A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

Numerical radius distance-preserving maps on

Let be a complex Hilbert space, be the algebra of all bounded linear operators on , be the subset of all selfadjoint operators in and or . Denote by the numerical radius of . We characterize surjective maps that satisfy for all without the linearity assumption.

     

Hermite-Biehler functions with zeros close to the imaginary axis

A Hermite-Biehler function gives rise to a de Branges Hilbert space consisting of entire functions. We are going to show that for Hermite-Biehler functions of sufficiently small growth and a certain distribution of zeros every proper de Branges subspace of coincides for some with the -dimensional linear space of all polynomials of degree at most .

     

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 .

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Single elements and module isomorphisms of some operator algebra modules

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the Alg-module is initiated, where is a completely distributive subspace lattice on a Hilbert space . Furthermore, as an application of single elements, we study module isomorphisms between norm closed Alg-modules, where is a nest, and obtain the following result: Suppose that are norm closed Alg-modules and that is a module isomorphism. Then and there exists a non-zero complex number such that .

     

Equicompact sets of operators defined on Banach spaces

Let and be Banach spaces. We say that a set denotes the space of all compact operators from into ) is equicompact if there exists a null sequence in such that for all and all . It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: is equicompact iff is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set is equicompact iff each bounded sequence in has a subsequence such that is a converging sequence uniformly for ; 2) if does not have finite cotype and is a maximal equicompact set, then, given and a finite set in , there is an operator such that for and all .

     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

A characterization of discrete groups

The purpose of this article is to prove the following result. Let be a locally compact group, the Fourier algebra of and ~0$"> such that . Then is a discrete group .

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

Some finiteness conditions on the set of overrings of an integral domain

Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing have finite length. The corresponding statement for (FO) is also valid.

     

Fuchs' problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group is determined by the homomorphism groups . A class of Abelian groups is a Fuchs 34 class if and in are isomorphic if and only if for all . Two -groups and satisfy for all groups if and only if they have the same -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class introduced by Glaz and Wickless. While is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

Numerical peak points and numerical Šilov boundary for holomorphic functions

In this paper, we characterize the numerical and numerical strong-peak points for when is the complex space or . We also prove that for all is the numerical Šilov boundary for

     

Essential numerical range of elementary operators

Let and denote two -tuples of operators with and Let denote the elementary operators defined on the Hilbert-Schmidt class by We show that

Here is the essential numerical range, is the joint numerical range and is the joint essential numerical range.

     

Compactness of Floquet isospectral sets for the matrix Hill's equation

Let denote the set of self adjoint potentials for the matrix Hill's equation having the same Floquet multipliers as . Elementary methods are used to show that has compact closure in the space of continuous matrix valued functions.

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

Nonabelian theta functions of positive genus

Let be a smooth projective irreducible curve over of genus and let be a set of distinct points on . We fix a nonnegative integer and denote by the moduli space of parabolic semistable vector bundles of rank on with trivial determinant and fixed parabolic structure of type at , where each weight is in . On there is a canonical line bundle , whose global sections are called generalized parabolic -theta functions of order . In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.