MATHEMATICS

G. KANGRO

ON THE GENERALIZATION OF A THEOREM OF MOORE

(Presented by Academician S. L. Sobolev, 25 IV 1958)

A sequence \(\{\varepsilon_k\}\) is called a sequence of convergence multipliers of the second (first) kind with respect to a given set of series \(\mathfrak X\), if every series\(^*\) \(\sum x_k\) from \(\mathfrak X\) generates a convergent series \(\sum \varepsilon_k x_k\) \(\left(\sum \varepsilon_k (x_0+x_1+\cdots+x_k)\right)\). In the case of absolute convergence of the series \(\sum \varepsilon_k x_k\) \(\left(\sum \varepsilon_k (x_0+x_1+\cdots+x_k)\right)\), one speaks of multipliers of absolute convergence of the second (first) kind.

Let \(A=(a_{nk})\) be a normal matrix, i.e. \(a_{nk}=0\) for \(k>n\), \(a_{kk}\ne0\). A series \(\sum x_k\) is said to be \(A\)-summable (\(A\)-bounded, \(|A|\)-summable) if the sequence \(\{x'_n\}\), defined by the formula

\[ x'_n=\sum_{k=0}^{n} a_{nk}x_k,\qquad (n=0,1,\ldots), \tag{1} \]

converges (is bounded, is absolutely convergent).

In the case of the Voronoi–Nörlund summability method, for which \(a_{nk}=P_n^{-1}P_{n-k}\) (\(\{P_k\}\) is a given sequence, \(P_k\ne0\)), Moore \((^1)\) proved that if the Voronoi–Nörlund method is regular and satisfies the condition

\[ \sum k\,|d_k|<\infty, \tag{2} \]

where \(d_k\) is determined from the relation

\[ \sum k\,|d_k|<\infty, \]

then \(\{\varepsilon_k\}\) is a sequence of convergence multipliers of the second kind with respect to the set of series summable by the Voronoi–Nörlund method if and only if:

1) \(\varepsilon_k=O(P_k^{-1})\), \((k=0,1,\ldots)\),

2) \(\displaystyle \sum \left|P_k\sum_{\nu=k}^{\infty} d_{\nu-k}\varepsilon_\nu\right|<\infty.\)

In the present note Moore’s theorem is generalized to the case when \(x_k\) are elements of some Banach space \(X\); \(\varepsilon_k\) are bounded linear operators from \(X\) into a Banach space \(Y\), and \((a_{nk})\) is any normal numerical matrix subject to the condition

\[ \sum^{\nu} D_\nu<\infty, \tag{3} \]

\(^*\) Unless otherwise stated, the summation indices run through all nonnegative integer values \(0,1,\ldots\).

where \(D_\nu=\sup\limits_k |a_{\nu+k,\nu+k}a'_{\nu+k,k}|\), \((a'_{nk})=(a_{nk})^{-1}\). In this case the set of all \(A\)-summable * (\(A\)-bounded, \(|A|\)-summable) series in \(X\) is denoted simply by \(A\) (\(A_0, |A|\)).

Theorem 1. In order that \(\{\varepsilon_k\}\) be a sequence of convergence multipliers of the second kind with respect to \(A\), it is necessary and sufficient that the following conditions be fulfilled:

1) \(\sum a'_k \varepsilon_k x\) converges for \(x\in X\);

2) \(\|\varepsilon_k\|=O(a_{kk})\) \((k=0,1,\ldots)\);

3) \(\left\|\sum\limits_{k=0}^{p} A_k x_k\right\|=O(1)\) \((\|x_k\|\leqslant 1;\ p=0,1,\ldots)\),

where
\[ a'_k=\sum_{\nu=0}^{k} a'_{k\nu}, \qquad A_k=\sum_{\nu=k}^{\infty} a'_{\nu k}\varepsilon_\nu . \]

