A new note on generalized absolute matrix summability

This paper gives necessary and sufficient conditions in order that a series should be summable , , whenever is summable . Some new results have also been obtained.

MSC: 40D25, 40F05, 40G99.

Let be a given infinite series with the partial sums . Let be a sequence of positive numbers such that

The sequence-to-sequence transformation

defines the sequence of the Riesz means of the sequence generated by the sequence of coefficients (see [3]). The series is said to be summable , if (see [1])

Let be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence to , where

The series is said to be summable , if (see [7])

where

If we take , then summability is the same as summability.

Before stating the main theorem we must first introduce some further notations.

Given a normal matrix , we associate two lover semimatrices and as follows:

and

It may be noted that and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have

and

If A is a normal matrix, then will denote the inverse of A. Clearly if A is normal, then is normal and has two-sided inverse , which is also normal (see [2]).

Sarıgöl [6] has proved the following theorem for summability method.

Theorem ASuppose thatandare positive sequences withandas. Thenis summable, wheneveris summable, if and only if

provided that

Theorem BThesummability implies the, , summability if and only if the following conditions hold:

where

and we regarded that the above series converges for eachvand Δ is the forward difference operator.

It may be remarked that the above theorem has been proved by Orhan and Sarıgöl [5].

Lemma ([4])

if and only if

for the cases, wheredenotes the set of all matricesAwhich mapinto.

The aim of this paper is to generalize Theorem A for the and summabilities. Therefore we shall prove the following theorem.

TheoremLet, andbe two positive normal matrices such that

Then, in order thatis summablewheneveris summable, it is necessary that

Also (19)-(21) and

are sufficient for the consequent to hold.

It should be noted that if we take and , then we get Theorem A. Also if we take , then we get Theorem B.

Proof of the Theorem Necessity. Let and denote A-transform and B-transform of the series and , respectively. Then, by (8) and (9), we have

For , we define

Then it is routine to verify that these are BK-spaces, if normed by

and

respectively. Since is summable implies is summable , by the hypothesis of the theorem,

Now consider the inclusion map defined by . This is continuous, which is immediate as A and B are BK-spaces. Thus there exists a constant M such that

By applying (25) to ( is the vth coordinate vector), we have

and

So (26) and (27) give us

and

Hence it follows from (28) that

Using (17), we can find

The above inequality will be true if and only if each term on the left-hand side is . Taking the first term,

then

which verifies that (19) is necessary. Using the second term, we have

which is condition (20). Now, if we apply (22) to , we have

and

respectively. Hence

Hence it follows from (28) that

Using (18) we can find

which is condition (21).

Sufficiency. We use the notations of necessity. Then

which implies

In this case

On the other hand, since

by (22), we have

By considering the equality

where is the Kronecker delta, we have that

and so

Let

Since

to complete the proof of Theorem, it is sufficient to show that

Then

where

Now

is equivalently

by Lemma. But it follows from conditions (20), (21) and (23) that

Finally,

where

Now

is equivalently

by Lemma. But it follows from conditions (21) and (24) that

Therefore, we have

This completes the proof of the Theorem. □

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