In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

In a recent paper [7], Rooney used a technique involving the Mellin transform to obtain solutions in certain spaces ℒμ, ρ of an integral equation which had been studied previously by Šub-Sizonenko [9]. The integral equation in question can be written as

where I denotes the identity operator and G0.1/2 is given by

with the inversion formula obtained by Rooney taking the form

Rooney verified that (1.1) and (1.2) formed an inversion pair in ℒμ, ρ for 1 ≤ p < ∞ and μ > 0.

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?

Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

For each natural number n, let un(x)=(1—cos nx)/πnx2(xɛℝ). It is well–known that a bounded continuous function f on the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined by

form a Cauchy sequence in the space L1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the space L1(ℝ) is complete with respect to convergence in mean.

R. C. James [2] (or see p. 7ff of [3]) gave a useful representation of the bidual of any space with a shrinking basis. This note gives a representation of the bidual of any space with a basis.

Our notation follows that of [3], where undefined terms can be found. Let {en} be a basic sequence with coefficient functionals {fn}. We will assume {en} is bimonotone; that is

. The space {en}LIM is the set of scalar sequences {an} so that ∥{an}∥ = . We will abuse notation and squate such {an} with the formal sum Σ anen.

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?

It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.

Peter May kindly tells me that the proof of the Nishida relations in §8 of this paper works only for E∞ spaces, not for H∞ ring spectra. The reason is that there is no suitable “diagonal” map d for H∞ ring spectra. The result is correct all the same (it is not mine), and its formulation in §3 is also correct.