1,225 research outputs found
Reflexivity of the isometry group of some classical spaces
We investigate the reflexivity of the isometry group and the automorphism
group of some important metric linear spaces and algebras. The paper consists
of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of
measurable functions. 4. Hardy spaces. 5. Banach algebras of holomorphic
functions. 6. Frechet algebras of holomorphic functions. 7. Spaces of
continuous functions.Comment: 18 pages. To appear in Rev. Mat. Iberoa
A Seidel-Walsh theorem with linear differential operators
Assume that {Sn}∞1 is a sequence of automorphisms of the open unit disk D and that {Tn}∞1 is a sequence of linear differential operators with constant coefficients, both of them satisfying suitable conditions. We prove that for certain spaces X of holomorphic
functions in the open unit disk, the set of functions f ∈ X such that {(Tnf) ◦ Sn : n ∈ N} is dense in H(D) is residual in X. This extends
the Seidel-Walsh theorem together with some subsequent results.Dirección General de Enseñanza Superior (DGES). EspañaJunta de AndalucÃ
Generalized DPW method and an application to isometric immersions of space forms
Let be a complex Lie group and denote the group of maps from
the unit circle into , of a suitable class. A differentiable
map from a manifold into , is said to be of \emph{connection
order } if the Fourier expansion in the loop parameter of the
-family of Maurer-Cartan forms for , namely F_\lambda^{-1}
\dd F_\lambda, is of the form . Most
integrable systems in geometry are associated to such a map. Roughly speaking,
the DPW method used a Birkhoff type splitting to reduce a harmonic map into a
symmetric space, which can be represented by a certain order map,
into a pair of simpler maps of order and respectively.
Conversely, one could construct such a harmonic map from any pair of
and maps. This allowed a Weierstrass type description
of harmonic maps into symmetric spaces. We extend this method to show that, for
a large class of loop groups, a connection order map, for ,
splits uniquely into a pair of and maps. As an
application, we show that constant non-zero curvature submanifolds with flat
normal bundle of a sphere or hyperbolic space split into pairs of flat
submanifolds, reducing the problem (at least locally) to the flat case. To
extend the DPW method sufficiently to handle this problem requires a more
general Iwasawa type splitting of the loop group, which we prove always holds
at least locally.Comment: Some typographical correction
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