Eigenvalues and s-numbers
Full length article: Approximation schemes satisfying Shapiro's Theorem
Journal of Approximation Theory
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For an operator T@?B(X,Y), we denote by a"m(T), c"m(T), d"m(T), and t"m(T) its approximation, Gelfand, Kolmogorov, and absolute numbers, respectively. We show that, for any infinite-dimensional Banach spaces X and Y, and any sequence @a"m@?0, there exists T@?B(X,Y) for which the inequality 3@a"@?"m"/"6"@?=a"m(T)=max{c"m(t),d"m(T)}=min{c"m(t),d"m(T)}=t"m(T)=@a"m/9 holds for every m@?N. Similar results are obtained for other s-scales.