Diagonal Operators and Bernstein Pairs

Replacing the nested sequence of ""nite" 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 < 1 and 1 q < 1: PRELIMINARIES s-Numbers. Let X and Y be Banach spaces and B(X; Y) denote the space of all bounded linear maps from X into Y: According to A. Pietsch 10], a map s which to each bounded linear map T from one Banach space to another such space assigns a unique sequence (s n (T)) is called a s-function if for all Banach spaces W; X; Y; Z : i) kTk = s 1 (T) s 2 (T) ::: 0 for all T 2 B(X; Y) ii) s n (S + T) s n (S) + kTk for all S; T 2 B(X; Y); and all n 2 N iii) s n (RST) kRks n (S)kTk for all T 2 B