# 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.

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