Proof. Since we have
\[ \sum_{k=0}^{n}\varepsilon_k x_k = \sum_{k=0}^{n} A_{nk}x'_k, \]
where \(x'_k\) is defined by formula (1) and
\[ A_{nk}=\sum_{\nu=k}^{n} a'_{\nu k}\varepsilon_\nu , \]
it follows that \(\{\varepsilon_k\}\) is a sequence of convergence multipliers with respect to \(A\) if and only if the transformation
\[ y_n=\sum_{k=0}^{n} A_{nk}x'_k \qquad (n=0,1,\ldots) \tag{4} \]
takes every convergent sequence \(\{x'_k\}\subset X\) into a convergent sequence \(\{y_n\}\subset Y\). For this, according to (²), it is necessary and sufficient that:

a) there exist \(\lim\limits_{n\to\infty} A_{nk}x=A_kx\) \((x\in X;\ k=0,1,\ldots)\);

b) there exist \(\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n} A_{nk}x\) \((x\in X)\);

c) \[ \left\|\sum_{k=0}^{p} A_{nk}x_k\right\|=O(1) \]
\[ (\|x_k\|\leqslant 1;\ p\leqslant n;\ n=0,1,\ldots). \]

Condition b) is equivalent to condition 1 of Theorem 1. From c) it follows that \(\|A_{nn}\|=O(1)\), i.e. condition 2). Taking a) into account, from c), as \(n\to\infty\), we obtain condition 3) of Theorem 1.

Conversely, from condition 2) on the basis of (3) follows the fulfillment of condition a). Further, in view of condition 2, we find
\[ \sum_{k=0}^{p}\|A_k-A_{pk}\| = \sum_{k=0}^{p}\left\|\sum_{\nu=p+1-k}^{\infty} a'_{\nu+k,k}\varepsilon_{\nu+k}\right\| \leqslant \]
\[ \leqslant M\left( \sum_{\nu=1}^{p+1}\sum_{k=p+1-\nu}^{p} D_\nu + \sum_{\nu=p+2}^{\infty}\sum_{k=0}^{p}D_\nu \right) < M\sum \nu D_\nu, \]
where \(M\) is a constant. By virtue of (3), now from condition 3) follows c). Theorem 1 is proved.

Let us note that if \(a_{n0}=1\) \((n=0,1,\ldots)\), then \(a'_k=0\) for \(k>0\), \(a'_0=1\), and condition 1) of Theorem 1 is fulfilled. In particular, if \(A\) is a regular

* Limit processes in \(X\) (and also in \(Y\)) are understood in the sense of strong convergence.

the Voronoi–Nörlund summation method, then (2) ensures the fulfillment of condition (3), and from Theorem 1 one obtains a generalization of Moore’s theorem to abstract convergence sets.

Theorem 1 is also applicable if \(A\) is the summation method of weighted arithmetic means \((^3)\). Then condition (3) is fulfilled if \(A\) is regular and satisfies the condition
\(P_{k+1}^{-1}p_{k+1}=O(P_k^{-1}p_k)\).

Using the theorems giving necessary and sufficient conditions for (4) to transform every bounded, respectively absolutely convergent, sequence of the space \(X\) into a convergent sequence of the space \(Y\), or every absolutely convergent sequence of the space \(X\) into an absolutely convergent sequence of \(Y\) \((^4)\), one can prove the validity of the following two theorems. As before, it is assumed that the matrix \(A=(a_{nk})\) is normal and satisfies condition (3).

Theorem 2. In order that \(\{\varepsilon_k\}\) be a sequence of convergence factors of the second kind with respect to \(A_0\), it is necessary and sufficient that the following conditions be satisfied:

1) \(\|\varepsilon_k\|=o(a_{kk})\) \((k=0,1,\ldots)\);

2) \(\sum A_k x_k\) converges uniformly for \(\|x_k\|\leq 1\).

Theorem 3. In order that \(\{\varepsilon_k\}\) be a sequence of factors of (absolute) convergence of the second kind with respect to \(|A|\), it is necessary and sufficient that the following conditions be satisfied:

1) \(\sum a'_k \varepsilon_k x\) converges (absolutely) for \(x\in X\);

2) \(\|\varepsilon_k\|=O(a_{kk})\) \((k=0,1,\ldots)\).

Remark. Theorems 1–3 turn out to be valid for factors of convergence, respectively of absolute convergence of the first kind, if in these theorems (and also in condition (3)) \(a_{nk}\) is replaced by \(a_{nk}-a_{n,k+1}\).

Tartu
State University

Received
8 X 1956

CITED LITERATURE

\(^1\) C. N. Moore, Summable Series and Convergence Factors, 1938.
\(^2\) K. Zeller, Math. Zs., 56, 18 (1952).
\(^3\) G. Kangro, DAN, 99, 9 (1954).
\(^4\) G. Kangro, Izv. AN Estonian SSR, Ser. Technical and Physico-Mathematical Sciences, 5, No. 2, 109 (1956).