Duality decompositions of some matrix sequence spaces

Abstract

Let [Formula: see text] be fixed. For each [Formula: see text], we define the set [Formula: see text] of sequences of infinite complex matrices and the norm [Formula: see text] on [Formula: see text] analogously to the classical sequence space [Formula: see text] and the [Formula: see text]-norm as follows: [Formula: see text] and [Formula: see text] where [Formula: see text] is referred to as the operator norm on [Formula: see text]. We first study the completeness of sequence space [Formula: see text] equipped with the norm [Formula: see text]. Some duality results are also discussed. The aim of this paper is to show that for the case where [Formula: see text], the dual [Formula: see text] of [Formula: see text] can be decomposed as an [Formula: see text] direct-sum of its two closed subspaces. This is done by a way analogous to the theorem of Dixmier on decomposing the dual [Formula: see text] of [Formula: see text].