Banach-Stone type theorems

LASSALLE, S.

University College Dublin

Congreso; Complex Analysis on Infinite Dimensional Spaces; 2009

This talk sumarises results relating Banach spaces E and F with function spaces defined on them.  First, we present to what extent does the dual space E´ determine the polynomials over E. Namely, we study under which conditions, if E´ is (isometrically) isomorphic to F´, the spaces of homogeneous polynomials over E and F are (isometrically) isomorphic. Sean Dineen introduced Nacho Zalduendo and the speaker to the problem.   The results obtained, for the scalar valued case, respond, in part, to the question posed by Días and Dineen. In order to tackle the problem we construct a natural operator between the spaces of polynomials via the Aron-Berner extension and the operator which relates the dual spaces. Later, in collaboration with Daniel Carando the vector valued case was studied. The converse, is a Banach-Stone type problem. Suppose the spaces of homogeneous polynomials over E and F are isometric, which is the relation between the underling spaces? With Christopher Boyd, the question for scalar and vector valued polynomials was discussed. We gave characterisation of such isometries in terms of the operator constructed by Lassalle and Zalduendo. This allowed us to conclude, that any isometry between spaces of homogeneous approximable polynomials on E and F is determined by an isometry between the duals E´ and F´. Also, the decomposable symmetric mappings between spaces of tensor products were described via injective operators linking the underlying spaces.