I nternat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 257-261
257
NONTRIVIAL ISOMETRIES ON Sp()
STEPHEN L. CAMPBELL
Department of Mathematics
North Carolina State University
Raleigh, North Carolina 27650 U.S.A.
(Received January 12, 1981)
ABSTRACT.
llxll
.
I.= 0
s () is a Banach space of sequences x with
P
"IxiIp + i=OI Ixi+I
xi Ip"
For i < p
there are no nontrivial surjective isometries in s ().
P
that there are no nontrivial isometries.
,
2, 0
p
< a <
,
a
# i,
It has been conjectured
This note gives two distinct counterexam-
ples to this conjecture and a partial affirmative answer for the case of isometries
with finite codimension.
KEY WORDS AND PHRASES.
Isometry, sequential Banach space, Banach space.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
Primay 47B37.
INTRODUCTION.
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
258
S.L. CAMPBELL
For i < p < =, p # 2,
complex sequences x
k}
{x
> 0, let s (e) denote the linear space of all real or
p
for which
llxll:
s (0)
<.
P
isometric to a subspace of
P
,
In [3], it is shown that for I < p
Sp(e)
metries in
p
If a > 0, then s (a) is isomorphic but not isometric to E
P
P
I01I p+I Ix+-xl
>
s (a) is
P
P
0, e # i, the only surjective isoIn [3], it is conjectured
are scalar multiples of the identity.
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
>
P
0, then
one kind cannot have finite codimension.
It is always assumed that p # 2.
Unless stated otherwise, we allow
i,
I.
e
0, or p
2.
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.
For x E
EXAMPLE i.
Tx
3
Sp (),
-I/p{xo’ x
x
i < p <
=,
define T by
(2.1)
I ’Xo’Xl’X2’Xl’X2’X3’X2’X3’X4’X3’X4
Then T is a Lamperti isometry.
Note, that if T
is an isometry on s
P
() for two distinct values of a, then T
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)
259
NONTRIVIAL ISOMETRIES
where each string of
. .
peated n times.
is repeated m times and each string of
8x.z
x
i)
is re-
Then
IITxIIP= mB p Ixi lp + ny [ lxi+1
(m + 2)8 p
Define S
Y(xi+I
Ixi Ip +
8-1(m + 2a)-I/PT.
xi Ip
(n + 2e)y p
.
+ 2aB p
Ixil p + 2yP I lxi+I
[.
xi Ip
xi Ip.
Ixi+I
Then S is an isometry on s (a) if and only if
P
m
+
(2.3)
a"
2
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
Thus each s (e) has isometries which are not isometries for
not a Lamperti isometry.
P
any other value of e.
3.
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
This section will show that there are no Lamperti isometries of
finite codimension.
finite codimension that are no surjective.
[3].
For x, y 6
The key will be the following fact from
let xy be the sequence (xy)
Sp(a),
i
{0,Xo,Xl
Let Vx
xiY i.
}"
Then, for any isometry T on s (),
P
xy
0
x(Vy)
0
(Vx)y
0
implies that
Let e. be the standard basis for
z
E.I
codimension.
Let
Ei N Ei+k
Ei, Ej adjoin
if i
p
(Tx) (Ty)
0
(Tx) (VTy)
0
(VTx)(Ty)
0
Suppose that T is an isometry of finite
{k
be the support of Te
that is, E
i,
i
_> O,
k
_> 2. Note
if there exists i
I
e
in Example 2 that E
Ei, J l
i
e E. such that
3
metry T, the following hold:
(i)
E
(ii)
E
i
i
and
Ei+ I
and
Ei+k,
either adjoin or intersect
k
(3.1)
2 neither adjoin nor intersect.
N
i
(Tel) k
#
El+ 1
I
j
l
.
# 0}.
By (3.1)
We shall say
i.
For any iso-
S.L. CAMPBELL
260
To see (i), suppose that (i) does not hold.
(ii) follows immediately from (3.1).
If i
i,
JIBe i + ei+lll p= IBI p
+ I + a(IBl p +
BI p
Ii
+ i)
(3.2)
whereas
IIBTe i + Tei+lll p= 181ITeilIP+ llTei+IIl p= IBIP(I +
But
l[Se i +
depends on
liT(Be i + ei+l) ll,
ei+ll
,
2a) + i + 2a.
so that (3.2) and (3.3) are equal.
II,
whereas (3.3) depends on
(3.3)
Since
(3.2)
A
we have a contradiction if a > 0.
O.
similar proof works if i
If T is a Lamperti Isometry of finite codimension in s (), then T
THEOREM i.
P
is surjective.
Since T is a Lamperti
PROOF.
singletons.
Isometry, all but a finite number of the E i are
What’s more, since Eo must adjoin, but not intersect
hk, O, ...)
Let (h
0 if r < k.
allow some h
r
I1
p
+ 1 +
+
a
.
+
.
k-i
+
-IBlPlhk Ip + alBhk
m
i=O
ii p
# 0 and
1
+
l] p +
{k + i}.
llBTej + Tej+lll p,
I IlPlhi
i=0
p
1
IlP]hi+l- hi Ip + IlPlhk
+
a
i
+
IIPlITejll p- clslPlhk Ip + clShk
1
+
[lP(1
+ 2c)
p
Ej+I
k
l)
[B]P]hi+I hilP+]hk-
Y. lmlPlhi lp
i=O
k
IIBej + ej+iIl
Now, if j _> i,
k
i
where h
Tej,
p
c(lel p + I1- el +
i=O
after
Let E. be the last set which is not a
some index, the sets are listed in order.
singleton.
El_ I, Ei+ I,
cllPlhk Ip + ctlBh k
iIp
+
c
iIp +
a-
We
or
261
NONTRIVIAL ISOMETRIES
Working with the first and last terms gives
ll- 81 p
181 p
ls{Plhk Ip +
IBhk llP-or
Ii- 81 p
181 p
181Plhk ]p +
ii p.
]Bh k
(3.4)
Equation (3.4) holds for all 8, and both sides of (3.4) are differentiable with respect
O, i, i/h k.
to 8, except at 8
Differentiating with respect to 8 gives for p > i,
and 8 > 0, 8 small,
PlSI p -I
81p-i
_pll
PIll p-I lhk Ip
But (3.5) holds in an interval of the form 0
<
ii p-I
PhkIShk
B
< e
(3.5)
Hence, it holds for 8
0
+
k
Thus -p
-ph
Hence, E_.
k
or h
{k},
k
i.
But T is also an isometry in
which is a contradiction.
p
h
i=O
0 is similar.
so that
P
The proof for j
I il
i.
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
too hard).
we can see.
Ei,
The difficulty is that h
k
for i greater than some k, singletons is not
I no longer provides any contradiction that
On the other hand, numerical counterexamples seem quite messy.
REFERENCES
I.
CAMPBELL, S. L., FAULKNER, G. D., and SINE,
2.
DAVIS, W. J. Separable Banach Spaces with Only Trivial Isometries, Rev. Roum.
Pures Appl. XVI, 1971, 1051-1054.
3.
FLEMING, R. J. and JAMISON, M. E.
(to appear).
4.
LAMPERTI, J.
R. Isometries, projections, and
Wold Decompositions, in Operator Theory and Functional Analysis, I. Erdelyi
Editor, Pitman Publishing Company, 85-114.
The Isometries of s
P
(a), Canadian J. Math.,
On the isometries of certain function spaces, Pacific J. Math.
2 (1958), 459-566.