Almost-finite, compact, and inessential operators

Abstract

We are concerned here with four closely related ideals in the Banach algebra of endomorphisms of a Banach space (cf. [1, pp. 51 ff. ]). T he term operator will be restricted to members of this algebra. Ideals are assumed to be two-sided. The most tractable operators, in the sense of actual computation, are those whose range is spanned by a finite set of elements. Such operators can be called, briefly, finite operators (the term degenerate operator is also used). The finite operators form an ideal which is not closed in any conventional topology. The other three ideals mentioned above are closed extensions of the ideal of finite operators. An operator will be called almost-finite if it is the uniform limit of a sequence of finite operators. An operator will be called compact if it takes bounded sets into sets with a compact closure. The term completely continuous operator is also used for these operators. An operator will be called inessential if its image in the quotient algebra of the algebra of all operators over the almost-finite operators belongs to the (Jacobson) radical. For a definition of the radical see [1, Chapter 24], or below in the proof of Theorem 1. It follows at once from these definitions that the sets of almostfinite, compact, and inessential operators are all ideals. M\loreover, each of these ideals is closed; the first by definition, the second by a well known theorem (cf. [1, p. 49]), and the third because the radical of a Banach algebra is closed and the mapping into a quotient algebra is continuous. It is well known that almost-finite operators are compact (cf. [1, p. 49]). In many cases these two ideals are equal; however, it is still an open question whether or not this is true in general. It is an immediate consequence of Theorem 1 that compact operators are inessential. In some spaces (for example, separable Hilbert space which has only one nontrivial closed ideal) inessential operators are compact; there are others in which they are not equal (cf. the introduction of [2]), but this question does not appear to have been investigated in detail.