In a recent paper (13), we introduced the class of strongly E-reflexive inversesemigroups. This class was shown to coincide with the class of those inverse semigroups which are semilattices of E-unitary inverse semigroups. In particular, therefore, E-unitary inverse semigroups and semilattices of groups are strongly E-reflexive, and in fact so are subdirect products of these two types of semigroups.

The Hardy-Littlewood-Pólya inequality in question can be written in the form

Here and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean that

defines a locally integrable function Kf for which (1.1) holds.

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if

(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient

In this paper we shall examine the relationship between the numerical range of aninner derivation, and that of its implementing element.

Much of this paper is taken from the author's doctoral thesis (5) written at theUniversity of Newcastle upon Tyne with the helpful guidance of Professor B. E.Johnson. The author also acknowledges the financial support of the Science ResearchCouncil during the period of this research.

In this paper we show that for each n∈{2,3,4,5,} the topological separationproperty Tn can be decomposed

where C, N2,…,Nn are purely lattice theoretic properties with the expected im-plications holding between them.

Let

be an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

Let R be a ring and X a right R-module (all rings have identities and all modulesare unitary). The intersection of all non-zero submodules of X is denoted by μ(X). The module X is called monolithic if and only if μ(X)≠0 and in this case μ(X) is anessential simple submodule of X. (Recall that a submodule Y of X is essential if and only if Y ⊂ A ≠ 0 for every non-zero submodule A of X.) It is well known that a module X is monolithic if and only if there is a simple right R-module U such that X is a submodule of the injective hull E(U) of U. If x is a non-zero element of an arbitrary right. R-module X then by Zorn's Lemma there is a submodule Yx of X maximal with the property x ∉ Yx. It can easily be checked that X/Yx is monolithic and ⊂ Yx = 0, where the intersection is taken over all non-zero elements x of X.

Throughout this paper A denotes an operator function, holomorphic on a deletedneighborhood of a complex number λo, with values in the space ℒ(X,Y) of boundedlinear operators between two complex Banach spaces X and Y. In his survey article(7), I. C. Gohberg has defined for such an arbitrary operator function A the algebraic multiplicity RM(A;λo) and the reduced algebraic multiplicity RM(A;λo) of A at λo. In earlier papers (e.g., (8, 16)) these notions have been defined and studied for morerestricted classes of operator functions. In (8) Gohberg and Sigal treated the case when A is finite-meromorphic at λo, A(λ) is bijective for λ in some deleted neighbor-hood of λo and the constant term A0 in the Laurent expansion of A at λo is aFredholm operator. They proved that in this case

Strong summability has been studied by many authors, including Borwein and Cass (1, 2) who have studied sequence-to-sequence transforms. Here we studyintegral transforms; and due to the lack of a limitation theorem for such transforms, some results do not follow directly as in the sequence cases. The strong methods defined here can be applied to construct known and new strong summability methods. We do not give details here, but refer the reader to (3) with the suggestion that the natural scale operator method be used for Q and the named method for P. For example, with the Cesà methods, let P=(C,k,δ) and Q=(C,δ) to obtain .

We prove the following theorem, which settles a conjecture made by J. S. Lew (2, p.379).

Let A be a complex Banach algebra. By an ideal in A we mean a two-sided idealunless otherwise specified. As in (7, p. 59) by the strong radical of A we mean theintersection of the modular maximal ideals of A (if there are no such ideals we set =A). Our aim is to discuss the nature of and the relation of to A for a specialclass of Banach algebras. Henceforth A will denote a semi-simple modular annihilatorBanach algebra (one for which the left (right) annihilator of each modular maximalright (left) ideal is not (0)). For the theory of such algebras see (2) and (9).

Throughout this paper a near-ring N will satisfy the distributive law α(β + γ)=αβ + αγ for all α, β and γ in N. We shall also assume that 0α = 0 for all α in N

We prove the following theorem.