The purpose of this paper is to introduce and study the notion of a vector-valued π-invariant mean associated to a unitary representation π of a locally compact groupG on S, a self-adjoint linear subspace containing I of B(Hπ). We obtain, among other results, an extension theorem for π-invariant completely positive maps and π-invariant means which characterizes amenability of G. We also study v...

We prove that one-dimensional Schrodinger operators with even almost periodic potential have no point spectrum for a dense Gδ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a dense Gδ in Θ even if the frequency is an irrational with good Diophantine properties.

Let {r(Z):Z>0} be a C0-semigroup on a Banach space X with generator A , and let x € X. If a (A) n ;'R is empty and t »-> T(t)x is uniformly continuous, then ||7"(Z)jc|| —> 0 as t —» oo . If the semigroup is sun-reflexive, o(A)CiiR is countable, Pa(A)DiS. is empty, and 1 >-> T(t)x is uniformly weakly continuous, then T(t)x —► 0 weakly as t —» oo . Questions of almost periodicity and of stabiliza...

The spectrum containment of almost periodic solution of second-order neutral delay differential equations with piecewise constant of argument EPCA, for short of the form x t px t − 1 ′′ qx 2 t 1 /2 f t is considered. The main result obtained in this paper is different from that given by some authors for ordinary differential equations ODE, for short and clearly shows the differences between ODE...

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set X in a metric space, it is proved the existence of an almost periodic sequence {ψk}k∈Z such that {ψk; k ∈ Z} = X and ψk = ψk+lq(k), l ∈ Z for all...

In this paper, the averaging principle for quasi-geostrophic motions with rapidly oscillating forcing is proved, both on nite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between quasi-geostrophic motions and its averaged motions. Furthermore, the existence of almost periodic quasi-geostrophic motions and att...

The solvability of the Riemann–Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener–Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions...

A crystallographic bar-joint framework, C in R, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω(C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, T, determined by a full rank t...

We consider operators of convolution generated by L func tions in L and various spaces of almost periodic functions It turns out to be that if such an operator is invertible in one of these spaces then it is in vertible in all the spaces we consider Further we prove that any convolution has identical norms in many natural couples of functional spaces

This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of (N, +) and (Z, +) are closed.