Eve Oja

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.

Geometry of Banach spaces having shrinking approximations of the identity

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M∗(a, B, c) of Banach spaces X by the requirement that lim sup ν ‖axν + bx∗ + cy∗‖ ≤ lim sup ν ‖xν‖ ∀b ∈ B whenever (xν) is a bounded net converging weak ∗ to x∗ in X∗ and ‖y∗‖ ≤ ‖x∗‖. Using M∗(a, B, c) with max |B|+c > 1, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to M -, u-, and h-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

Ideals of compact operators

We give an example of a Banach space such that is not an ideal in . We prove that if is a weak denting point in the unit ball of and if is a closed subspace of a Banach space , then the set of norm-preserving extensions of a functional is equal to the set . Using this result, we show that if is an -ideal in and is a reflexive Banach space, then is an -ideal in whenever is an ideal in . We also show that is an ideal (respectively, an -ideal) in for all Banach spaces whenever is an ideal (respectively, an -ideal) in and has the compact approximation property with conjugate operators.

Bounded approximation properties via integral and nuclear operators

Published version of an article in the journal:Proceedings of the American Mathematical Society. Also available from the publisher, Open Access

Principle of local reflexivity revisited

We give, departing from Grothendiecks description of the dual of the space of weak*-weak continuous finite-rank operators, a clear proof for the principle of local reflexivity in a general form.

Géométrie des espaces de Banach ayant des approximations de l'identité contractantes

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M* (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M*) due to Kalton. Using M*(a, B, c) with max ¦B¦ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that the existence of these approximations of the identity is separably determined. As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators.

OPERATORS THAT ARE NUCLEAR WHENEVER THEY ARE NUCLEAR FOR A LARGER RANGE SPACE

Let

Hahn-Banach extension operators and spaces of operators

X

On Commuting Approximation Properties of Banach Spaces

be a Banach space and let

Lifting of the approximation property from Banach spaces to their dual spaces

Y