Physical Sciences and Mathematics Commons^™

Diagonal Operators, S-Numbers, And Bernstein Pairs, Asuman Güven Aksoy, Grzegorz Lewicki

CMC Faculty Publications and Research

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 [O, 1] , L_q[O, 1]) form a Bernstein pair with respect to any sequence of s-numbers (s_n) ,for 1 < p < ∞ and 1 ≤ q < ∞ …