Frank Smithies was born in Edinburgh, on 10 March 1912. His father, also Frank Smithies, and his mother, Mary (née Blakemore) had met through their involvement in the socialist movement. They had two children, Frank and his sister Violet, some four years younger.

In an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This is used to estimate the exponential rank of certain algebras of operator-valued continuous functions.

Let T be a compact linear operator acting in a complex Hilbert space H. Then there is a simple resolution of the identity {Eλ} in H which reduces T (for a proof of this statement, and for definitions of the terms used, see (l), section 5, especially Theorem 6). In the present paper we give constructions for the resolvent of T, and for a complete set of principal (that is, eigen and adjoined) vectors, in terms of T, {Eλ}, and the diagonal coefficients {αλ} of T. These constructions require no technique more involved than the expansion of a resolvent operator in a Neumann series.

This note is concerned with a question arising from some work of D. R. Smart (2). In the introduction we indicate the nature of the problem. Notations will be explained only in as far as they differ from those of (2).

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.

The Riesz–Schauder theory for the equations

where T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.

1·1. Throughout this paper, all functions considered are assumed to be measurable, and an equation between two functions indicates equality almost everywhere. When the space Lp(a, b) is considered, the interval (a, b) may be finite, semi-infinite, or infinite.