%0 Journal Article
%T Generalized K£¿the-Toeplitz Duals of Some Vector-Valued Sequence Spaces
%A Y£¿lmaz Y£¿lmaz
%J International Journal of Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/862949
%X We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces. 1. Introduction The idea of dual sequence space was introduced by K£¿the and Toeplitz [1]. Then, Maddox, [2], generalized this notion to -valued sequence classes where is a Banach space. This brings an important contribution to the operator matrix transformation of Banach space-valued sequence spaces. Remember that - and -duals of a (complex-valued) sequence space , denoted by and , respectively, are defined to be where is the space of all complex-valued sequences. The in the classical definitions of K£¿the-Toeplitz duals is replaced by a sequence of linear operators, not necessarily continuous, from into another Banach space . Thus, if is a nonempty set of sequences with , then generalized - and -duals of are defined to be respectively. It is clear that this notion depends on the space and if , then where is the space of all linear operators from into . Without the loss of generality we can restrict ourselves in this work to continuous operators and being the space of all continuous linear operators from into and being the space of all -valued sequences which is a natural generalization of . We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a sequence space whenever it has the AK-property. Related results are also expressed in [3, page 176]. Here, we are going to show that there is an analogue relationship for -valued sequence spaces in the context of generalized -duals with respect to another fixed Banach space . Further, by applying this result, we obtain generalized -duals of some vector-valued Orlicz sequence spaces. We think that our results give a fruitful way to find generalized duals of this kind of vector-valued sequence spaces. 2. Prerequisites We use the notations , and for the sets of all positive integers, complex numbers, and real numbers, respectively. For some locally convex (lc, for short) space denotes the continuous dual of and we denote by and the closed unit ball and the sphere of some normed space , respectively. An FH-space is an lc Fr¨¦chet space such that is a vector subspace of a Hausdorff topological vector space
%U http://www.hindawi.com/journals/ijanal/2013/862949/