A separable Banach space X is said to be ergodic if the relation E₀ of eventual agreement between sequences of 0's and 1's is Borel reducible to isomorphism between subspaces of X. This means that there exists a Borel map f mapping elements of 2^w to subspaces of X such that aE₀ b if and only if f(a) @ f(b). In particular, an ergodic Banach space X must contain 2^w mutually non-isomorphic subspaces.

We present a constructive version of a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic l_p spaces.

We study the existence of pointwise Kadec renormings for Banach spaces of the form C(K). (A pointwise Kadec renorming is a norm equivalent to the sup norm for which the norm topology and the topology of pointwise convergence agree on the unit sphere.) We show in particular that such a renorming exists when K is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if C(K₁) has a pointwise Kadec renorming and K₂ belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then C(K₁×K₂) has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.

This is joint work with W. Kubis and S. Todorcevic.

We will discuss the solvability of a class of nonlinear differential equations on Banach spaces that relate to variational inequalities and complementarity problems. Such equations have been recently formulated in B-spaces, but their solvability has not yet been discussed. We offer a first insight into the question of existence of solutions for such equations and its implications for the study of applied problems related to such equations. They are a generalization of similar equations in Hilbert spaces, now widely used in applied equilibrium problems in networks, game theoretic and economic problems.

Let (e_i) be a bounded sequence in a Banach space X such that the additive group G generated by (e_i) is r-dense in the unit ball of X, where r < 1. When (e_i) is a bounded minimal system then l₁ embeds into X^*. If, in addition, the approximant from G may be chosen by a natural algorithm then c₀ embeds into X. We discuss these results and attempt to generalize them to redundant systems such as frames.

This is joint work with Casazza, Odell, Schlumprecht, and Zsák.

We will describe some recent work on the problem of extending linear maps into C(K)-spaces. We will also discuss separable Banach spaces X which can only be embedded into a C(K)-space (with K compact metric) in one way up to automorphism; these spaces include c₀ and l₁ not l_p when p > 1.

In 1934, H. Whitney showed that continuous, real-valued functions defined on open sets can be uniformly approximated by analytic maps. Subsequently (1953), Kurzweil demonstrated that in a separable Banach space admitting a separating polynomial, any continuous function can be uniformly approximated by maps analytic on the space. In the spirit of Kurzweil, we show that in such spaces any uniformly continuous, real-valued function can be uniformly approximated by Lipschitz analytic maps on bounded sets.

Let f Î L_p(R) and for l Î R let f_(l) be the translation of f by l,

Some results we obtain are

(1) If (f_(l))_{l Î L} is a basis (or Schauder frame) for X(f,L) Í L₁(R) then X (f,L) embeds isomorphically into l₁.
(2) If 1 < p £ 2 and (f_(l))_{l Î L} is unconditional basic then it is equivalent to the unit vector basis of l_p.
(3) If 2 < p £ 4 and (f_(l))_{l Î L} is unconditional basic then X(f,L) embeds into l_p.
(4) For 4 < p < ¥ there exists f Î L_p(R) and L Í N so that (f_(l))_{l Î L} is unconditional basic and X(f,L) contains an isomorph of L_p(R).

We report on recent joint work with Th. Schlumprecht, B. Sari and B. Zheng.

In this talk we will show that the strictly singular operator without invariant subspaces constructed by C. J. Read is finitely strictly singular. This result is obtained from the following fact: if k £ n then every k-dimensional subspace of Rⁿ contains a zigzag of order k, that is, a vector x=(x_i)_i=1ⁿ with |x_i| £ 1 for all i such that x_mi=(-1)ⁱ for some m₁ < m₂ < ¼ < m_k.

Let l be a positive number, and let (x_j:j Î Z) Ì R be a fixed Riesz-basis sequence, namely, (x_j) is strictly increasing, and the set of functions {R ' t® e^{ix_j t} : j Î Z} is a Riesz basis (i.e., unconditional basis) for L₂(-p,p). Given a function f Î L₂ (R) whose Fourier transform is zero almost everywhere outside the interval [-p,p], there is a unique sequence (a_j : j Î Z) in l₂ (Z), depending on l and f, such that the function

We characterise those spaces X with an (uncountable) unconditional basis which admit an equivalent norm, the dual norm of which is strictly convex. The problem is essentially topological, and the notion of adequate families, introduced by Talagrand, plays a central role. We discuss the corresponding situation for Gâteaux smooth norms and related questions.

This is joint work with S. Troyanski of the Universidad de Murcia in Murcia, Spain.

We say that a bounded linear operator T on a Banach space X admits an almost invariant half-space if there exists an infinite dimensional and infinite codimensional closed subspace Y (a half-space) and a finite dimensional subspace F such that T(Y) Ì Y+F. The question whether every bounded linear operator admits an almost invariant half-space is connected to the invariant subspace problem, but it is not necessarily weaker.

In this talk we introduce a promising technique for approaching this question and prove several positive results for weighted shifts operators. In particular we show that Donoghue operators, which do not have invariant half-spaces, admit almost invariant half-spaces with one dimensional "error".

This is joint work with G. Androulakis, A. Popov and V. Troitsky.