In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identity

where f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X then

is a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollák [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

Let C(T) be the Banach algebra of all complex-valued continuous functions on the compact set T of all complex numbers with modulus one. As usual we may suppose that A is embedded in C(T), where A is the disc algebra, i.e., the algebra of all complex-valued functions f(λ) continuous for |λ| ≦ 1. and analytic for |λ| < 1. We set Mλ = {f ∈ A: f(λ) = 0} and Mλ = {f ∈ A: f(λ) ≧ 0}.

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

Peller [4, 5] has proved that a Hankel operator S on the Hardy space H2 is in the trace class if and only if with h analytic on the open unit disc Dand with its second derivative belonging to the Bergman space L1a. This theorem does not include an estimate for the trace class norm ∥S∥1, of the operator in terms of the symbol function. In fact it is clear that cannot give an estimate for since the first two terms in the coefficient sequence of the Hankel operator have been removed by differentiation.

Riemann's method for solving the Cauchy problem for hyperbolic differential equations in two independent variables has been extended in a number of papers [4], [5], [2] to the wave equation in space of higher dimensions. The method, which consists in the determination of a so-called Riemann function, hinges on the solution of a characteristic value problem. Accordingly, if Riemann's method is to be used in solving a characteristic value problem, one will have to consider another characteristic value problem and thus the process becomes circular. This difficulty was first overcome by Protter [7] in solving the characteristic value problem for the wave equation in three variables. There he employed a variation of Riemann's method developed by Martin [5]. Martin's result was later extended by Diaz and Martin [2] to the wave equation in an arbitrary number of variables. This made it possible to extend Protter's result to the wave equation in space of higher dimensions [8].

Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.

Let A be a Banach *-algebra. We assume neither that A has an identity element, nor that the involution on A is continuous.

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).

The easiest way to construct automorphic functions is by means of the Poincaré series. If G is a Kleinian group with ∞ an ordinary point of G and if k ≧ 4, then

where Vz=(az+b)/(cz+d) and ad-bc=1. The convergence of this series is the crucial step in showing that the Poincaré series converges and is an automorphic form on G If ∞ is a limit point of ∞ the series in (1) may diverge and one can derive automorphic forms on ∞ from the Poincaré series of some conjugate group. These constructions are described in greater detail in /3, pp. 155–165].

In a paper published recently in this Journal, Watson obtained a reduction formul for the integral

He showed that this reduction formula could be written in the form:

If

then

where

and 2σ = α + β + γ + δ, 2θ3 = α − β − γ + δ.

By applying Gauss's Theorem it can be seen that, if n is a positive integer and α is not integral,

so that

In section 2 it will be proved that, if

Where α and β are not integers

Where μ and M are constants independent of n.

In contrast to single-product pricing models, multi-product pricing models have been much less studied because of the complexity of multi-product demand functions. It is highly non-trivial to construct a multi-product demand function on the entire set of non-negative prices, not to mention approximating the real market demands to a desirable accuracy. Thus, many decision makers use incomplete demand functions which are defined only on a restricted domain, e.g. the set where all components of demand functions are non-negative. In the first part of this paper, we demonstrate the necessity of defining demand functions on the entire set of non-negative prices through some examples. Indeed, these examples show that incomplete demand functions may lead to inferior pricing models. Then we formulate a type of demand functions using a Nonlinear Complementarity Problem (NCP). We call it a Complementarity-Constrained Demand Function (CCDF). We will show that such demand functions possess certain desirable properties, such as monotonicity. In the second part of the paper, we consider an oligopolistic market, where producers/sellers are playing a non-cooperative game to determine the prices of their products. When a CCDF is incorporated into the best response problem of each producer/seller involved, it leads to a complementarity constrained pricing problem facing each producer/seller. Some basic properties of the pricing models are presented. In particular, we show that, under certain conditions, the complementarity constraints in this pricing model can be eliminated, which tremendously simplifies the computation and theoretical analysis.

Let K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

Let M be an n-dimensional totally real minimal submanifold in CPn. We prove that if M is semi-parallel and the scalar curvature τ, , then M is an open part of the Clifford torus Tn ⊂ CPn. If M is semi-parallel and the scalar curvature τ, , then M is an open part of the real projective space RPn.

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.