The purpose of this paper is to provide a vector version of the characterisations of the multipliers for L1(G) (see (9), p. 2). This problem was considered by Akinyele (1). However, the Banach algebras involved in that paper are commutative semi-simple Banach algebras and the proof of the main theorem seems to be incomplete. Indeed we give at the end of this paper an example which shows that statements (i) and (ii) of Theorem 3.2 of Akinyele ((1), p. 490) are not equivalent in general.

In (3) it is shown that, for a locally compact abelian group G and x∈G, δx has a logarithm in M(G) if and only if x has finite order. Since M(G) can be identified with the multipliers of L1(G), one might expect a similar result for the algebras of multipliers on Lp(G) for 1 < p < ∞. However, in contrast, it is shown in (2) that for a locally compact abelian group G and 1 < p < ∞, every translation operator on Lp(G) has a logarithm in the multiplier algebra. Here we consider whether the same results are true for non-abelian groups.

The Carter subgroups of a finite soluble group may be characterised either as theself-normalising nilpotent subgroups or as the nilpotent projectors. Subgroups with properties analogous to both of these have been considered by Newell (2, 3) in the class of -groups. The results obtained are necessarily less satisfactory than in the finite case, the subgroups either being almost self-normalising (i.e. having finite index in their normaliser) or having an almost-covering property. Also the subgroups are not necessarily conjugate but lie in finitely many conjugacy classes.

If G is a group, let

m(G)

be the least cardinal such that G may be generated by m(G) elements, and let

m,(G)

In (2) Hickin and Phillips establish various results connecting the ideas of local systems and serial subgroups in group theory. In (4) Petty shows that if a group has a local factor system of -groups, then it may be embedded in an ultraproduct of -groups and he uses this result to extend some local theorems which were originally due to Mal'cev (3) and at the same time providing an elementary proof. We shall show that Petty's method may be used to prove results about seriality and local systems.

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the condition

for all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

In this paper, we solve the following dual integral equations

where δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.

The kernel function method of Bergman and Schiffer (see (1) and (2)) has recently been used by Colton and Gilbert (see (3) and (4)) in connection with approximation theory and the numerical treatment of elliptic differential equations. In (5) Gilbert and Weinacht have successfully extended the kernel function method to elliptic systems of differential equations. Essential to their work is the concept of a matrix kernel satisfying the reproducing property. This reproducing kernel is defined initially as the difference of the Neumann matrix and the Dirichlet matrix. Thus actually to obtain the kernel matrix from this definition one has to solve both a Neumann problem and a Dirichlet problem. In view of this restriction, Gilbert and Weinacht derive an ingenious representation for the reproducing kernel in terms of purely geometric quantities which are obtained directly from the fundamental matrix for the differential system.

It is a most implausible fact that a one-to-one operator from c0 into a Banach space which maps the unit ball of c0 onto a closed set is necessarily an isomorphism.

In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semi-embeddings defined on C(X), X compact.

Throughout this paper all near-rings considered will be zero-symmetric and left distributive. All groups will be written additively, but this does not imply commutativity. The near-ring of all zero-fixing maps of a group V into itself will be denoted by Mo(V). If N is a near-ring withan identity and α ≠ 1 is an element of N such that α2 = 1, then α will be called an involution of N. Let V be a group. An involution a of Mo(V) will be called an involution on V.

In this article we prove that in every infinite dimensional separable Fréchet space there is a dense barrelled subspace which is not the inductive limit of Bairehyperplane spaces.

In this article some properties on subseries convergent sequences in locally convex topological vector spaces are studied and some open questions of (2) are answered.

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

If A is a complex Banach algebra (not necessarily unital) and x∈A, σ(x) will denote the spectrum and spectral radius of x in A. If I is a closed two-sided ideal in A let x + I denote the coset in the quotient algebra A/I containing x. Then

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation