Diagonal operators, s-numbers and Bernstein pairs
Abstract
Replacing the nested sequence of "finite" dimensional subspaces by the nested sequence of "closed" subspaces in the classical Bernstein lethargy theorem, we obtain a version of this theorem for the space
of all bounded linear maps. Using this result and some properties of diagonal operators, we investigate conditions under which a suitable pair of Banach spaces form an exact Bernstein pair. We also show that many "classical" Banach spaces, including the couple
form a Bernstein pair with respect to any sequence of s- numbers
, for
and
.
![B (X,Y)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/0a4051747b6590817aa850d8a5d1a224.png)
![(L<sub>p</sub>[0,1], L<sub>q</sub>[0,1])](http://siba-ese.unile.it/plugins/generic/latexRender/cache/39aebaa7f1923c4bf4bd64a1efea4f71.png)
![(s<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e71bd795b08cd70aeba60dec7aa5fdcd.png)
![1< p < ∈fty](http://siba-ese.unile.it/plugins/generic/latexRender/cache/750276e77137f9a9fa672f13457a6649.png)
![1≤ q < ∈fty](http://siba-ese.unile.it/plugins/generic/latexRender/cache/baccc169e820be32d4e240b7acd978b5.png)
DOI Code:
10.1285/i15900932v17p209
Full Text: PDF