Injektive Operatorenideale über der Gesamtheit aller Banachräume und ihre topologische Erzeugung
We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space to a space has the 3.2 intersection property if and only if and have the 3.2 intersection property and...
We study conditions on an infinite dimensional separable Banach space implying that is the only non-trivial invariant subspace of under the action of the algebra of biconjugates of bounded operators on : . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of , and show in particular that any space which does not contain and has property (u) of Pelczynski is simple.
By using the concepts of limited -converging operators between two Banach spaces and , -sets and -limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as -Dunford–Pettis property of order and Pelczyński’s property of order , .
We show for and subspaces of quotients of with a -unconditional finite-dimensional Schauder decomposition that is an -ideal in .
We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces and the subspace of all compact operators is an -ideal in the space of all continuous linear operators whenever and are - and -ideals in and , respectively, with and . We also prove that the -ideal in is separably determined. Among others, our results complete and improve some well-known results...
We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.
The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz-Räbiger net is equivalent to quasi-compactness of some operator . We prove that strong convergence of a quasi-compact uniform Lotz-Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.