Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

The optimality of James's distortion theorems

A renorming of , explored here in detail, shows that the copies of produced in the proof of the Kadec-Pelczynski theorem inside nonreflexive subspaces of cannot be produced inside general nonreflexive spaces that contain copies of . Put differently, James's distortion theorem producing one-plus-epsilon-isomorphic copies of inside any isomorphic copy of is, in a certain sense, optimal. A similar renorming of shows that James's distortion theorem for is likewise optimal.

     

Perturbations of the Haar wavelet

Let be given. For any we construct a function having the following properties: (a) has support in . (b) . (c) If denotes the Haar function and , then . (d) generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to as .

     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .

     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or in .

     

A Cantor-Lebesgue theorem with variable ``coefficients'

If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown.

     

Central units of the integral group ring

There are very few cases known of nonabelian groups where the group of central units of , denoted , is nontrivial and where the structure of , including a complete set of generators, has been determined. In this note, we show that the central units of augmentation 1 in the integral group ring form an infinite cyclic group , and we explicitly find the generator .

     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

Combinatorial aspects of F-sets

We discuss F filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F filter is the better known bounded evasion number . An application to -sets from trigonometric series is given by showing that if is an -set and has size less than , then is again an -set.

     

On invertibility in non-selfadjoint operator algebras

Let be a complete commutative subspace lattice on a Hilbert space. When is purely atomic, we give a necessary and sufficient condition for for every in , where and denote the spectrum of in and respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in .

     

Covering by complements of subspaces, II

Let be an -dimensional vector space over an algebraically closed field . Define to be the least positive integer for which there exists a family of -dimensional subspaces of such that every -dimensional subspace of has at least one complement among the 's. Using algebraic geometry we prove that .

     

On the Fredholm Alternative for the -Laplacian

Consider

where and and let be the principal eigenvalue of the problem with . For , we discuss for which values of and the Fredholm alternative holds.

     

Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

The rank stable topology of instantons on

Let be the moduli space of based (anti-self-dual) instantons on of charge and rank . There is a natural inclusion . We show that the direct limit space is homotopy equivalent to . Let be a line in the complex projective plane and let be the blow-up at a point away from . can be alternatively described as the moduli space of rank holomorphic bundles on with and and with a fixed holomorphic trivialization on .

     

On -summable sequences in the range of a vector measure

Let . Among other results, we prove that a Banach space has the property that every sequence lies inside the range of an -valued measure if and only if, for all sequences in satisfying that the operator is 1-summing, the operator is nuclear, being the conjugate number for . We also prove that, if is an infinite-dimensional -space for , then can't have the above property for any .

     

Wavelet bases in rearrangement invariant function spaces

We point out that the well known characterization of spaces () in terms of orthogonal wavelet bases extends to any separable rearrangement invariant Banach function space on (equipped with Lebesgue measure) with nontrivial Boyd's indices. Moreover we show that such bases are unconditional bases of .

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

and interpolation between Banach lattices

We study the possibility of obtaining the -norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in such that the -norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the -norm by any method which guarantees interpolation of not only linear operators (= bilinear forms on but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the -norm, or both have to be weighted -norms.

     

A formula with nonnegative terms for the degree of the dual variety of a homogeneous space

Let be a reductive group and a parabolic subgroup. For every -regular dominant weight let denote the variety embedded in the projective space by the embedding corresponding to the ample line bundle . Writing , we prove that the degree of the dual variety to is a polynomial with nonnegative coefficients in . In the case of homogeneous spaces we find an expression for the constant term of this polynomial.