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 .