence for the presence of an intermediate-mass black hole in some of them. The high sensitivity of the SKA will allow us to study the faintest ULX radio counterparts in the Local Universe as well as to detect new sources at much larger distances. It will thus perform a leap step in understanding ULXs, their accretion physics, and their possible role as seed black holes in supermassive black hole and galaxy growth.