SUBMATRICES OF SUMMABILITY MATRICES

It is proved that a matrix that maps l1 into l1 can be obtained from any regular matrix by the deletion of rows. Similarly, a conservative matrix can be obtained by deletion of rows from a matrix that preserves boundedness. These techniques are also used to derive a simple sufficient condition for a matrix to sum an unbounded sequence.

J.A. FRIDY the well-known theorem of Kojima and Schur [6, p. 43] that characterizes those matrices A that map the set c (convergent sequences) into c by the three con- ditions: (i) for each k, lim a n nk (ii) lim S n lk=0 ank (iii) SUPn {Zk= 0 lank Such a matrix is called a conservative matrix.A regular method preserves limit values as well as convergence, and such matrices are characterized by the Silverman-Toepiitz conditions (i), (ii), (iii) in which S=I and o k -= 0. Some of the well-known summability matrices are bothand regular methods [5].The main purpose of this paper is to establish a general corre- spondence between regular matrices andmatrices by showing that every regular matrix gives rise to anmatrix by the deletion of an appropriate set of rows.A similar theorem is proved that asserts that a matrix that maps the set m (bounded sequences) into m contains a row-submatrix that is conservative.In the final section, the row-selection technique is replaced by a column-selection technique in order to prove a simple criterion for the summability of an unbounded sequence.

THE MAIN RESULTS.
Although our primary motivation is concerned with regular matrices, we can relax considerably the Silverman-Toeplitz conditions and still select the row- submatrix that we seek.THEOREM i.If A is a summability matrix in which each row and each column converge to zero and SUPn,k lank < =, then A contains a row-submatrix that is anmatrix.
PROOF.First choose a positive integer (0) satisfying a <= 1; then (o) ,o using the assumption that lira k a(0),k 0, choose K(0) so that k > K(0) implies lav(0),k _s I. Having selected v(i) and K(i) for i < m, we choose v(m) greater than 9(m-l) so that Now define the submatrix B by bmk a(m),k" The above construction guarantees that each column sequence of B is dominated, except for at most one term, by the sequence {2-m}; i.e., if K(m-l) < k <-(m) and i # m, then Hence, by (1.2), B is anmatrix.
We can now state our principle objective as an immediate consequence of this theorem.
COROLLARY i.Every regular matrix contains a row-submatrix that is an matrix.
It is easy to see that if A is regular, then the submatrix B of the pre- ceding proof is bothand regular; for, any matrix method is included by a method determined by one of its row-submatrices.Also, it is obvious that in Corollary 1 it is not sufficient to assume only that A is conservative; for if # 0 for some k, then Zm=0 la(m),k for any choice of {(m) m= 0. Yurther- a k more, it is easy to see that not everymatrix is a submatrix of a regular matrix; e.g., if b0,k 1 and bmk 0 (when m 0) for every k, then B is but SUPnZk=0]bnkl .
Another way of ensuring that the hypotheses of Theorem 1 hold is to assue that A maps P into q, where p > and q > i.Although explicit row/column conditions that characterize such a matrix are not known, it is easy to see that the columns of A must be in q and the rows must be uniformly bounded in P', where i/p + i/p' I. Thus we state this formally in the following result.
COROLI&RY 2. If A maps P into ,tq, where p > 1 and q > 1, then A con- tains a row-submatrix that is anmatrix.
For the next theorem, we prove a variant of Corollary 2 in which P and are replaced by m and c, respectively.THEOREM 2. If A maps m into m, then A contains a row-submatrix B that is conservative.
PROOF.Since A maps m into m, we have sup n Ek=0]ankl < .Therefore the sequence of row sums {Ek 0 ank}n= 0 is bounded, so we can choose a convergent subsequence.This yields a row-submatrix A' of A that satisfies properties (ii)   and (iii).It remains to choose a row-submatrix of A' whose columns are con- vergent sequences.But this is simply a special case of the familiar diagonal process that is used in the proof of the Helley Selection Principle (see, e.g., [2, p. 227]); for we have a family of functions (the rows of A') that are uni- formly bounded by SUPnY.k=01ank on their countable domain {0, I, 2, ...}.There- fore we can select a sequence of these "functions" that converges at each k.
This sequence of rows of A' are then the rows of B.
[i], R. P. Agnew proved that if A is a regular matrix such that lim 0 n, k+ ank (3.1) then there exists a nonconvergent sequence of zeros and ones that is summable by A. It then follows by the well-known theorem of Maur and Orlicz [8] that A sums an unbounded sequence.Because the Mazur-Orliez Theorem requires the development of Fk-spaces, it would be useful to have a direct construction of an unbounded sequence that is summed by such an A. By modifying the proof of Theorem 1 from row selection to column selection, we can prove a theorem in which we relax the regularity of A, weaken property (3.1), and construct an unbounded sequence that is summed by A. Then choose increasing row indices {m)}m= 0 so that if k= < K(m) and n > (m), then lank < 2-m.Now define the sequence x by m + I, if k <(m) for some m, O, otherwise.

THEOREM 2 .
If A is a summability matrix whose column sequences tend to zero and lim inf k{max nlankl O, (3.2) then A sums an unbounded sequence.PROOF.Using (3.2), we choose an increasing sequence of column indices (m) }m= 0 such that for each m, maXn la n, (m) < 2-m-(3.3) j >m (m + i) (m + 2)2 -m-I +R m where lim R 0. Hence limn --(AX)n 0. m m