Olav Nygaard

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Peŀczyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.

Slices in the unit ball of a uniform algebra

Abstract. We show that every nonvoid relatively weakly open subset, in particular every slice, of the unit ball of an infinite-dimensional uniform algebra has diameter 2.

BOUNDEDNESS AND SURJECTIVITY IN NORMED SPACES

We define the (w* -) boundedness property and the (w* -) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w* -) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of two w*-thick sets in X** and Y* is a w*-thick subset in L(X,Y)* and obtain as a consequence that the set w*-expBK(l2)* is w*-thick.

ALMOST ISOMETRIC IDEALS IN BANACH SPACES

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

Restricted uniform boundedness in Banach spaces

Precise conditions for a subset A of a Banach space X are known in order that pointwise bounded on A sequences of bounded linear functionals on X are uniformly bounded. In this paper, we study such conditions under the extra assumption that the functionals belong to a given linear subspace Γ of X*. When Γ = X*, these conditions are known to be the same ones assuring a bounded linear operator into X, having A in its image, to be onto. We prove that, for A, deciding uniform boundedness of sequences in Γ is the same property as deciding surjectivity for certain classes of operators.

Two properties of Müntz spaces

Abstract We show that Müntz spaces, as subspaces of C[0, 1], contain asymptotically isometric copies of c0 and that their dual spaces are octahedral.