N. Yannakakis | Searcharxiv

arXiv2008

Subspaces with a common complement in a Banach space

We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution S on X exchanging the two subspaces, such that I+S is bounded from below on their union. Moreover we show that, in a separable Hilbert space, the only pairs of subspaces with a common complement are those which are either equivalently positioned or not completely asymptotic to one another. We also obtain characterizations for the existence of a common complement for subspaces with closed sum.

math.FA↗

arXiv2008

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.

math.FA↗

arXiv2008

Generalizations of the Lax-Milgram theorem

We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations.

math.FA↗