Grothendieck Numbers and Volume Ratios of Operators on Banach Spaces.
Haar null and non-dominating sets
We study the σ-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this σ-ideal is closely related, in a precise sense, to the σ-ideal of non-dominating subsets of . Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris. We also obtain...
Hajłasz-Sobolev type spaces and -energy on the Sierpinski gasket.
Hereditarily finitely decomposable Banach spaces
A Banach space is said to be if the maximal number of subspaces of X forming a direct sum is finite and equal to n. We study some properties of spaces, and their links with hereditarily indecomposable spaces; in particular, we show that if X is complex , then dim , where S(X) denotes the space of strictly singular operators on X. It follows that if X is a real hereditarily indecomposable space, then ℒ(X)/S(X) is a division ring isomorphic either to ℝ, ℂ, or ℍ, the quaternionic division ring....
Hermitian Operators on C (X, E) and the Banach-Stone Theorem.
Hilbert Spaces have the Strong Banach-Stone Property for Bochner Spaces.
How small are the sets where the metric projection fails to be continuous
Hyers-Ulam constants of Hilbert spaces
The best constant in the Hyers-Ulam theorem on isometric approximation in Hilbert spaces is equal to the Jung constant of the space.
-measures in Minkowski planes.
Ideals induced by Tsirelson submeasures
We use Tsirelson’s Banach space ([2]) to define an P-ideal which refutes a conjecture of Mazur and Kechris (see [12, 9, 8]).
Ideals of finite rank operators, intersection properties of balls, and the approximation property
We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of , the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).
(I)-envelopes of unit balls and James' characterization of reflexivity
We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing.
Inégalités isopérimétriques en analyse et probabilités
Infinite-dimensional sets of constant width and their applications.
Sets of constant width appear as a curiosity in the context of finite-dimensional Euclidean spaces. These sets are convex bodies of such an space with the property that the distance between any two distinct parallel supporting hyperplanes is constant. The easiest example of a set of constant width which is not a ball is the so called Reuleaux triangle in the Euclidean plane. This is the intersection of three closed discs of radius r, whose centers are the vertices of an equilateral triangle of side...
Integral polynomials on Banach spaces not containing
We give new characterizations of Banach spaces not containing in terms of integral and -dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.
Interpolation of Banach spaces, differential geometry and differential equations.
In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.
Interpolation of Cesàro sequence and function spaces
The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that is an interpolation space between and for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, is not an interpolation space between Ces₁[0,1] and .
Interpolation of Orlicz-valued function spaces and U.M.D. property
Intersection properties of balls in tensor products of some Banach spaces.
Intersections of closed balls and geometry of Banach spaces.