A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.

     

Irreducible representations of the Cuntz algebra

In this paper, we establish formulas for the configuration of a special class of irreducible representations of the Cuntz algebra , . These irreducible representations arise as subrepresentations of naturally occurring representations of acting in and arise from consideration of multiresolution wavelet filters.

     

Characterizing indecomposable plane continua from their complements

We show that a plane continuum is indecomposable iff has a sequence of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence of open arcs, with and , there is a sequence of shadows , where each is a shadow of , such that . Such an open arc divides into disjoint subdomains and , and a shadow (of ) is one of the sets .

     

Iterating the Cesàro operators

The discrete Cesàro operator associates to a given complex sequence the sequence , where . When is a convergent sequence we show that converges under the sup-norm if, and only if, . For its adjoint operator , we establish that converges for any .

The continuous Cesàro operator, , has two versions: the finite range case is defined for and the infinite range case for . In the first situation, when is continuous we prove that converges under the sup-norm to the constant function . In the second situation, when is a continuous function having a limit at infinity, we prove that converges under the sup-norm if, and only if, .

     

Extreme points of lattice intervals in the Minkowski-Rådström-Hörmander lattice

In this paper we characterize extreme points of any symmetric interval in the Minkowski-Rådström-Hörmander lattice over any Hausdorff topological vector space (Theorem 1). Then we prove that the unit ball in the Minkowski-Rådström-Hörmander lattice over any normed space , dim has exactly two extreme points (Theorem 2).

     

A note on -bases of a regular affine domain extension

Let be a tower of commutative rings where is a regular affine domain over an algebraically closed field of prime characteristic and is a regular domain. Suppose has a -basis over and . For a subset of whose elements satisfy a certain condition on linear independence, let be a set of maximal ideals of such that is a -basis of over . We shall characterize this set in a geometrical aspect.

     

On the isolated points of the surjective spectrum of a bounded operator

For a bounded operator acting on a complex Banach space, we show that if is not surjective, then is an isolated point of the surjective spectrum of if and only if , where is the quasinilpotent part of and is the analytic core for . Moreover, we study the operators for which . We show that for each of these operators , there exists a finite set consisting of Riesz points for such that and is connected, and derive some consequences.

     

On the convergence in capacity on compact Kahler manifolds and its applications

The main aim of the present note is to study the convergence in on a compact Kahler mainfold . The obtained results are used to study global extremal functions and describe the -pluripolar hull of an -pluripolar subset in .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

The Hausdorff operator is bounded on the real Hardy space

We prove that the Hausdorff operator generated by a function is bounded on the real Hardy space . The proof is based on the closed graph theorem and on the fact that if a function in is such that its Fourier transform equals for (or for ), then .

     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .

     

Global behavior of the branch of positive solutions to a logistic equation of population dynamics

We consider the nonlinear problem arising in population dynamics:

where is a constant and is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions in -framework as , where ( ). Especially, for the original logistic equation, namely the case where and , we obtain not only the asymptotic expansion formula for but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

A polynomial analogue of the twin prime conjecture

We consider the problem of counting the number of (not necessarily monic) `twin prime pairs' of degree , where is a polynomial of degree . We formulate an asymptotic prediction for the number of such pairs as and then prove an explicit estimate confirming the conjecture in those cases where is large compared with . When has degree , our theorem implies the validity of a result conditionally proved by Hayes in 1963. When has degree zero, our theorem refines a result of Effinger, Hicks and Mullen.

     

Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.