Internat. J. Hath. & Hath. Sci.
VOL. 13 NO. 2 (1990) 227-232
227
UNIFORM TOEPLITZ MATRICES
l.J. MADDOX
Department of Pure Mathematics
Queen’s University of Belfast
Belfast BT7 INN
Northern Ireland
(Received October 19, 1988)
ABSTRACT.
We characterize all infinite matrices of bounded linear operators
on a Banach space which preserve the limits of uniformly convergent sequences
defined on an infinite set.
Also, we give a Tauberian theorem for uniform
summability by the Kuttner-Maddox matrix.
KEY WORDS AND PHRASES.
Taubian theorems.
Uniform Toeplitz mx, strong
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
INTRODUCTION.
fk
T
slow
oscon,
40CO5, 4OEO5.
By T we denote any infinite set of objects and we consider functions
X for k
where (X,
I, 2
II) is a Banach space.
II.
The notation
fk
=> f will be used to signify that
on T, which is to say that there exists f
exists k
T
f as k
fk
,
uniformly
X such that for all e > 0 there
k (e) > O with
o
o
..llfk(t)
f(t)
ll
Now suppose that for n,k
bounded linear operator on X.
< e, for all k > k
1,2
each
Ank
o
E
and all t
E
T.
B(X), i.e. each
Then we shall say that A
(Ank)
Ank
is a
is a uniform
Toeplitz matrix of operators if and only if:
r.
k=l
for each n
Z
fk
converges in the norm of X
{1,2,3,4,...
N
k=l
whenever
Ankfk(t)
Ankfk
and each t
T and
=> f
=> f"
Following Robinson [i] and Lorentz and Macphail [2], if (Bk) is a sequence in
B(X) we denote the group norm of
by
()
228
I.J. MADDOX
P
E N and all x
in the closed unit sphere of X.
k
By C we shall denote the (C,I) matrix of arithmetic means, given by
where the supremum is over all p
i
O
i
I
2
2
1
3
1
3
0
O
O
O
O
O
1
3
0
0
By D we denote the Kuttner-Maddox matrix, used extensively in the theory of
strong summability
1
0
O
[3, 4, 5]:
O
O
i
I
2
2
0
0
O
O
O
O
O
O
0
0
0
0
0
0
I
i
I
I
4
4
4
4
0
0
In work on strong sunnability it is often advantageous to use the fact that,
for non-negative
sense that
Pk
(pk) the sunnability methods C and D are equivalent, in the
O(D).
O(C) if and only if
Pk
In connection with Tauberian theorems we now introduce the idea of uniform
strong slow oscillation.
Then we say that (s
T
Let s
X for each k E N.
k
slow oscillation if and only if
n/k
s
Sn
O(1)-
k
k)
has uniform strong
and n > k with
=> 0 whenever k
In what follows we shall regard s as the k-th partial sum of a given
k
series of functions
2.
r.ak
a
+ a
I
2
+
each a
k
T
X.
UNIFORM TOEPLITZ MATRICES.
The following theorem characterizes the uniform Toeplitz matrices of
operators which were defined in Section I.
THEOREM i.
A
(Ank)
SUPnIl(Anl, An2,
is a uniform Toeplitz matrix if and only if
...)II
<
,
(2.1)
(2.2)
A is column-finite,
for each n
A
n
PROOF.
N, An
Y.
k= I
Ank
converges,
I, ultimately in n.
We remark that in (2.3) the convergence
(2.3)
(2.4)
is in the strong operator
topology, and in (2.4), I is the identity operator on X.
For the sufficiency, let H denote the value of the supremum in (2.1), let
UNIFORM TOEPLITZ MATRICES
n
T.
N and t
..llfk(t)
Then, for any e > O there exists k
f(t)
l_
e for all k > k
P
E
P
E
k=l
Ankfk(t)
k=l
where we assume that
k=l
such that
o
o
N,
Now for each p
P
Z
229
fk
f(t)) +
Ank(fk(t)
By (2.3), as p
=> f"
Ankf(t)
P
E
k=l
Ankf(t),
,
we have
Anf(t).
