Let T be a bounded linear operator on a Banach space W, assume W and Y are in normed duality, and assume that T has adjoint T† relative to Y. In this paper, conditions are given that imply that for all λ≠0, λ−T and λ −T† maintain important standard operator relationships. For example, under the conditions given, λ −T has closed range if, and only if, λ −T† has closed range.

These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h(p, α), h(p, log(α)) of Hardy–Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.

Let H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of and H. Moreover, we show that is Kähler and find an orthogonal almost complex structure on .

Let be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

In this paper, we show that the semilinear elliptic systems of the form(0.1) possess at least one positive solution pair (u, v) ∈ H 1 0(Ω) × H 1 0(Ω), where Ω is a smooth bounded domain in , f(x,t) and g(x, t) are continuous functions on and asymptotically linear at infinity.

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

Certain generalized Samelson products of SO(3) are calculated and applications to the homotopy of gauge groups are given.

We prove that if G is a locally finite group admitting an automorphism φ of order four such that C G (φ) is Chernikov, then G has a soluble subgroup of finite index.