In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

For integers a, N and H > 0 write

where ϰ denotes a non-principal Dirichlet character modulo the positive integer k and e(y) denotes e2πiy. By a well-known generalisation of the Póya–Vinogradov inequality

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Let u be the Thorn class of ξ in mod 2 cohomology. Then u is stably spherical by [2] and therefore all stable cohomology operations vanish on u, showing that wi(ξ)u = Sqiu = 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong mod p conditions for every prime p.

We define two functions F(p), G(p) in terms of a third function ψ(ζ) by the equations

A standard problem in the theory of dual integral equations is to determine the function ψ(ζ) such that

when the functions f(p), g(p) are prescribed.

In a recent paper [1] we showed that there is a (1,) -correspondence between the homomorphisms of an inverse semigroup S and its normal subsemigroups. The normal subsemigroup of S corresponding to and determining the homomorphism μ of S is the inverse image under μ of the set of idempotents of Sμ and is called the kernel of the homomorphism μ. The inverse image of each idempotent of Sμ is itself an inverse semigroup [1], and each such inverse semigroup is said to be a component of the normal subsemigroup determined by μ.

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.

Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that

)O:Mabs)=(O:Mbs+l).

An element x of A is said to be integrally dependent on a relative to M if there exists n y ℕ(where ℕ denotes the set of positive integers) such that

It is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreover

ᾱ={x ɛ A: xis integrally dependent on a relative to M}

is an ideal of A called the integral closure of a relative to M and is the unique maximal member of

℘ = {b: b is an ideal of A which has a as a reduction relative to M}.

Inthis note, we present a thorough investigation of convolution operators that are naturally associated to vector measures. We characterize those convolution operators that are weakly compact and compact on Ll(G) and C(G) as well as those that are p summing, (1 ≤ p ≤ ∞) and nuclear on C(G).

The formula

was stated by Bateman ([2], p. 457); a proof is sketched in [3], p. 144. Here

the Laguerre polynomial of degree

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(ifh)‖ = 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.

This paper deals with the following problem: Can an arbitrary continuous function on [0, 1], which vanishes at the origin, be represented in some sense as a series of constant multiples of indefinite integrals of a complete orthonormal set of functions on [0,1]? Four contexts in which this problem arises naturally will be given in the introduction and the remainder of the paper will be devoted to giving a partial answer to the specific problem formulated in one of these contexts.

The purpose of this note is to present a unified treatment of the material contained in Chapter 10 of [2] on roots and logarithms of prespectral operators. Our main result gives a sufficient condition for an analytic function of a prespectral operator of class Γ to be prespectral of class Γ. A result in the opposite direction for spectral operators has been obtained by Apostol [1]. Terminology and notation in this paper are as in [2].

The theory of algebraic curves associated with subgroups of finite index in the modular group Γ is highly developed for such subgroups of Γ as may be defined by means of congruences in the ring ℤ of rational integers. The situation in he case of non-congruence subgroups of Γ, on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in [12]. Related research by A. O. L. Atkin and H. P. F. Swinnerton-Dyer began in [1].

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.

Recent work [2, 6] on subalgebras of matrix algebras leads naturally to the following situation. Let A be a C*-subalgebra of the C*-algebra B andM be a weakly closed *-subalgebra of the von Neumann algebra N. Consider the following Conditions.

Condition 1. For every b≠ 0 in B there exists a ∈ A such that O≠ab ∈ A.

Condition 2. For every b∈B there exists a ≠ 0 in A such that ab ∈ A.

If we replace A by M and B by N in Conditions 1 and 2 we get von Neumann algebra versions which we shall call Condition 1'and Condition 2'. Clearly Condition 1 implies Condition 2, and both conditions suggest that A is some kind of weak ideal of B. This paper explores the extent to which this is true. The paper grew out of the author's attempts [1, 3] to generalize the Stone-Weierstrass theorem to C*-algebras.

Throughout the paper n denotes a fixed positive integer unless otherwise specified. Let B = Bn denote the open unit ball of ℂn and let S = Sn denote its boundary, the unit sphere. The unique rotation-invariant probability measure on 5 will be denoted by σ = σn. For n = l, we use more customary notations D = B1, T = S1 and dσ1= dθ/2π. The Hardy space on B, denoted by H2(B), is then the space of functions f holomorphic on B for which

Let s = s(a1, a2,...., ar) denote the number of integer solutions of the equation

subject to the conditions

the ai being given positive integers, and square brackets denoting the integral part. Clearly s (a1,..., ar) is also the number s = s(m) of divisors of which contain exactly λ prime factors counted according to multiplicity, and is therefore, as is proved in [1], the cardinality of the largest possible set of divisors of m, no one of which divides another.

Let {ak} be a sequence of non-negative real numbers satisfying a1 = l and

(1)

Brannan [1] proved that the function

(2)

is close-to-convex univalent in the unit disc D. The example

shows that the conclusion in Brannan's theorem is sharp in that sense that “close-to-convex” cannot be replaced by the stronger one: “starlike”. It is therefore of interest to see which additional condition can guarantee this stronger conclusion.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.