The Nevanlinna counting functions for Rudin's orthogonal functions

and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .

     

On a characterization of the maximal ideal spaces of algebraically closed commutative -algebras

Let be the algebra of all complex-valued continuous functions on a compact Hausdorff space . We say that is algebraically closed if each monic polynomial equation over has a continuous solution. We give a necessary and sufficient condition for to be algebraically closed for a locally connected compact Hausdorff space . In this case, it is proved that is algebraically closed if each element of is the square of another. We also give a characterization of a first-countable compact Hausdorff space such that is algebraically closed.

     

Catalan paths and quasi-symmetric functions

We investigate the quotient ring of the ring of formal power series over the closure of the ideal generated by non-constant quasi-symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from and above the line . We investigate as well the quotient ring of polynomial ring in variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of is bounded above by the th Catalan number.

     

The alternative Dunford-Pettis property in -algebras and von Neumann preduals

A Banach space is said to have the alternative Dunford-Pettis property if, whenever a sequence weakly in with , we have for each weakly null sequence in X. We show that a -algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for -algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.

     

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if , are finite modules over a Gorenstein local ring of codimension at most , then the vanishing of for is equivalent to the vanishing of for . Furthermore, if has no embedded deformation, then such vanishing occurs if and only if or has finite projective dimension.

     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

Some finiteness conditions on the set of overrings of an integral domain

Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing have finite length. The corresponding statement for (FO) is also valid.

     

Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

     

Ordered group invariants for nonorientable one-dimensional generalized solenoids

Let be an edge-wrapping rule which presents a one-dimensional generalized solenoid , and let be the adjacency matrix of . When is a wedge of circles and leaves the unique branch point fixed, we show that the stationary dimension group of is an invariant of homeomorphism of even if is not orientable.

     

Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide

A set is -straight if has finite Hausdorff -measure equal to its Hausdorff -content, where is continuous and non-decreasing with . Here, if satisfies the standard doubling condition, then every set of finite Hausdorff -measure in is shown to be a countable union of -straight sets. This also settles a conjecture of Foran that when , every set of finite -measure is a countable union of -straight sets.

     

Borel subrings of the reals

A Borel (or even analytic) subring of either has Hausdorff dimension or is all of . Extensions of the method of proof yield (among other things) that any analytic subring of having positive Hausdorff dimension is equal to either or .

     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .

     

Fully transitive -groups with finite first Ulm subgroup

An abelian -group is called (fully) transitive if for all with ( ) there exists an automorphism (endomorphism) of which maps onto . It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive -groups with finite first Ulm subgroup. In this paper we restrict ourselves to -groups of type , this is to say -groups satisfying . We show that the answer to Corner's question is no if is finite and is of type .

     

Fuchs' problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group is determined by the homomorphism groups . A class of Abelian groups is a Fuchs 34 class if and in are isomorphic if and only if for all . Two -groups and satisfy for all groups if and only if they have the same -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class introduced by Glaz and Wickless. While is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

Perfect cliques and colorings of Polish spaces

A coloring of a set is any subset of , where 1$"> is a natural number. We give some sufficient conditions for the existence of a perfect -homogeneous set, in the case where is and is a Polish space. In particular, we show that it is sufficient that there exist -homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect -clique if it contains any uncountable -clique, where is a natural number or (a set is a -clique in if the convex hull of any of its -element subsets is not contained in ).

     

Characterizations of paracompactness and Lindelöfness by the separation property

The separation property in our title is that, for two spaces and , any two disjoint closed copies of in are separated by open sets in . It is proved that a Tychonoff space is paracompact if and only if this separation property holds for the space and every Tychonoff space which is a perfect image of (where denotes the Stone-Cech compactification of ). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.

     

On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces

In this paper, we prove the following strong convergence theorem: Let be a closed convex subset of a Hilbert space . Let be a strongly continuous semigroup of nonexpansive mappings on such that . Let and be sequences of real numbers satisfying , 0$"> and . Fix and define a sequence in by for . Then converges strongly to the element of nearest to .

     

Bernstein-Walsh inequalities and the exponential curve in

It is shown that for the pluripolar set in there is a global Bernstein-Walsh inequality: If is a polynomial of degree on and on , this inequality gives an upper bound for which grows like . The result is used to obtain sharp estimates for .

     

A characterization of quasiconvex vector-valued functions

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are -quasiconvex with respect to a closed convex cone in a Banach space. Our main result extends a well-known characterization of -quasiconvexity by means of extreme directions of the polar cone of , obtained by Dinh The Luc in the particular case when is a polyhedral cone generated by exactly linearly independent vectors in the Euclidean space .