Also, if s > r > k o
s
Z Ank(fk(t)
II k=r
whence
Z
k=l
Ankfk (t)
f(t))ll
n
N such that
o
n > m + n
o
I for all n > m, and by (2.2)
N such that A
By (2.4) there exists m
k=l
He,
converges.
there exists n (e)
Taking
<
O for I < k < k
Ank
o
and for n > n
o
we have
Ankfk(t)
Ank(fk(t)
f(t) +
k=l+k
f(t)).
o
Since
II k--l+k Ank(fk(t)
f(t))ll
<
)II,
ell(Anl, An2
o
it follows by (2.1) that
ZAnkfk
=> f, which proves the sufficiency.
Take any convergent sequence (x
k)
Now consider the necessity.
fk(t)
x. Define
x
k
T.
all t
Then
fk
x
k
for all k
=> f and so
ZAnkXk
in X, with
T, and define f(t)
N and all t
x for
converges for each n and tends to x,
whence the usual Toeplitz theorem for operators, see Robinson [i] or Maddox
[6],
yields (2.1) and (2.3) of our present theorem.
Then there exist natural numbers
Next, suppose that (2.4) is false.
X with
Hence there exist x.
with A
n(1) < n(2) <
# I for all i N
n(i)
(2.5)
for all i
N.
Let us write y(i) for the expression inside the norm bars in (2.5).
Since T is an infinite set we may choose any countably infinite subset
{tl, t2, t3,
...} of T.
f(ti)
Then we define f
xi/I ly(i) ll
T
X by
(2.6)
230
l.J. MADDOX
for all i
N, and f(t)
we certainly have
fk
0 otherwise.
If we define
N then
f for all k
fk
But A is not a uniform Toeplitz matrix, since for
=> f"
n --n(i) we have by (2.6),
II k= IAnkf(t i) f(ti) ll
A
x.
n i
I
II/I ly(i) ll
Hence, if A is a uniform Toeplitz matrix then (2.4) must hold, and a similar
argument shows that (2.2) is necessary, which completes the proof of the theorem.
Since C, the (C,I) matrix, is not column-finite we immediately obtain:
COROLLARY 2.
C is a Toeplitz matrix but not a uniform Toeplitz matrix.
However, since the elements of the Kuttner-Maddox matrix D are non-negative
and its row sums all equal I it is clear that the conditions of Theorem I hold,
whence D is a uniform Toeplitz matrix.
=> f it follows that
Thus, whenever
fk
2
-r
Irfk=>
where the sum in (2.7) is over 2
express (2.7) by writing
fk
(2.7)
f,
r <
r+l
k < 2
for r
We also
O,1,2,
=> f(D).
The relation between C and D for uniform summability is given by:
THEOREM 3.
PROOF.
fk
=>
f(C) implies
fk
=>
f(D), but
not conversely in general.
Write
c(n)
n
-I
n
r.
k=l
fk(t)
and
d(r)
2
-rr.rfk (t).
Then we find that
d(r)
(2
2 -r) c( 2
r+l
i)
(i-
2-r)c(2 r
(2.8)
I),
and it is clear that the right-hand side of (2.8) defines a uniform Toeplitz
transformation between the c and d sequences.
fk(t)
Then
fk
For the last part of the theorem we may define real-valued functions on T by
r
r
r
r
2 when k
2 and
-2 when k
i + 2
0 otherwise
and
fk(t)
fk
=>
=> f(D).
O(D).
Hence f
c(2 r)
O.
(i-
fk
=>
f(C), which implies
But
2-r)c(2 r
contrary to the fact that c(n)
3.
fk(t)
Now suppose, if possible, that
i)
i,
O.
A UNIFORM TAUBERIAN THEOREM.
