Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod dual Steenrod algebra, , with a view to determining the subspace, , of elements invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting , it becomes polynomial on a natural set of generators. Finally we note that, without inverting , is far from being polynomial.

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Holomorphic sections of pre-quantum line bundles on

Let be the Iwasawa decomposition of a complex connected semi-simple Lie group . Let be a parabolic subgroup containing , and let be its commutator subgroup. In this paper, we characterize the -invariant Kähler structures on , and study the holomorphic sections of their corresponding pre-quantum line bundles.

     

On the parametrization of self-similar and other fractal sets

We prove for many self-similar, and some more general, sets that if is the Hausdorff dimension of and is Hölder continuous with exponent , then the -dimensional Hausdorff measure of is .

     

On some properties of the Gamma function

Anderson and Qiu (1997) conjectured that the function is concave for 1$">. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function and derive inequalities for and .

     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Integral representation of linear functionals on spaces of unbounded functions

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

     

Semigroups and weights for group representations

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

On the pointwise maximum of convex functions

We study the conjugate of the maximum, , of and when and are proper convex lower semicontinuous functions on a Banach space . We show that on the bidual, , of provided that and satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for and for the ``preconjugate' of .

     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

-semigroups generated by second order differential operators with general Wentzell boundary conditions

Let us consider the operator where is positive and continuous in and is equipped with the so-called generalized Wentzell boundary condition which is of the form at each boundary point, where This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions.

Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.

     

A counterexample for approximable functions

Let be the unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.

     

A note on tensor products of ample line bundles on abelian varieties

Let be a polarized abelian variety defined over the complex number field. Then we classify with such that is not -jet ample nor -very ample.

     

The Mardesic factorization theorem for extension theory and c-separation

We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .

     

Strongly asymptotically stable Frobenius-Perron operators

Let be a -finite measure space and let be a Frobenius-Perron operator.

In 1997 Bartoszek and Brown proved that if overlaps supports and if there exists , 0$"> on , such that , then is (strongly) asymptotically stable.

In the note we prove that instead of assuming that 0$"> on , it is enough to assume that and . More precisely, we prove that is asymptotically stable if and only if overlaps supports and there exists , , , such that .

     

Composition operators on Dirichlet-type spaces

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.