A Banach space (X, v) is said to be a member of the class S? if there exists a Banach sequence space (E, n) with absolute norm such that (i) (E, n) possesses a sufficiently P-like semi-inner product, (ii) there exists a sequence of Hilbert spaces {X t } which forms a decomposition of X, and (iii) if x = £ Xj e X, then { ||AJ ||} e E and V(JC) = /4l|x,.||}. In this case we write (X, {X t }, E, ii)eS?. We note that the coordinate vectors u t = (0, 0, ..., I 1 ' 1 *, 0,0, ...) form a normalized basis for (E, fi). In the preceding definition and in the sequel, || • || denotes a Hilbert space norm. The exact meaning of " sufficiently /Mike semi-inner product" is given in [1] but will not be used explicitly in what follows. The class Sf does include l p spaces (1 ^ p < oo) and i p -sums of not necessarily separable Hilbert spaces as well as discrete Banach function spaces with absolutely continuous norm. In fact, the class Sf includes any Banach space (X, v) with hyperorthogonal basis; that is, a basis {«,} with the property that if x = £ <*< w,-, then v(x) = vQ2 |a f | u t ).If X = (X, {Xf}, E, fi) e $f, then an operator U on X may be described by an operator matrix U = (U (J ) where t / y is an operator on Xj to X t . Using the characterization of Hermitian operators obtained in [1], we characterize the onto isometries of X in terms of the operator matrix representation. This result is then applied to extend the theorems of Tarn [8] on isometries of discrete Banach function spaces and reflexive Orlicz spaces. In connection with this we use the solution of a particular functional equation to determine when the defining function for an Orlicz sequence space must be quadratic.We remark here that our Theorem 1 has essentially been obtained in the finitedimensional case by Schneider and Turner [6] and that our work in the present paper as well as in [1] has been inspired by that work.
