Let$X$and$Y$be closed subspaces of the Lorentz sequence space$d(v, p)$and the Orlicz sequence space$l$_{$M$}, respectively. It is proved that every bounded linear operator from$X$to$Y$is compact whenever $$p > \beta _M : = \inf \{ q > 0:\inf \{ M(\lambda t)/M(\lambda )t^q :0< \lambda ,t \leqslant 1\} > 0.$$ As an application, the reflexivity of the space of bounded linear operators acting from$d(v, p)$to$l$_{$M$}is characterized.

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In this paper, I study the group of analytic automorphisms of a bounded product domain in the space$C(S,C)$of continuous functions on a compact space$S$. I prove that its automorphism group is a Lie group and I am able to prove which are the bounded symmetric ones.

First we define the dyadic Hardy space$H$_{$X$}$(d)$for an arbitrary rearrangement invariant space$X$on [0, 1]. We remark that previously only a definition of$H$_{$X$}$(d)$for$X$with the upper Boyd index$q$_{$x$}<∞ was available. Then we get a natural description of the dual space of$H$_{$x$}, in the case$X$having the property 1<-$p$_{$X$}<-$q$_{$X$}<2, imporoving an earlier result [P1].