We prove that intermediate Banach spaces $\mathcal{A}$ and $\mathcal{B}$ with respect to arbitrary Hilbert couples $\bar {H}$ and $\bar {K}$ are exact interpolation if and onlyif they are exact K-monotonic, i.e. the condition $f^0 \in \mathcal{A}$ and the inequality $K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$ , imply g⁰∈B and ‖g⁰‖B≤‖f⁰‖_A (K is Peetre’s K-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator $T:\bar {H} \to \bar {K}$ such that Tf⁰=g⁰, and $K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$ . Verifying the latter property, it suffices to consider the “diagonal case” where $\bar {H} = \bar {K}$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.

We prove that if T is a strictly singular one-to-one operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of X such that Z∩Y is infinite dimensional, Z contains orbits of T of every finite length and the restriction of T to Z is a compact operator.

Completely invariant components of the Fatou sets of meromorphic maps are discussed. Positive answers are given to Baker’s and Bergweiler’s problems that such components are the only Fatou components for certain classes of meromorphic maps.

In this paper we study the spectral counting function for the weighted p-laplacian in one dimension. First, we prove that all the eigenvalues can be obtained by a minimax characterization and then we show the existence of a Weyl-type leading term. Finally we find estimates for the remainder term.

We consider the operator, f(Δ) for Δ the Laplacian, on spaces of measures on the sphere in R^d, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the L²-norm of f(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on R^d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

We find the exact value of the best possible constant associated with a covering problem on the real line.

We show that if E is a complex Banach space which contains no subspace isomorphic to l₁, then each infinite dimensional subspace of E′ contains a normalized sequence which converges to zero for the weak star topology.