Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 527360 | https://doi.org/10.1155/2011/527360

T. Selmanogullari, E. Savaş, B. E. Rhoades, "Some E-J Generalized Hausdorff Matrices Not of Type 𝑀 ", Abstract and Applied Analysis, vol. 2011, Article ID 527360, 5 pages, 2011. https://doi.org/10.1155/2011/527360

Some E-J Generalized Hausdorff Matrices Not of Type 𝑀

Academic Editor: Alberto d'Onofrio
Received23 Apr 2011
Accepted04 Aug 2011
Published01 Oct 2011

Abstract

We show that there exists a regular E-J generalized Hausdorff matrix which has no zero elements on the main diagonal and which is not of type and establish several other related theorems.

1. Introduction

The convergence domain of an infinite matrix will be denoted by and is defined by , where denotes the space of convergence sequences and . If for two matrices and , we have the relation , we say that is not weaker than . The necessary and sufficient conditions of Silverman and Toeplitz for a matrix to be conservative (some authors use the word convergence-preserving instead of conservative) are as exists for each , exists, follows: . A conservative matrix is called multiplicative if each and regular if, in addition, . If is a conservative matrix, then is called the characteristic of . A conservative matrix is called coregular if and conull if . Regular matrices are coregular, since .

A matrix is called triangular if for all , and it is called a normal if it is, triangular and for all .

Let denote an infinite matrix. Then is said to be of type if the conditions always imply .

Matrices of type were first introduced by Mazur [1] and named by Hill [2]. Hill [2] developed several sufficient conditions for a Hausdorff matrix to be of Type . He showed that there exists a regular Hausdorff matrix which has a zero on the main diagonal, not of type . He also posed the following question: does there exist a regular Hausdorff matrix which has no zero elements on the main diagonal and which is not of type ? Rhoades [3] answered the above question in the affirmative and established several other related theorems. In this paper, we answer the above question for E-J generalized Hausdorff matrices.

We use the words finite sequence to describe a sequence which is containing only a finite number of nonzero terms. It is clear that a triangular matrix which is not a normal cannot be of type , since a finite sequence can be found satisfying (1.1). Also, if a matrix is a normal, there can be no finite sequence as a solution of (1.1). All diagonal matrices with nonzero diagonal elements are of type .

Hausdorff matrices were shown by Hurwitz and Silverman [4] to be the class of triangular matrices that commute with , the Cesáro matrix of order one. Hausdorff [5] reexamined this class, in the process of solving the moment problem over a finite interval, and developed many of the properties of the matrices that now bear his name.

Several generalizations of Hausdorff matrices have been made. In this paper we will be concerned with the generalized Hausdorff matrices as defined independently by Endl ([6, 7]) and Jakimovski [8]. A generalized Hausdorff matrix is a lower triangular infinite matrix with entries where is real number, is a real sequence, and is forward difference operator defined by . We will consider here only nonnegative . For one obtains an ordinary Hausdorff matrix.

From [6] or [8], a generalized Hausdorff matrix (for ) is regular if and only if there exists a function with such that in which case is called the moment sequence for and is called the moment generating function, or mass function, for .

For ordinary Hausdorff summability (see, e.g., [9]), the necessary and sufficient conditions for regularity are that the function , and (1.3) is satisfied with .

The purpose of this paper is to show that there exists a regular E-J generalized Hausdorff matrix which has no zero elements on the main diagonal and which is not of type and establish several other related theorems.

The following is a consequence of [10, Theorem 3.2.1(d)].

Theorem 1.1 (see [10]). If is a normal, conservative and coregular, then is of type if and only if .

The following is a consequence of [11, Theorem 1((a) and (c))].

Theorem 1.2 (see [11]). Let be conservative. is closed in if and only if sums no bounded divergent sequences.

Theorem 1.3. Let Then the corresponding regular E-J generalized Hausdorff matrix is not of type .

Proof. If is a positive integer, then is not of type as remarked above, since it has a zero on its diagonal.
Assume is not a positive integer. From [12], the convergence domains for E-J generalized Hausdorff matrices with moment generating sequence as defined above are , where Since sums no bounded divergent sequences, from Theorem 1.2, is closed in . From Theorem 1.1, since is not dense in the convergence domain of each , none of them is of type .

Let be a conservative matrix. If is dense in then is called perfect. For certain classes of matrices, perfectness and type are closely related. Note that, from Theorem 1.1, type and perfectness are equivalent for normal, conservative, and coregular matrices.

If one examines the sequences of Theorem 1.3 for an integer, then one notes that the corresponding matrices are not normal. It remains to determine if each such matrix is perfect.

Theorem 1.4. Let Then the corresponding regular E-J generalized Hausdorff matrix is not perfect.

Proof. In [12], it is proved that is a regular E-J generalized Hausdorff matrix with , where As mentioned in the proof of Theorem 1.3, is not dense in the convergence domain of each . Hence each is not perfect.

Theorem 1.5 (see [2]). The product of two triangular perfect methods and is also a triangular perfect method.

Theorem 1.6 (see [2]). If the product of two triangular convergence-preserving methods and is of type , then must be of type .

Theorem 1.7 (see [2]). If is normal and is triangular, then is not weaker than if and only if is convergence preserving.

Theorem 1.8. If is normal, conservative, and not of type , and if is not weaker than , then is not of type .

Proof. By Theorem 1.7, is conservative. From the definition of E-J generalized Hausdorff matrix, it is easily shown that multiplication of an E-J generalized Hausdorff matrix is commutative and the result is again an E-J generalized Hausdorff matrix. Also the inverse of a normal E-J generalized Hausdorff matrix is also a normal E-J generalized Hausdorff matrix. Thus and , and the result follows at once from Theorem 1.6.

Theorem 1.9. If is normal and conservative and if is of type and not weaker than , then is of type .

Proof. As in the previous theorem we have , where is conservative. Then , and the conclusion follows again from Theorem 1.6.

From Theorem 1.5 and the multiplication facts of E-J generalized Hausdorff matrix, we obtain the following theorem.

Theorem 1.10. The product of a finite number of perfect E-J generalized Hausdorff methods is likewise a perfect E-J generalized Hausdorff method.

Acknowledgment

The first author acknowledges support from the Scientific and Technical Research Council of Turkey in the preparation of this paper.

References

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Copyright © 2011 T. Selmanogullari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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