N. J. Kalton

An F-space sampler

1. Preliminaries 2. Some of the classic results 3. Hardy spaces 4. The Hahn-Banach extension property 5. Three space problems 6. Lifting Theorems 7. Transitive spaces and small operators 8. Operators between LP spaces 9. Compact convex sets with no extreme points 10. Notes on other directions of research.

The H ∞ −calculus and sums of closed operators

We develop a very general operator-valued functional calcu- lus for operators with an H 1 −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H 1 calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Ba- nach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting.

Spaces of Compact Operators

In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather different problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F. It is also worth stressing that in general the theorems of the paper do not depend on the approximation property, which is now known to fail in some Banach spaces; the paper is constructed independently of the theory of tensor products. These results were presented at the Gregynog Colloquium in May

Twisted sums of sequence spaces and the three space problem

In this paper we study the following problem: given a complete locally bounded sequence space Y, construct a locally bounded space Z with a subspace X such that both X and Z/X are isomorphic to Y, and such that X is uncomplemented in Z. We give a method for constructing Z under quite general conditions on Y, and we investigate some of the properties of Z. In particular, when Y is lp (1 <p < oo), we identify the dual space of Z, we study the structure of basic sequences in Z, and we study the endomorphisms of Z and the projections of Z on infinite-dimensional subspaces.

Stability results on interpolation scales of quasi-Banach spaces and applications

We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented.

Some remarks on multilinear maps and interpolation

Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasi-normed rearrangement-invariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integrals.

Nonlinear equations and weighted norm inequalities

We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem −∆u = v u + w, u ≥ 0 on Ω, u = 0 on ∂Ω, on a regular domain Ω in Rn in the “superlinear case” q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green’s kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary “integration by parts” argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

Symmetric norms and spaces of operators

Abstract We show that if (E, ∥ · ∥ E ) is a symmetric Banach sequence space then the corresponding space of operators on a separable Hilbert space, defined by if and only if , is a Banach space under the norm . Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.

Szlenk indices and uniform homeomorphisms

We prove some rather precise renorming theorems for Banach spaces with Szlenk index wo. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

Uniformly exhaustive submeasures and nearly additive set functions

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that co (or any E.-space) is a T;space, i.e. cannot be realized as the quotient of a nonlocally convex F-space by a one-dimensional subspace.