Some problems in operator theory and the geometry of Banach spaces.

摘要

One of the fundamental problems in Operator Theory is the Invariant Subspace Problem asking whether every bounded linear operator on an infinite dimensional complex Banach space admits a closed nontrivial invariant subspace. The study of invariant subspaces can be seen as a study of particular properties of orbits of operators. We study the orbits of a class of isometries of L1[0, 1]. Every isometry of Lp, 1 ≤ p infinity, p ≠ 2, can be written as Tf = h(f ∘ tau). When tau is not measure preserving, we show that the set of functions f in L1[0, 1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis of l1 is an open dense set. A similar problem is also studied for other classical Banach spaces.; In 1996 P. Enflo introduced the concept of extremal vectors and their connection to the Invariant Subspace Problem. We continue studying the properties and behaviour of backward minimal vectors, give some new formulas and improve results from papers by S. Ansari and P. Enflo.; Finally we turn to the application of Functional Analysis and the geometry of Banach spaces to mathematical economics. We study the simplest form of economic activity called a pure exchange economy together with the existence of equilibrium prices problem. More precisely, we study how well the equilibrium price for the subeconomy En approximates the equilibrium price for a larger economy EN when these two economies have the same distribution of agents' characteristics.