$p$-Convergent sequences and Banach spaces in which $p$-compact sets are $q$-compact

On p-Compact Sets in Classical Banach Spaces

Given p ≥ 1, we denote by Cp the class of all Banach spaces X satisfying the equality Kp(Y,X) = Πp(Y,X) for every Banach space Y , Kp (respectively, Πp) being the operator ideal of p-compact operators (respectively, of operators with p-summing adjoint). If X belongs to Cp, a bounded set A ⊂ X is relatively p-compact if and only if the evaluation map U∗ A : X ∗ −→ ∞(A) is p-summing. We obtain p-...

Banach-Like Distances and Metric Spaces of Compact Sets

In the first part we study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space; we obtain a rigidity result. We then discuss the Hausdorff distance, proposing some less–known but important results: a closed–form formula for geodesics; generically two compact sets are connected by a continuum of geodesics. In the second p...

Pre-compact Families of Finite Sets of Integers and Weakly Null Sequences in Banach Spaces

In this paper we provide a somewhat general framework for studying weakly null sequences in Banach spaces using Ramsey theory of families of finite subsets of N. Recall that the Ramsey theory on families of finite subsets of N was developed in a series of papers of Nash-Williams in the 60’s, a theory that is today naturally embedded in the more familiar infinite-dimensional Ramsey theory. The a...

P-Amenable Locally Compact Hypergroups

Compact operators on Banach spaces

In this note I prove several things about compact linear operators from one Banach space to another, especially from a Banach space to itself. Some of these may things be simpler to prove for compact operators on a Hilbert space, but since often in analysis we deal with compact operators from one Banach space to another, such as from a Sobolev space to an L space, and since the proofs here are ...