NONTRIVIAL ISOMETRIES ON Sp ( )

s p ( α ) is a Banach space of sequences x with ‖ x ‖ = ( ∑ i = 0 ∞ | x i | p + α ∑ i = 0 ∞ | x i + 1 − x i | p ) 1 / p . For 1 p ∞ , p ≠ 2 , 0 α ∞ , α ≠ 1 , there are no nontrivial surjective isometries in s p ( α ) . It has been conjectured that there are no nontrivial isometries. This note gives two distinct counterexamples to this conjecture and a partial affirmative answer for the case of isometries with finite codimension.

. "Ixi Ip + I Ixi+I xi Ip" For i < p , p 2, 0 < a < , a # i, .=I 0 i=O there are no nontrivial surjective isometries in s ().It has been conjectured P that there are no nontrivial isometries.This note gives two distinct counterexam- ples to this conjecture and a partial affirmative answer for the case of isometries with finite codimension.
Banach spaces provide a natural setting for a variety of pure and applied problems in functional analysis.The geometry of a Banach space is much more complex then that of the simpler Hilbert spaces.The geometry of a Banach space is closely related to the types of linear isometries that the Banach space admits [i].
There are examples of Banach spaces for which the only isometries from the space into itself are the trivial ones, I [2].These examples are constructed by placing extreme points on the unit ball in an asyvmnetrical pattern.While these examples are of interest it is also helpful to have examples which closely resemble the Banach spaces commonly encountered.
As one step in this direction, Jamison and Fleming [3] studied a discrete ana- logue of the classical Sobolov spaces, which they denoted as s (a).P S.L. CAMPBELL For i < p < =, p # 2, > 0, let s (e) denote the linear space of all real or p complex sequences x {x k} for which llxll: I01I p+I Ix+-xl p <. s (0) If a > 0, then s (a) is isomorphic but not isometric to E s (a) is P P P P P isometric to a subspace of P In [3], it is shown that for I < p , > 0, e # i, the only surjective iso- metries in Sp(e) are scalar multiples of the identity.In [3], it is conjectured that all isometries in s () must be surjective and hence scalar multiples of the P identity.
In this note we exhibit non-surjective isometries for all > 0. We also show that there are essentially two types of isometries in s () and that if > 0, then P one kind cannot have finite codimension.
It is always assumed that p # 2. Unless stated otherwise, we allow i, e 0, or p I.

EXAMPLES AND TERMINOLOGY.
The simplest type of isometry on s () is one that preserves both sums in (i.I).P Such an isometry is an isometry independent of and is an isometry on Thus, it P has the structure developed in [4].We shall call such an isometry a Lamperti iso- metry.
(), i < p < =, define T by Then T is a Lamperti isometry.
Note, that if T is an isometry on s () for two distinct values of a, then T P is a Lamperti isometry.EXAMPLE 2. Suppose that 0 and m > 0, n > 0 are integers.For x E s (), P define T as Tx--{0,BXo,...,BXo,0,y(Xl Xo) Y(xl Xo)'0'8Xl BXl'0' (2.2)  where each string of 8x.z is repeated m times and each string of Y(xi+ I x i) (2.3) But then for any a > 0, any m > I, n > i, S will be an isometry on s (a) provided p satisfies (2.3).Since (2.3) is inconsistent for a 0, we see that this S is not a Lamperti isometry.Thus each s (e) has isometries which are not isometries for P any other value of e.

FINITE CODIMENSION.
The isometry in both Example i and Example 2 has a range of infinite codimension.
This suggests that perhaps there are no isometries on s (a) which have a range with P finite codimension.This section will show that there are no Lamperti isometries of finite codimension that are no surjective.The key will be the following fact from [3].For x, y 6 Sp(a), let xy be the sequence (xy) i xiYi.Let Vx {0,Xo,Xl }" Then, for any isometry T on s (), Let e. be the standard basis for Suppose that T is an isometry of finite z p codimension.Let E.I be the support of Te i, that is, E i {k (Tel) k # 0}.By (3.1) Ei N Ei+k if i _> O, k _> 2. Note in Example 2 that E i N El+1 # .We shall say Ei, Ej adjoin if there exists i I e Ei, J l e E. such that i I j l i.For any iso- (ii) follows immediately from (3.1).To see (i), suppose that (i) does not hold.
If T is a Lamperti Isometry of finite codimension in s (), then T P PROOF.Since T is a Lamperti Isometry, all but a finite number of the E i are singletons.What's more, since Eo must adjoin, but not intersect El_I, Ei+I, after some index, the sets are listed in order.Let E. be the last set which is not a singleton.Let (h h k, O, ...) Tej, where h k # 0 and Ej+I {k + i}.We allow some h r 0 if r < k.Now, if j _> i, IIBej + ej+iIl p llBTej + Tej+lll p, or I1 p + 1 + c(lel p + I1-el p + {k}, which is a contradiction.The proof for j 0 is similar.

Hence,
We conjecture that Theorem 1 is true also for non-Lamperti isometries on s (), P but we have been unable to prove it.With minor modifications, one can proceed exactly as in Theorem i (to get the Ei, for i greater than some k, singletons is not too hard).The difficulty is that h k I no longer provides any contradiction that we can see.On the other hand, numerical counterexamples seem quite messy.
Working with the first and last terms gives ll-81 p