For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E/E) is defined by R(S)(x + E) = Sx + E (x ∈ E). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W (E) (here W (E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non–zero compact...

‎We present streamlined proofs of certain maximality principles studied by Hamkins‎ ‎and Woodin‎. ‎Moreover‎, ‎we formulate an intermediate maximality principle‎, ‎which is‎ ‎shown here to be equiconsistent with the existence of a weakly compact cardinal $kappa$ such that $V_{kappa}prec V$‎.

Let λ be an infinite cardinal number and let `∞(λ) denote the subspace of `∞(λ) consisting of all functions which assume at most countably many non zero values. We classify all infinite dimensional complemented subspaces of `∞(λ), proving that they are isomorphic to `∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on `∞(λ) or `∞(λ) has the u...

We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(μ) spaces for finite measures μ. The purpose of this note is to show that a simple transfer renorming theorem explains why L1(μ)-spaces, for finite measures μ, share some properties with superreflexive spac...

We study on a product Banach space the properties of a class of saddle functions called partially ball weakly inf-compact. For such a function we prove that the domain of the subdifferential is nonempty, that the operator naturally associated with the subdifferential is maximal monotone, and that the subdifferential of the function is integrable. For a function in a large subclass of that class...

A unitary operator U acting on the weakly compact unit ball B of a Hilbert space H gives a topological dynamical system (B, V). We prove in this note that for every ergodic u-invariant measure m, the dynamical system (B, U, m) is conjugated to a strictly ergodic group translation. Especially, the topological entropy of (B, U) is zero. As an application, it follows that any flow on the unit ball...