Diagonal operators, s-numbers and Bernstein pairs


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<sub>p</sub>[0,1], L<sub>q</sub>[0,1]) form a Bernstein pair with respect to any sequence of s- numbers (s<sub>n</sub>), for 1< p < ∈fty and 1≤ q < ∈fty.

DOI Code: 10.1285/i15900932v17p209

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