Abstract
In this paper we propose a different (and equivalent) norm on which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of for a class of composition operators. We completely characterize multiplication operators which are m-isometries. As an application of the 3-isometry, we describe the reducing subspaces of on when is a finite Blaschke product of order 2.
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