Approximate antilinear eigenvalue problems and related inequalities

If is a complex symmetric operator on a separable complex Hilbert space , then the spectrum of can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator ), in terms of the spectra of the selfadjoint operators and , restricting the possible location of elements of . A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

     

Isomorphisms of subalgebras of nest algebras

Let be a subalgebra of a nest algebra . If contains all rank one operators in , then is said to be large; if the set of rank one operators in coincides with that in the Jacobson radical of , is said to be radical-type. In this paper, algebraic isomorphisms of large subalgebras and of radical-type subalgebras are characterized. Let be a nest of subspaces of a Hilbert space and be a subalgebra of the nest algebra associated to (). Let be an algebraic isomorphism from onto . It is proved that is spatial if one of the following occurs: (1) () is large and contains a masa; (2) () is large and closed; (3) () is a closed radical-type subalgebra and ( is quasi-continuous (i.e. the trivial elements of are limit points); (4) () is large and one of and is not quasi-continuous.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

A new construction of the unstable manifold for the measure-preserving Hénon map

Let denote the measure-preserving Hénon map with the parameter . The map has a hyperbolic fixed point . The main result of this paper is that the unstable mainfold of is the iterated limit of a very simple set. Informally,

where is the line and denotes the unstable manifold of .

     

Nests with the partial factorization property

It is proved that a nest on a separable complex Hilbert space has the left (resp. right) partial factorization property, which means that for every invertible operator from onto a Hilbert space there exists an isometry (resp. a coisometry) from into such that both and are in the associated nest algebra if and only if it is atomic (resp. countable).

     

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  .

     

On the topological centre problem for weighted convolution algebras and semigroup compactifications

Let be a locally compact, non-compact group (we make the non-compactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function , the weighted group algebra is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of equal . Also, we show that the topological centre of the algebra equals the weighted measure algebra . Moreover, still in the same situation, we prove that every linear (left) -module map on is automatically bounded, and even --continuous, hence given by convolution with an element in . To this end, we derive a general factorization theorem for bounded families in the -module . Finally, using this result in the case where , we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup is empty, where denotes the -compactification of . This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.

     

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 .

     

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.