The paper shows that, if the operator T[ratio ]A(γ)→B(γ) is compact for almost every γ∈Γ, then T[ratio ]F(Ā)→F([Bmacron]) is compact when F([Xmacron])=(X)SK or F([Xmacron])=(X)SJ is the interpolation functor constructed for infinite families of Banach spaces and S satisfies certain conditions.

The paper proves two univalence criteria involving harmonic mappings in multiply connected domains. These generalise a classical result of Radó, Kneser and Choquet and a recent result of Bshouty and Hengartner.

We characterize the minimal isometric dilation of a non-commutative contractive sequence of operators as a universal object for certain diagrams of completely positive maps. A non-spatial construction of the minimal isometric dilation is also given, using Hilbert modules over C*-algebras.

It is shown that the non-commutative disc algebras [Ascr ]n (n[ges ]2) are the universal algebras generated by contractive sequences of operators and the identity, and C*(S1, …, Sn) (n[ges ]2), the extension through compact operators of the Cuntz algebra [Oscr ]n, is the universal C*-algebra generated by a contractive sequence of isometries. It is also shown that the algebras [Ascr ]n and C*(S1, …, Sn) are completely isometrically isomorphic to some free operator algebras considered by D. Blecher. In particular, the universal operator algebra of a row (respectively column) contraction is identified with a subalgebra of C*(S1, …, Sn). The internal characterization of the matrix norm on a universal algebra leads to some factorization theorems.

It is shown that the composition operator w[map ]w∘u generated by an inner function u is a self-map of the Muckenhoupt class A2, but that it need not preserve the Muckenhoupt classes Ap for p∈(1, 2)∪(2, ∞).

We prove Landesman–Lazer type existence conditions for the solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equations of the form x″+cx′+f(t, x)=0. The proofs use upper and lower solutions techniques together with some limiting arguments.

Combining Le Cam's operator approach with Bergström's identity, estimates are obtained for convolutions of compound measures. The results are exemplified by consideration of asymptotic expansions for Compound Poisson approximation in Le Cam's theorem, approximation of nearly homogeneous portfolios and convergence to Compound Poisson laws.

Suppose that K is a closed, total cone in a real Banach space X, that A[ratio ]X→X is a bounded linear operator which maps K into itself, and that A′ denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x0≠0 and m[ges ]1 with Am(x0) =rmx0 (or, more generally, that there exist x0∉(−K) and m[ges ]1 with Am(x0) [ges ]rmx0). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u∈K−{0} and θ∈K′−{0} with A(u)=ru, A′(θ)=rθ and θ(u)>0. The support boundary of K is used to discuss the algebraic simplicity of the eigenvalue r. The relation of the support boundary to H. Schaefer's ideas of quasi-interior elements of K and irreducible operators A is treated, and it is noted that, if dim(X)>1, then there exists an x∈K−{0} which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which X is a Hilbert space and A is self-adjoint; the theorems in the paper generalize several of Toland's propositions.

Let ω=(ωn)n[ges ]1 be a log concave sequence such that lim infn→+∞ ωn/nc>0 for some c>0 and ((log ωn)/nα)n[ges ]1 is nonincreasing for some α<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ∥T−n∥=O(ωn) as n tends to +∞ with [sum ]n[ges ]11/(n log ωn)=+∞, then T is unitary. On the other hand, if [sum ]n[ges ]11/(n log ωn)<+∞, then there exists a (non-unitary) contraction T on the Hilbert space such that the spectrum of T is a Carleson set, ∥T−n∥=O(ωn) as n tends to +∞, and lim supn→+∞ ∥T−n∥=+∞.

It is shown that if S is an aperiodic random walk on the integers, S* is the Markov chain that arises when S is killed when it leaves the non-negative integers, and H+ is the renewal process of weak increasing ladder heights in S, then there is a 1[ratio ]1 correspondence between functions which are non-negative and superregular for S* and H+. This allows all the regular functions for S* to be described, and thus a result due to Spitzer to be completed for the recurrent case. This result is then applied to give a ratio limit theorem for Px(τ* =n)/P0{τ*=n}, where τ* is the lifetime of S*, in the case when S drifts to −∞, and the right-hand tail of its step distribution is ‘locally sub-exponential’.

The paper gives a condition that is sufficient for E. Bishop's property (β), and it applies this result to the Cesàro operator on the Hardy spaces Hp, 1<p<∞. In the process, it identifies the approximate point spectrum of the Cesàro operator.

The discrete spectrum of the Schrödinger operator is studied for a system of three identical particles with short-range interactions in a homogeneous magnetic field. All the two-particle subsystems are supposed to be unstable. Finiteness of the discrete spectrum is established under some assumptions about the solutions of the corresponding two-particle Schrödinger equation.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.

The paper gives simple necessary and sufficient conditions for a closed 4-manifold to be homotopy equivalent to the mapping torus of a self homotopy equivalence of a PD3-complex. This is a homotopy analogue of the Stallings and Farrell fibration theorems available in other dimensions. The paper also considers 4-manifolds which admit a geometry of Euclidean factor type and complex surfaces which fibre over S1.

The paper shows that the times spent in [0, +∞) by certain processes Y which are defined by perturbations of Brownian motion involving reflection at maxima and minima are beta distributed. This result relies heavily on Ray–Knight theorems for such perturbed Brownian motions.