It is shown that, if F and G are inner functions, (H2 ⊖ FH2)/(H2 ⊖ FH2) ∩ GH2 is n-dimensional if and only if G is a Blaschke product of degree n. This is an extension of the well known result for the case (H2 ⊖ FH2) ∩ GH2 = {0}.

Let (αn.k) be a sequence of positive numbers. We define a regular sequence (resp. a weakly regular sequence) and then show the existence of a unitary operator (resp. a contraction T) L2[0, 1] → L2[0, 1] and a function f ∊ L2[0, 1] such that the pointwise convergence of the sequence of functions is not satisfied almost surely. As a first corollary the pointwise convergence of the Abel means of a contraction from L2 into L2 does not hold necessarily almost surely. As a second corollary there exists a contraction T for which the means (and powers) of Brunei's operator A do not converge pointwise a.s. We also show that, for P > 1 fixed, there exists a sequence of positive numbers αn.k for which we have the pointwise convergence in LP of the sequence of polynomials where T is a contraction of L1 and Lα. The dominated theorem does not, however, always hold for such LP-contractions.

We show that a locally compact semigroup S is topological left amenable iff a certain space of left uniformly continuous functions on the convolution semigroup of probability measures M0(S) on S is left amenable or equivalently iff the convolution semigroup M0(S) has the fixed point property for uniformly continuous affine actions on compact convex sets.

Milnor showed that non-singular bilinear products on Rn exist only if n = 1, 2, 4, 8 using topological methods. In this note, we give a proof of this result by purely analytical methods.

The paper describes polyhedral realizations for Felix Klein's map {3, 7}8 of genus 3, for Gordan's map {4, 5}6 of genus 4, and for two maps of genus 5, the Klein-Fricke map of type {3, 8} and Sherk's map of type {4, 6}. The polyhedra have self-intersections but high symmetry and thus are close analogues to the Kepler-Poinsot-polyhedra.

Let A be a local ring with maximal ideal m, let N(m ) be the order of the residue field A/m and let N be a subgroup of GLn (A) which is normalized by SLn(A). It follows from results of Klingenberg that N is normal in GLn(A) when n ≥ 3 or . Results of Lacroix show that this is also true when n = 2 and N(m) = 3, provided that N∩2(A) ≠ SL2(A)'.

The principal aim of this paper is to provide examples of non-normal subgroups of GL2(A) which are normalized by SL2(A). In the process we extend results of Lacroix and Levesque on GL2(A)-normalized subgroups of GL2(A), where 2 ∊ m and N(m) > 2.

Let A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

We construct an orthomodular inner product space to answer the questions posed by R. P. Morash in his paper "Angle bisection and orthoautomorphisms in Hilbert lattices" [6]. For example we show that every automorphism of the Hilbert lattice belonging to our inner product space has the property, that no atom is orthogonal to its image.

In this paper we prove some properties of the so-called modulus of noncompact convexity. This notion was recently introduced by K. Goebel and T. Sȩkowski [6] and it appears to be an interesting and useful generalization of the classical Clarkson modulus of convexity. We extend the results obtained in [6] showing that the modulus of noncompact convexity is continuous and has some extra properties in reflexive Banach spaces. The properties applicable in the fixed point theory are also stated.

If G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

Let be a separable Hilbert space, the algebra of all bounded operators on and Ai, Bi, ∊ , i = 1,. . . , r. It is shown that if no nontrivial linear combination of the operators Ai, is compact, then there exist X, Y ∊ such that X Ai, Y = Bi, for all i. A related (but much milder) result is discussed in other algebras with the unique maximal ideal and an application to elementary operators is given.

In this paper the ergodic Hilbert transform is investigated at the operator theoretic level. Let T be an invertible positive operator on Lp = Lp(X, , μ) for some fixed p, 1 < p < ∞, such that sup{||Tn||p: — ∞ < n < ∞} < ∞. It is proved that the limit

exists almost everywhere and in the strong operator topology, where the prime denotes that the term with zero denominator is omitted. Related results are also proved.

L'objet de cet article est de construire un modèle pour l'élément β1, de Toda (premier élément non nul de la composante p - primaire de l'homotopie stable des sphères qui n'est pas dans l'image du J-homomorphisme, p ≠ 2). Le modèle construit possède en outre la propriété de se plonger, comme variété différentiable, en codimension 1.

La construction, basée sur la J-théorie et la chirurgie, exhibe en outre des complexes cellulaires satisfaisant à certaines conditions de plongement, répondant partiellement à un problème posé par le 2e et le 3e auteur.

In this paper we extend and greatly generalize, with some new information, the well known results that an approximately continuous function is Darboux, and that a finite approximate derivative has the mean value property and is Darboux. Our theorems on Darboux and mean value properties of derivatives include also those of selective derivatives and I-approximate derivatives.

Let p be odd prime and suppose that G = 〈a, b〉 where ap-1 = bp = 1, a-1 ba = br, and r is a generator of the multiplicative group of integers mod p. An explicit characterization of the group of normalized units V of the group ring ZG is given in terms of a subgroup of GL(p - 1, Z). This characterization is used to exhibit a normal complement for G in V.

The intricacy of the problem of (2s + 1 )-factorizating Kn is determined. Some generalizations are also given.

Let R be a commutative ring with identity. An R-module M is said to be distributive if the lattice of submodules of M is distributive. We characterize such modules and study their properties.

This paper constructs a class of examples to show that for torsion-free groups H with finite nilstufe v(H) = n < ∞ there can be divisible torsion groups D with v(H ⊕ D) - n + k for all k ≤ n + 1. This answers a question of Feigelstock. The construction is based on a proposition which bounds v(H ⊕ D) in terms of v(H) and rank (D).