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.
Highlights
We know from the classical sequence spaces theory that there is a useful relationship between continuous and ββ-duals of a scalarvalued FK-space EE originated by the AK-property
E idea of dual sequence space was introduced by Köthe and Toeplitz [1]. en, Maddox, [2], generalized this notion to XX-valued sequence classes where XX is a Banach space. is brings an important contribution to the operator matrix transformation of Banach space-valued sequence spaces
EEαα = aakk ∈ ww w aakkxxkk < ∞ for all xx xxx, kkkk where ww is the space of all complex-valued sequences. e in the replaced by classical de nitions of Köthe-Toeplitz duals is a sequence (AAkk)∞ kkkk of linear operators, not necessarily continuous, from XX into another Banach space
Summary
E idea of dual sequence space was introduced by Köthe and Toeplitz [1]. en, Maddox, [2], generalized this notion to XX-valued sequence classes where XX is a Banach space. is brings an important contribution to the operator matrix transformation of Banach space-valued sequence spaces. E (aakk) in the replaced by classical de nitions of Köthe-Toeplitz duals is a sequence (AAkk)∞ kkkk of linear operators, not necessarily continuous, from XX into another Banach space. Us, if EE is a nonempty set of sequences xx xxxxkk) with xxkk ∈ XX, generalized ββ- and αα-duals of EE are de ned to be EEββ = AAkk ∶ AAkkxxkk convergent in the YY-norm kkkk (2). International Journal of Analysis to show that there is an analogue relationship for XX-valued sequence spaces in the context of generalized ββ-duals with respect to another xed Banach space YY. By applying this result, we obtain generalized ββ-duals of some vectorvalued Orlicz sequence spaces. We think that our results give a fruitful way to nd generalized duals of this kind of vectorvalued sequence spaces
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