Approximation Schemes, Related S-Numbers and Applications.

A generalized approximation scheme via a sequence (p(,n)) of properties on a Banach space X is introduced. Approximation numbers and Kolmogorov diameters are defined in this context and comparison of approximation numbers and Kolmogorov diameters of T (ELEM) L(X) with those of T' and JT are studied. Compact sets and compact maps are defined with respect to this approximation scheme (we call them Q-compact sets and Q-compact maps) and a Dieudonne-Schwartz-type characterization of Q-compact sets is obtained. A relation between Q-compact sets and Q-compact maps and a representation theorem for Q-compact maps are proved. Q-compact maps are genuine generalization of compact maps is shown by means of an example of a Q-compact map which is not a compact map. Replacing the role of l(,u) in the classical approximation spaces with a nuclear infinite type power series space an approximation space is defined. Representation and transformation theorems for such spaces are obtained. Also, using K and J-functionals and stable nuclear infinite type power series spaces, discrete intermediate spaces are defined and their interpolation are examined. The notion of an operator measure s on L(X) is defined and a formula for the essential spectral radius r(,e)(T) is obtained in terms of (s(T('n))).Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/160252/1/8502751.pd