Generating classes of perfect Banach sequence spaces

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A perfect sequence space A is said to be a step if 11 'Z cJ'10 and A is a Banach space in its strong topology from A<. In this paper a method is given to generate additional steps from a step A. Precisely, A2 is a step where AP-{x=(x)jx, E C and Ix|P=(IxJP)eA}, for 1_p<oo, with norm IxIx1AP=(JI JxJPIIA)1/P. It is shown that AP, 1 <p< oo, is reflexive iff A has a Schauder basis. The space of diagonal maps of AP into A is characterized, as is the space of diagonal nuclear maps of A into AP when A has a Schauder basis. If 2 is a perfect sequence space which is a Banach space under the strong topology from AX, and contains 11 and is contained in lZ, we say that 2 is a step. Examples of steps in general include the Kothe dual of the usual sequence space associated with a Banach space possessing a normalized unconditional basis; see [4]. More specifically the lP' spaces, the spaces Ia,P and va,p of Garling [5], and the spaces mn(+) and n(b) of Sargent [13], are steps. In this paper we generate additional steps from a step 2 by paralleling a method of generating the IP spaces from 11. In the cases where the usual coordinate vectors form a basis for 2, these generated spaces are reflexive. Others results paralleling the known properties of tP spaces are obtained under this additional hypothesis. 1. Definitions and preliminary results. The general terminology of this paper is as in [9]. Throughout we will assume that the sequence spaces 2 are normal and equipped with the topology b(Ax), unless we specifically state otherwise. For sequences x=(xi), y=(y) we denote by xy the sequence (xjyi). Using the notation of Ruckle [12], we denote by UIt the {u|ux e 2 for each Presented to the Society, September 2, 1971 under the title Some properties of a special class of perfect Banach sequence spaces; received by the editors October 6, 1971 and, in revised form, March 10, 1972. AMS 1970 subject class{flcations. Primary 46A45, 46B99.