Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition be an IN-group cannot be entirely removed.

     

On the Gelfand-Kirillov conjecture for quantum algebras

Let be a complex not a root of unity and be a semi-simple Lie -algebra. Let be the quantized enveloping algebra of Drinfeld and Jimbo, be its triangular decomposition, and the associated quantum group. We describe explicitly and as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.

     

Multidimensional analogues of Bohr's theorem on power series

Generalizing the classical result of Bohr, we show that if an -variable power series converges in -circular bounded complete domain and its sum has modulus less than 1, then the sum of the maximum of the modulii of the terms is less than 1 in the homothetic domain , where . This constant is near to the best one for the domain

     

Hilbert coefficients and the associated graded rings

Let be a -dimensional Cohen-Macaulay local ring with infinite residue field. Let be an -primary ideal of . In this paper, we prove that if for some minimal reduction of , then depth .

     

The spectral properties of certain linear operators and their extensions

Let be a Hilbert space with inner-product , and let be a bounded positive operator on which determines an inner-product, . Denote by the completion of with respect to the norm . In this paper, operators having certain relationships with are studied. In particular, if where , then has an extension , and and have essentially the same spectral and Fredholm properties.

     

On semisimple Hopf algebras of dimension

We consider the problem of the classification of semisimple Hopf algebras of dimension where are two prime numbers. First we prove that the order of the group of grouplike elements of is not , and that if it is , then . We use it to prove that if and its dual Hopf algebra are of Frobenius type, then is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension , and a partial classification in dimensions and .

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

Curvature restrictions on convex, timelike surfaces in Minkowski 3-space

Suppose that and are Minkowski Gauss curvature and Minkowski mean curvature respectively on a timelike surface that is immersed in Minkowski 3-space . Suppose also that and that is complete as a surface in the underlying Euclidean 3-space . It is shown that neither nor can be bounded away from zero on such a surface .

     

Higher order symmetric spaces and the roots of the identity in a Lie group

Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

A generalization of the Lefschetz fixed point theorem and detection of chaos

We consider the problem of existence of fixed points of a continuous map in (possibly) noninvariant subsets. A pair of subsets of induces a map given by if and elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If is metrizable, and are compact ANRs, and is continuous, then has a fixed point in provided the Lefschetz number of is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.

     

The residues of the resolvent on Damek-Ricci spaces

We determine the poles and residues of the resolvent kernel of the Laplacian on a Damek-Ricci space We show that all poles are simple and the residues define convolution operators of finite rank. This generalizes a result of Guillopé-Zworski for the real hyperbolic -space. If corresponds to a symmetric space of negative curvature , the image of each residue is a -module with a specific highest weight. We compute the dimension by the Weyl dimension formula.

     

A characterization of Möbius transformations

Let be an integer and let be a domain of . Let be an injective mapping which takes hyperspheres whose interior is contained in to hyperspheres in . Then is the restriction of a Möbius transformation.

     

Sampling sequences for Hardy spaces of the ball

We show that a sequence in the unit ball of is sampling for the Hardy spaces , , if and only if the admissible accumulation set of in the unit sphere has full measure. For the situation is quite different. While this condition is still sufficient, when (in contrast to the one dimensional situation) there exist sampling sequences for whose admissible accumulation set has measure 0. We also consider the sequence obtained by applying to each a random rotation, and give a necessary and sufficient condition on so that, with probability one, is of sampling for , .

     

On the boundary of attractors with non-void interior

Let be a family of contractive mappings on such that the attractor has nonvoid interior. We show that if the 's are injective, have non-vanishing Jacobian on , and have zero Lebesgue measure for then the boundary of has measure zero. In addition if the 's are affine maps, then the conclusion can be strengthened to . These improve a result of Lagarias and Wang on self-affine tiles.

     

An extension of a non-commutative Choquet-Deny theorem

Let be a discrete group, and let be a normal subgroup of . Then the quotient map induces a group algebra homomorphism . It is shown that the kernel of this map may be decomposed as , where is a closed right ideal with a bounded left approximate identity and is a closed left ideal with a bounded right approximate identity. It follows from this fact that, if is a closed two-sided ideal in , then is closed in . This answers a question of Reiter.

     

On a theorem of Barbara Schmid

Let be a finite group and a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions on the vector space is generated by homogeneous polynomials of degree at most , where is the largest degree of a generator in a minimal generating set for , and is the complex regular representation of . In this note we give a new proof of this result, and at the same time extend it to fields whose characteristic is larger than , the order of the group .