By the remark following Corollary 2 we know that
fk
=> f implies
fk
but the example of Theorem 3 shows that the converse is generally false.
=>
f(D),
The
next result shows that uniform strong slow oscillation is a Tauberian condition
for uniform D summability.
231
UNIFORM TOEPLITZ MATRICES
If (s
THEOREM 4.
then s
k)
has uniform strong slow oscillation and s
k
-->
f(D)
--> f.
k
PROOF.
Without loss of generality we may suppose that f
O.
r
N and determine r such that 2 -< n < 2 r+l
If e > 0 there exists
r
r+l
such that if 2 < k < 2
then
Take n
r
o
lSk(t) Sn(t)l
whenever r > r
T.
and t
o
<
Since
2-rZrsk(t) Sn(t) + 2-rZr(Sk(t) Sn(t))
we see that s
--> O.
n
Our final result shows that the natural conditions ka => 0 or ka => O(C,I)
k
k
are both Tauberian conditions for uniform D summability, but that the restriction
ka => O cannot be relaxed to the uniform boundedness of
k
THEOREM 5. (i) If ka => O or ka => O(C,I) and s => f(D) then s => f.
k
k
k
k
(ii) There exists a divergent series la with (ka
uniformly
k
bounded and s => O(D).
k
PROOF. (i) First note that ka --> O does not generally imply ka --> O(C,I)
k
k
because (C,I) is not a uniform Toeplitz matrix by Corollary 2. We shall show
(kak).
k)
that ka => O(C,I) is a Tauberian condition for D, the proof for ka --> O being
k
k
has
similar.
In fact we shall show that kak => O(C,I) implies that (s
uniform strong slow oscillation.
k)
ak(t), Sn Sn(t)
Let us write a
k
and
n
A
n
n
with the assumption that A
n
s
s
n
A
k
k
n
Z
ka.,
k=l
Ak
+
=> O.
n-I
Z
=k+l
Then for n > k > i, by partial summation,
(A- kAk)/(+l)
whence
llSn Skll <max{llAv II
If n/k
=0(I)
n
n
(ii)
s
n}(l +
k
n
+ 2
Z
=-k+l
__).i
then
k
+ 2
I +
and so s
k < u <
k
n
l
=k+l
=>
1<2+
+I
2
0(i),
O, as required.
Define a numerical sequence (s
k)
by s
k
--0 when 1 < k < 4, and for n >2
232
l.J. MADDOX
define s
and s
k
2
0 when k
k
2
-i when k
the graph of (s
k)
n
n
and when k
+ 3
2
n-2
2
n
+ 2
2
n-2;
s
Otherwise define s
is a triangular-shaped wave.
Then
1 when k
k
k
(Sk)
2
n
+ 2
n-2
linearly, so that
diverges and it is
klakl
O for all r e O.
8
Also, it is easy to check that
k
for all k e i, whence our result follows on defining
s for all k e I
k
T.
and all t
clear that
ErS
Sk(t)
REFERENCES
i.
ROBINSON, A.
2.
LORENTZ, G.G. and MACPHAIL, M.S. Unbounded operators and a theorem of
Trans. Royal Soc. of Canada. XLVI (1952) 33-37.
A Robinson.
3.
KUTTNER, B. and MADDOX, l.J.
4.
MADDOX, l.J.
On functional transformations and summability.
Math. Soc. 52 (1950) 132-160.
On strong convergence factors.
J. of Math. (Oxford). 16 (1965) 165-182.
On
Kuttner’s
theorem.
Proc. London
Quarterly
J. London Math. Soc. 43 (1968)
285-290.
5.
Inclusions between FK spaces and Kuttner’s theorem.
MADDOX, l.J.
Math. Proc. Camb. Phil. Soc. IO1 (1987) 523-527.
6.
MADDOX, l.J.
Infinite matrices of operators,
Springer-Verlag, 1980.