Generalised nuclear maps in normed linear spaces
M. S. Ramanujan · Bulletin of the American Mathematical Society · 1970
Preliminary definitions and notations. Grothendieck [3] and Pietsch [ó] present an exhaustive study of nuclear operators and nuclear maps.The notion of a nuclear operator was extended by Persson and Pietsch in a recent paper [5] and they study in detail the ^-nuclear and quasi-^-nuclear maps.In this paper we define and study certain linear maps called X-nuclear and quasi-X-nuclear maps.Our definition and generalisation here are motivated by the Köthe sequence spaces and their duality theory.For the special case X = I 1 we obtain the nuclear operators and for X ~P we obtain the ^-nuclear maps; also, the special case X = c 0 yields the co-nuclear operators of Persson and Pietsch.Most of the results in this work are motivated by the work of Persson and Pietsch [5] and Köthe sequence spaces.We shall briefly outline our assumptions.For definitions not stated here see Garling [l], Köthe [4], Ruckle [7], Sargent [9] and Zeiler [l0].Let X be a symmetric sequence space of scalars and X* be its Köthe dual.We shall assume that X is provided with the Mackey topology of the duality (X, X*) and that this topology is provided by a norm p, p itself being an extended seminorm on co.We assume now that X is solid and that it is i£-sy mme trie, i.e., for each # (EX and for each permutation TT of I + we have # T £X and p(x)-p(x 1F ).X is also assumed to be a BK space with AK.We remark that our assumptions imply that X =co or X = /°° or XCc<>.The space X* is now considered as the topological dual of X and equipped with its natural norm topology.We pause now to point out that in addition to the spaces l p , l^p x n ÇzE for each n and such that the sequence ((#", a)) £X for each a£E'.Formally define € X (*) = sup p{\ (x n , a)|), where p is the norm on X.A MS Subject Classifications.Primary 4710.