Journal of Operator Theory

Volume 32, Issue 1, Summer 1994 pp. 31-46.

A characterization and equations for minimal projections and extensions

Authors: B.L. Chalmers (1) and F.T. Metcalf (2)
Author institution: (1) Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.
(2) Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.

Summary: In this paper we develop first a theory providing a characterization theorem (Theorem 1) for finite-rank minimal projections. In order to demonstrate the usefulness of this characterization we provide several examples. More generally, the theory characterizes operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace. Secondly, a characterization theorem (Theorem 2) is given for minimal linear operators in a general setting. This setting includes, as examples, minimal and co-minimal projections, optimal recovery and linear estimation, linear n-widths, and best linear approximation to continuous proximity maps. These characterization theorems lead to concrete equations from which the minimal operators can be obtained and, in some important cases, described geometrically.

Contents Full-Text PDF