Collectively compact sets of linear operators

Abstract : A set of linear operators from one normed linear space to another is collectively compact iff the union of the images of the unit ball is precompact. Several criteria for sets of operators to be collectively compact are given. It is shown that a compact set of compact operators is collectively compact, but not conversely. For a set H of compact normal operators on a Hilbert space, H is collectively compact iff H is totally bounded iff H* + (K* : K epsilon H) is collectively compact. For any set H of compact operators on a Hilbert space, H is totally bounded iff H and H* are collectively compact. The proof of these assertions depends on some interesting properties of the spectral decomposition of the operators. (Author)