All Title Author
Keywords Abstract


Generalized K?the-Toeplitz Duals of Some Vector-Valued Sequence Spaces

DOI: 10.1155/2013/862949

Full-Text   Cite this paper   Add to My Lib

Abstract:

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 Fréchet space such that is a vector subspace of a Hausdorff topological vector space

References

[1]  G. K?the and O. Toeplitz, “Lineare r?ume mit unendlich vielen koordinaten und reigne unendlicher matrizen, jour,” Journal für die reine und Angewandte Mathematik, vol. 171, pp. 193–226, 1934.
[2]  I. J. Maddox, Infinite Matrices of Operators, Springer, Heidelberg, Germany, 1980.
[3]  A. Wilansky, Summability Through Functional Analysis, North-Holland Mathematics Studies, Amsterdam, The Netherlands, 1984.
[4]  P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, New York, NY, USA, 1981.
[5]  M. I. Ostrovskii, “Hahn-Banach operators,” Proceedings of the American Mathematical Society, vol. 129, no. 10, pp. 2923–2930, 2001.
[6]  A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw Hill, New York, NY, USA, 1978.
[7]  Y. Y?lmaz, “Characterizations of some operator spaces by relative adjoint operators,” Nonlinear Analysis, Theory, Methods and Applications, vol. 65, no. 10, pp. 1833–1842, 2006.
[8]  B. Choudhary and S. Nanda, Functional Analysis With Applications, John Wiley & Sons, New York, NY, USA, 1989.
[9]  G. K?the, Topological Vector Spaces, Springer, Heidelberg, Germany, 1969.
[10]  Y. Y?lmaz, “Relative bases in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 5-6, pp. 2012–2021, 2009.

Full-Text

comments powered by Disqus