Andrzej Szankowski

Subspaces without the approximation property

It is proved that the Banach spacelp with 1≦p<2 contains a subspace without AP (the case 2<p≦∞ follows from the Enflo’s construction and also from the present one). The result generalizes to the following one: if the supremum of types ofX is strictly less than 2 or if the infimum of cotypes ofX is strictly more than 2 thenX contains a subspace without AP.

B(H) does not have the approximation propertydoes not have the approximation property

A method of improving resistance of ferrous parts to wear failure such as galling and seizure, and parts having such increased resistance are disclosed. The invention includes a two step process wherein the part is first nitrided to provide a nitride case. A coating of molybdenum-disulfide (MoS2) is then applied over the nitrided case.

On Dvoretzky’s theorem on almost spherical sections of convex bodies

A complete and rather simple proof of the famous Dvoretzky’s theorem is presented.

On the uniform approximation property in Banach spaces

We derive a criterion for a Banach space to fail the uniform approximation property (UAP). This criterion is applied to prove thatcp spaces of matrices fail UAP ifp>80.

Embedding Banach spaces with unconditional bases into spaces with symmetric bases

Every reflexive Banach space with unconditional basis is isomorphic to a complemented subspace of a reflexive Banach space with symmetric basis.

A stability property of the unit vector basis ofl p

Let {en} be the unit vector basis oflp, l<p<∞, and letxn=anen−bnen+1. Necessary and sufficient conditions are given for the operatorT:lp → span {xn} defined byTei=xi to be invertible.