Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

Approximation of measurable mappings by sequences of continuous functions

Let be a completely regular Hausdorff space, a positive, finite Baire measure on , and a separable metrizable locally convex space. Suppose is a measurable mapping. Then there exists a sequence of functions in which converges to a.e. . If the function is assumed to be weakly continuous and the measure is assumed to be -smooth, then a separability condition is not needed.

     

A characterization of compactly generated metric groups

Recall that a topological group is: (a) -compact if where each is compact, and (b) compactly generated if is algebraically generated by some compact subset of . Compactly generated groups are -compact, but the converse is not true: every countable non-finitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group is compactly generated if and only if is -compact and for every open subgroup of there exists a finite set such that algebraically generates . As a corollary, we obtain that a -compact metric group is compactly generated provided that one of the following conditions holds: (i) has no proper open subgroups, (ii) is dense in some connected group (in particular, if is connected itself), (iii) is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.

     

Small covers of the dodecahedron and the -cell

Let be the right-angled hyperbolic dodecahedron or -cell, and let be the group generated by reflections across codimension-one faces of . We prove that if is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold is determined up to homeomorphism by modulo symmetries of .

     

Weak compactness of certain sets of measures

For a compact Hausdorff space and a Montel Hausdorff locally convex space , let being the uniform topology. We determine the necessary and sufficient conditions for an equicontinuous to be -compact. Special results are obtained when is an -space or a -Stonian space.

     

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 .

     

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 .

     

On the Bieberbach conjecture and holomorphic dynamics

In this note we prove that when is a polynomial of degree with connected Julia set and when belongs to the filled-in Julia set , then . We also show that equality is achieved if and only if is a segment of which one extremity is . In that case, is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.

     

Poincaré duality in P.A. Smith theory

Let , or more generally be a finite -group, where is an odd prime. If acts on a space whose cohomology ring fulfills Poincaré duality (with appropriate coefficients ), we prove a mod congruence between the total Betti number of and a number which depends only on the -module structure of . This improves the well known mod congruences that hold for actions on general spaces.

     

A weakly Stegall space that is not a Stegall space

A topological space is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space and minimal usco , is single-valued at some point of . In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space whose dual, equipped with the weak topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space such that belongs to the class of weakly Stegall spaces but does not.

     

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.

     

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 ).

     

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.

     

Discrete spectra of -algebras and complemented submodules in Hilbert -modules

Let be a -algebra and let be a full (right) Hilbert -module. It is shown that if the spectrum of is discrete, then every closed --submodule of is complemented in , and conversely that if is a -space and if every closed --submodule of is complemented in , then is discrete.

     

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 .

     

Regularization of weights

We show how to approximate a given weight function in the class by weights that are bounded above by multiples of their infima in such a way that the constant is not increased. As an application, we show that the precise range of for which a given weight is in cannot be determined by extrapolating the constants.

     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.