Diagonal operators, <i>s</i>-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 $B (X,Y)$ 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 $(L_p[0,1], L_q[0,1])$ form a Bernstein pair with respect to any sequence of s- numbers $(s_n)$, for $1< p < ∈fty$ and $1≤ q < ∈fty$.