UNIFORM TOEPLITZ MATRICES

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.


T
X such that for all e > 0 there exists k k (e) > O with o o ..llfk(t) f(t) ll < e, for all k > k and all t E T.
o Now suppose that for n,k 1,2 each Ank E B(X), i.e. each Ank is a bounded linear operator on X.
Then we shall say that A (Ank) is a uniform Toeplitz matrix of operators if and only if: r.Ankf k(t) converges in the norm of X k=l for each n N {1,2,3,4,... and each t T and Z Ankfk => f k=l whenever fk => f" Following Robinson [i] and Lorentz and Macphail [2], if (B k) is a sequence in B(X) we denote the group norm of () by I.J.MADDOX P where the supremum is over all p E N and all x k in the closed unit sphere of X.
By C we shall denote the (C,I) matrix of arithmetic means, given by 1 1 1 0 0 3 3  3   By D we denote the Kuttner-Maddox matrix, used extensively in the theory of strong summability [3, 4, 5]: In work on strong sunnability it is often advantageous to use the fact that, for non-negative (pk) the sunnability methods C and D are equivalent, in the sense that Pk O(C) if and only if Pk O(D).
In connection with Tauberian theorems we now introduce the idea of uniform strong slow oscillation.
Let s k T X for each k E N.
Then we say that (s k) has uniform strong slow oscillation if and only if Sn s k => 0 whenever k and n > k with n/k O(1)- In what follows we shall regard s k as the k-th partial sum of a given series of functions r.a k a I + a 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) is a uniform Toeplitz matrix if and only if SUPnIl(Anl, An2, ...)II < , A is column-finite, n PROOF.We remark that in (2.3) the convergence 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 ( it follows by (2.1) that ZAnkfk => f, which proves the sufficiency.Now consider the necessity.
Take any convergent sequence (x k) in X, with x k x. Define fk(t) x k for all k N and all t T, and define f(t) x for all t T.
Then fk => f and so ZAnkXk 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.Next, suppose that (2.4) is false.
Then there exist natural numbers n(1) < n(2) < with A # I for all i N Hence there exist x.X with 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. Then we define f T X by f(ti) xi/I ly(i) ll (2.6) l.J. MADDOX for all i N, and f(t) 0 otherwise.
If we define fk f for all k N then we certainly have fk => f" But A is not a uniform Toeplitz matrix, since for n --n(i) we have by (2.6), II Ankf(ti) f(ti) ll A x. II/I ly(i) ll k= I n i I 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.Thus, whenever fk => f it follows that where the sum in (2.7) is over We also express (2.7) by writing fk => f(D).
The relation between C and D for uniform summability is given by: THEOREM 3. fk => f(C) implies fk => f(D), but not conversely in general.
and it is clear that the right-hand side of (2.8) defines a uniform Toeplitz transformation between the c and d sequences.
For the last part of the theorem we may define real-valued functions on T by fk(t) 2 r when k 2 r and fk(t) -2 r when k i + 2 r and fk(t) 0 otherwise Then fk => O(D).Now suppose, if possible, that fk => f(C), which implies fk => f(D).Hence f O.But c(2r contrary to the fact that c(n) O.
By the remark following Corollary 2 we know that fk => f implies fk => f(D), but the example of Theorem 3 shows that the converse is generally false.The next result shows that uniform strong slow oscillation is a Tauberian condition for uniform D summability.If (s k) has uniform strong slow oscillation and s k --> f(D) then s k --> f.
PROOF.Without loss of generality we may suppose that f O.
Take n N and determine r such that 2 r -< n < 2 r+l   (ii) Define a numerical sequence (s k) by s k --0 when 1 < k < 4, and for n >2 ,I) is not a uniform Toeplitz matrix by Corollary 2. We shall show that ka k => O(C,I) is a Tauberian condition for D, the proof for ka k --> O being similar.In fact we shall show that ka k => O(C,I) implies that (s k) has uniform strong slow oscillation.the assumption that A => O. Then for n > k > i, by partial summation, <max{llAv II k < u < n}(l + k + 2 Z __).i as required. n O cannot be relaxed to the uniform boundedness of (kak). =>