Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Common operator properties of the linear operators

Let and be bounded linear operators defined on Banach spaces, , . When , then the operators and have many basic operator properties in common. This situation is studied in this paper.

     

A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

For odd-dimensional hyperbolic space , we construct transforms between the cohomology of certain line bundles on (a twistor space for ) and eigenspaces of the Laplacian and of the Dirac operator on . The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of or on extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of . We also obtain vanishing theorems for the cohomology of a class of line bundles on .

     

Remarks on the results by Koskela concerning the radial uniqueness for Sobolev functions

In this note we aim to complete the results by Koskela concerning the radial uniqueness for Sobolev functions.

Let be a positive nonincreasing function on the interval , and let denote the unit ball of . Consider a -precise function on such that

where . We give conditions on which assure that whenever has vanishing fine boundary limits on a set of positive -capacity.

We are also concerned with the sharpness.

     

Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

The stability radius of a quasi-Fredholm operator

We extend the technique used by Kordula and Müller to show that the stability radius of a quasi-Fredholm operator is the limit of as . If is an isolated point of the Apostol spectrum , then the above limit is non-zero if and only if is quasi-Fredholm.

     

Remarks on the non-Cohen-Macaulay locus of Noetherian schemes

In this paper we give a notion of polynomial type of a Noetherian scheme and define the function by for all Then we show that if admits a dualizing complex and is equidimensional, is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM is not Cohen-Macaulay} is biequidimensional iff is constant on nCM

     

Ensembles sur lesquels les polynômes sont déterminés par leur image

Let be a non-empty subset of the complex plane , and , two complex polynomials. If and having the same image on implies , we say that is a generalized unicity set (for polynomials). We construct in this paper a subset of such that and are generalized unicity sets, and we give an example of a generalized unicity set which is open, connected and unbounded.

RÉSUMÉ. Soit un sous-ensemble non vide du plan complexe , et , deux fonctions polynômes à coefficients complexes. Si l'égalité entraîne , on dira que est un ensemble d'unicité généralisée (pour les polynômes). On construit dans cet article un sous-ensemble de tel que et sont d'unicité généralisée, et on donne aussi l'exemple d'un ensemble d'unicité généralisée qui est ouvert, connexe et non borné.

     

Coding nested mixing one-sided subshifts of finite type as Markov shifts having exactly the same alphabet

Let , be mixing one-sided subshifts of finite type such that . We show a necessary and sufficient condition for the existence of mixing Markov shifts , , , and a conjugacy with , such that the sets of letters appearing in both systems are the same, more precisely, .

     

Characterizations of contraction -semigroups

A -semigroup is of contractions if for , . Using the Hille-Yosida space, we completely characterize the generators of contraction -semigroups. We also give the Lumer-Phillips theorem for -semigroups in several special cases.

     

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's and classes of weights under a nonlinear operation, the -operation. We prove that the dyadic doubling reverse Hölder classes are not preserved under the -operation, but the dyadic doubling classes are preserved for . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

     

Reducibility of translates of Dickson polynomials

Let be a field and . The Dickson polynomial is characterized by the equation . We prove that is reducible if and only if there is a prime such that for some , or and for some . This result generalizes the well-known reducibility criterion for binomials; and it provides a reducibility criterion for where denotes the Chebyshev polynomial of degree .

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

Products of similar matrices

Let and be matrices of determinant over a field , with or . We show that if is not a scalar matrix, then is a product of matrices similar to . Analogously, we conjecture that if and are elements of a semisimple algebraic group over a field of characteristic zero, and if there is no normal subgroup of containing but not , then is a product of conjugates of . The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.

     

A characterization for spaces of sections

The space of smooth sections of a bundle over a compact smooth manifold can be equipped with a manifold structure, called an -manifold, where represents the Fréchet algebra of real valued smooth functions on . We prove that the -manifold structure characterizes the spaces of sections of bundles over and its open subspaces. We also describe the -maps between -manifolds.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Uniqueness for an overdetermined boundary value problem for the p-Laplacian

For set , and let be a measure with compact support. Suppose, for , there are functions and (bounded) domains , both containing the support of with the property that in (weakly) and in the complement of . If in addition is convex, then and .