On the number of Mather measures of Lagrangian systems

In 1996, Ricardo Ricardo Ma\~n\'e discovered that Mather measures are in fact the minimizers of a "universal" infinite dimensional linear programming problem. This fundamental result has many applications, one of the most important is to the estimates of the generic number of Mather measures. Ma\~n\'e obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able with Gonzalo Contreras to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest

INVARIANT SUBSPACES AND HYPER-REFLEXIVITY FOR FREE SEMIGROUP ALGEBRAS

In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner-outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn

Shapiroʼs Theorem for subspaces

AbstractIn the previous paper (Almira and Oikhberg, 2010 [4]), the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (An) (defined by E(x,An)=infa∈An‖x−an‖) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive