Let H be a real Hilbert space and let A: H→H be a nonlinear operator such that A(0) = 0. We consider the eigenvalue problem

Recall that λ0 ε ℝ is said to be a bifurcation point for (1.1) if every neighbourhood of (λ0, 0) in ℝ × H contains solutions of (1.1).

Let Pn be the semigroup of all partial transformations on the set Xn = {1,…, n}. As usual, we shall say that an element α in Pn is of type (k, r) or belongs to the set [k, r] if |dom α|=k and |lim α|. The completion α* of an element α ∈ [n − 1, n − 1] is an element in [n, n] defined by

where {i} = Xn∖dom α and {j} = Xn∖im α.

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

The following work establishes a new proof of the theorem: Every archimedean ordered group is abelian. This theorem has been proved differently by many authors. It was first proved by O. Hölder [2]. A second proof has been given by H. Cartan [1]: he uses the topology which is naturally introduced in the group by its order.

Two numerical characterizations of commutativity for C*-algebra (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in (Theorem 2), and the other the numerical index of (Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in is 0, then is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.

The object of this paper is to find the solutions of some dual equations involving the inverses of certain Mellin type transforms that were first introduced by D. Naylor in his paper [1]. Because these transforms are relatively unknown we shall begin by defining them and making a note of some of their properties. The main result is contained in the following theorem.

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.

Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.

In Schinzel [1] the following interesting question is asked: does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible in ℚ[x] which has at most K terms? The only known result appears to be that of Mrs. H. Smyczek, given in the above paper, that if K exists, then K ≥ 6. We here extend this bound to K ≥ 8 by exhibiting a trinomial in ℤ[x] which splits into the product of two irreducible factors, each having 8 terms.

The formulae

where γ = ½ (α + β + l), and

where γ = 1 + β, were given by Gauss (Ges. Werke, iii, pp. 225, 226). It is here proposed to find the corresponding expressions for the hypergeometric function when γ has general values (not zero or negative integral). These will be derived in section 2 by applying Lagrange's expansion

where

and that root of equation (4) in x is taken which is equal to λ when w = 0. Two generalisations of Whipple's Transformation will be obtained in section 3.

We show that the commutative V*-algebras with relatively weakly compact unit spheres are those that are representable by means of hermitian spectral measures. This provides a more unified approach to the results of [15], and allows us to generalise some of them.

All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B if

A simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part ½(C + C*) of an operator C.

We show that continuous bilinear forms on spaces of continuous functions can be approximated by norm attaining bilinear forms.

The study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. Subsequently these algebras were called distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [12]. In [9], M. S. Goldberg extended this theory and described the injective algebras in the subvarieties of the variety O of distributive Ockham algebras which are generated by a single subdirectly irreducible algebra. The aim here is to investigate some elementary properties of injective algebras in join reducible members of the lattice of subvarieties of Kn,1 and to give a complete description of injectivealgebras in the subvarieties of the Ockham subvariety defined by the identity x Λ f2n(x) = x.

K. Shoji has pointed out to me that construction [1] does not always yield a completion. In the notation of [1], the homomorphism from the strong semilattice of cancellative semigroups S to its purported completion T in Abian's order is not always a monomorphism. The difficulty arises when there is eɛ E, e=sup{e'ɛEe'<e>e} but {‐e,e'}e' is not faithful, i.e. there are x, y with x¬y in Se such that φe,e'(x)=φe,e'(y) for all e'<e. A modification of the construction saves all parts of Theorem 1 except the fact that the new embedding S⊆T need not preserve suprema existing in S; it does if S is a semilattice of groups. The sequel [2] also needs amodification in the form of an additional hypothesis.