In [9], R. S. Phillips gave a compactness criterion for subsets of a Banach space X, namely, that if (Ts) is a net of compact operators converging strongly to the identity then a bounded set A ⊆ X is relatively compact if and only if (Ts) converges uniformly on A to the identity. He then used this general result to give concrete compactness criteria in some specific spaces and to investigate compactness of operators. In this paper, we develop these ideas in two directions: firstly, to demonstrate that this method can be used as a unifying tool to derive many classical compactness criteria originally proved using other techniques, and secondly to extend the method from giving a criterion for compactness to giving a generalised measure of noncompactness. The conditions for compactness that result from this approach are not new (though they are possibly given in slightly more generality than can easily be found in the literature), but they all arise from a single, simple, and usually elementary, approach.

Throughout, vector operations applied to sets are all defined pointwise, so if A and B are subsets of a vector space, x is a vector and λ is a scalar, then

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The ball centred at x with radius r in a normed space X is denoted BX(x; r). More generally, if A is a subset of a normed space X then the r-neighbourhood BX(A; r) of A is defined by

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It is sometimes convenient to combine these notations, for example to use x+rBX(0; 1) in place of BX(x; r) or A+rBX(0; 1) in place of BX(A; r).

The ball measure of noncompactness in a normed space X will be denoted by βX: if A is a bounded subset of X then βX(A) is the infimum of r such that A can be covered with finitely many balls of radius r.

Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn([Copf ]) onto Mn([Copf ]) which preserves the spectrum has one of the forms Φ(x) = axa−1 or Φ(x) = atxa−1, for some invertible matrix a. (Incidentally the hypothesis that Φ is onto is superfluous; see Proposition 2.1(i).) This result was extended by J. Dieudonné [17] supposing Φ onto and satisfying SpΦ(x) ⊂ Sp x, for every n × n matrix x.

Several results of M. Nagasawa, S. Banach and M. Stone, R. V. Kadison, A. Gleason and J. P. Kahane and W. Żelazko led I. Kaplansky in [22] to the following problem: given two Banach algebras with unit and Φ a linear mapping from A into B such that Φ(1) = 1 and SpΦ(x) ⊂ Sp x, for every x ∈ A, is it true that Φ is a Jordan morphism? With this general formulation, this question cannot be true (see [2], p. 28).

The paper considers the heat kernel KΩ(t, x, y) of the operator – Δ on a proper Euclidean domain Ω, with Dirichlet boundary conditions. A general pointwise lower bound for KΩ, which is valid for t larger than a suitable t0(x,y), is proved (the short-time behaviour being well understood). The resulting non-Gaussian bounds describe simultaneously both the case of bounded domains and the case, modelled on the half-space example, of domains which satisfy a twisted infinite internal cone condition. Bounds for the Green's function are given as well.

The semi-classical asymptotic behaviour of the Riesz means of a distribution of eigenvalues is investigated at a non-critical energy level. For Schrödinger type operators, the second term related to the periodic trajectories of the classical Hamiltonian is obtained. This oscillating term is explained for Riesz means of order 1 with the aim of discussing a Lieb–Thirring conjecture.

The notion of stable isomorphism of bundle gerbes is considered. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H3(M, ℤ). Stable isomorphism sheds light on the local theory of bundle gerbes and enables a classifying theory for bundle gerbes to be developed using results of Gajer on B[Copf ]× bundles.

The bordism class of a self-transverse immersion f:Mn−k ℝn corresponds to an element α of the homotopy group πnΩ∞Σ∞MO(k). It is explained how the ℤ/2 Hurewicz image h(α)∈Hn(Ω∞Σ∞MO(k);ℤ/2) may be used to determine the characteristic numbers of the self-intersection manifolds Δr(f) of the immersion f.

Let {Xi}i[ges ]1 be a sequence of independent random variables taking the values ±1 with the probability ½, and let us set Sn = X1 + X2 +…+ Xn. A classical theorem of Hardy and Littlewood (1914) says that, for any C > 0 and for all n large enough, we have

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with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequality

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for any ε > 0. In view of Khinchin's result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this paper, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2).

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. The paper explains a method of computing the excellent component of the character variety of tunnel number 1 knots.

Let T be a self-affine tile in ℝn defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in ℝ2, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.

The paper is concerned with finding sufficient smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic diffusion to imply the existence and smoothness of a taboo density.

The singular homology groups of compact CW-complexes are finitely generated, but the groups of compact metric spaces in general are very easy to generate infinitely and our understanding of these groups is far from complete even for the following compact subset of the plane, called the Hawaiian earring:

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Griffiths [11] gave a presentation of the fundamental group of ℍ and the proof was completed by Morgan and Morrison [15]. The same group is presented as the free σ-product [smashp ]σℕℤ of integers ℤ in [4, Appendix]. Hence the first integral singular homology group H1(ℍ) is the abelianization of the group [smashp ]σℕℤ. These results have been generalized to non-metrizable counterparts ℍI of ℍ (see Section 3).

In Section 2 we prove that H1(X) is torsion-free and Hi(X) = 0 for each one-dimensional normal space X and for each i [ges ] 2. The result for i [ges ] 2 is a slight generalization of [2, Theorem 5]. In Section 3 we provide an explicit presentation of H1(ℍ) and also H1(ℍI) by using results of [4].

Throughout this paper, a continuum means a compact connected metric space and all maps are assumed to be continuous. All homology groups have the integers ℤ as the coefficients. The bouquet with n circles [xcup ]nj=1Cj is denoted by Bn. The base point (0, 0) of Bn is denoted by o for simplicity.

Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm [mid ]·[mid ]:

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It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V[rarrhk ]H[rarrhk ]V*, t [ges ] 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·)[ratio ]H×R+→R1 which satisfies the following conditions:

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where [Lscr ] is the infinitesimal generator of the Markov process Xt and ci, ki, μi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, μi = 0) are considered.