In this note we give some negative results concerning the question of whether certain integrable functions on the unit circle with mean value zero are expressible as finite sums of differences g – gα of integrable functions g, where gα denotes the translate of g by α.

We construct a semigroup S with an epimorphically embedded proper subband U. The band U furnishes an example of a regular semigroup which is not saturated, thus answering a question posed by Hall, Semigroup Forum (to appear).

In this paper it is shown that if p(z) is a polynomial of degree n satisfying then

The result is best possible.

A Banach algebra A with radical R is said to have property (S) if the natural mapping from the algebraic tensor product A ⊗ A onto A2 is open, when A ⊗ A is given the protective norm. The purpose of this note is to provide a counterexample to Zinde's claim that when A is commutative and R is one dimensional the fulfillment of property (S) in A implies its fulfillment in the quotient algebra A/R.

Let (ωi, σi, μi.) be two positive finite measure spaces, V a non-zero Hilbert space, and 1 ≤ p < ∞, p # 2. In this article it is shown that each surjective linear isometry between the Bochner spaces induces a Boolean isomorphism between the measure algebras , thus generalizing a result of Cambern's for separable Hilbert spaces.

This Banach–Stone type theorem is achieved via a description of the Lp-structure of .

It is shown that a transformation group (X, T, Π) is of strong characteristic 0 if and only if it is of P-strong characteristic 0 for some replete semigroup P in the phase group, provided that all orbit closures are compact. It is shown also that, under certain conditions, (X, T, Π) is of P-strong characteristic 0 if and only if (X × X, T, Π × Π) is Liapunov stable.

Let (T, T, μ) be a σ-finite measure space and X a Suslin space. Let A be a class of normal integrands on T × X. We discuss the existence of an essential supremum of A, namely, a normal integrand l with

where A0 is a countable subclass of A, and, for each α ∈ A,

In this way we obtain an extension of the classical essential supremum concept. The applications include a result on measurable selectors of nonmeasurable multifunctions.

We show that a semigroup satisfying a heterotypical identity of which at least one side has no repeated variable is saturated and find sufficient conditions on a homotypical identity which is not a permutation identity and of which at least one side has no repeated variable, to ensure that any semigroup satisfying the identity is saturated.

Let X, Y be rearrangement invariant spaces and let M = M(Y, X) be the space of all multipliers of Y into X. It is shown that if X = YM and some technical conditions are satisfied, then the K-functional K(t, f, X, Y) is equivalent to the expression

where ψ is the inverse of the fundamental function ϕM of M, defined by

Let R be a ring with Krull dimension α and let τα be defined on a module MR by . Equivalent conditions for τα to be a torsion radical are given. The relationship between τα-criticals and α-criticals is also explored.

Sufficient conditions which are verifiable in a finite number of arithmetical steps are derived for the existence and global asymptotic stability of a feasible steady state in an integro-differential system modelling the dynamics of n competing species in a constant environment with delayed interspecific interactions. A novel method involving a nested sequence of “asymptotic” upper and lower bounds is developed.

The adjoint of a composition operator CT on the L2-space of an atomic measure is computed and a characterization for an operator to be a composition operator is given in this short note. The dimensions of kernel and co-kernel of CT are calculated in order to characterise Fredholm composition operators. Finally, essentially normal composition operators are studied on l2.

In this paper, we consider an optimal control problem involving second-order hyperbolic systems with boundary controls. Necessary and sufficient conditions are derived and a result on the existence of optimal controls is obtained. Also, a computational algorithm which generated minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated.

Techniques from martingale theory are used to obtain the Skorokhod embedding of reverse martingales in Brownian motion. This result is then used to obtain a functional law of the iterated logarithm for reverse martingales.

The notion of accretiveness for multi-valued nonlinear maps is defined in locally convex spaces and it is used to obtain a locally convex space version of a result of M.G. Crandall and J.A. Nohel.

The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.