Hahn-Banach operators

Abstract

We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces (X, Y ) such that there exists a Hahn-Banach operator T : X → Y of rank k. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman. Everywhere in this paper we consider only real linear spaces. Our starting point is the classical Hahn-Banach theorem ([H], [B1]). The form of the Hahn-Banach theorem we are interested in can be stated in the following way. Hahn-Banach Theorem. Let X and Y be Banach spaces, T : X → Y a bounded linear operator of rank 1 and Z a Banach space containing X as a subspace. Then there exists a bounded linear operator T : Z → Y satisfying (a) ||T || = ||T ||; (b) T x = Tx for every x ∈ X. Definition 1. An operator T : Z → Y satisfying (a) and (b) for a bounded linear operator T : X → Y is called a norm-preserving extension of T to Z. The Hahn-Banach theorem is one of the basic principles of linear analysis. It is quite natural that there exists a vast literature on generalizations of the HahnBanach theorem for operators of higher rank. See papers by G. P. Akilov [A], J. M. Borwein [Bor], B. L. Chalmers and B. Shekhtman [CS], G. Elliott and I. Halperin [EH], D. B. Goodner [Go], A. D. Ioffe [I], S. Kakutani [Kak], J. L. Kelley [Kel], J. Lindenstrauss [L1], [L2], L. Nachbin [N1] and M. I. Ostrovskii [O], representing different directions of such generalizations, and references therein. There exist two interesting surveys devoted to the Hahn-Banach theorem and its generalizations; see G. Buskes [Bus] and L. Nachbin [N2]. We shall use the following natural definition. Definition 2. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. Received by the editors February 9, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 47A20.