# On the Relationship Between a Summability Matrix and its Transpose

Original Publication Date: 1976-Dec-31

Included in the Prior Art Database: 2005-Sep-15

## Publishing Venue

Software Patent Institute

## Related People

J. Swetits: AUTHOR [+3]

## Abstract

Let E, F be sequence spaces, A an infinite matrix of complex numbers, and A' the transpose of A. In this note the relationship between A as a mapping from E to F and A' as a mapping from F to 0 is investigated. A result of Jakimorski and Livne is improved.

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__Page 1 of 8__THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

**On the Relationship Between a Summability Matrix and its Transpose **

by J. Swetits

**Abstract **

Let E, F be sequence spaces, A an infinite matrix of complex numbers, and A' the transpose of

A. In this note the relationship between A as a mapping from E to F and A' as a mapping from F
to 0 is investigated. A result of Jakimorski and Livne is improved.

1970 A. M. S. subject classification; Primary, 40CO5, 40D25; Secondary, 40HO5.

Key words and phrases: summability matrix; FK space; transpose matrix; weak topology.

**Introduction and preliminaries: **

A sequence space is a vector subspace of the space w of all complex sequences. A sequence space E with a locally convex topology, 7, is a K space if the linear functionals

(Equation Omitted)

are continuous. In addition, if (E, -r) is complete and metriz- able, then E is an FK space. A normed FK space is a BK space, If E is a sequence space, we write

(Equation Omitted)

converges

(Equation Omitted)

(Equation Omitted)

(Equation Omitted)

(Equation Omitted)

f F is a subspace of 0 , then E and F form a dual pair

under the bilinear form

(Equation Omitted)

Old Dominion University Page 1 Dec 31, 1976

__Page 2 of 8__On the Relationship Between a Summability Matrix and its Transpose

If F contains the space of sequences with only finitely many non zero terms, ~, then the weak topology on E by F, a(E, F), is a K space topology. In this note it will be assumed that all sequence spaces contain Topologies for dual pairings of the type described above have been considered by Garling 131.

If

(Equation Omitted)

If E is an FK space such that

(Equation Omitted)

for each continuous linear functional, f, an E and each x 6 E, then E is an FK-SAK space. E is an AK space if

(Equation Omitted)

in E.

If

(Equation Omitted)

is an infinite matrix of complex numbers, the nk sequence

(Equation Omitted)

is defined by

(Equation Omitted)

If E, F are sequence spaces such that Ax e F whenever

(Equation Omitted)

, then A is called an E-F method. A' denotes the transpose of A.

In this note we investigate the relationship between A as an E-F method and A' as an F~-O method..

The following spaces will be used in the sequel:

(Equation Omitted)

(Equation Omitted)

the space of all absolutely p summable sequences;

(Equation Omitted)

Old Dominion University Page 2 Dec 31, 1976

__Page 3 of 8__On the Relationship Between a Summability Matrix and its Transpose

the space of bounded sequences;

(Equation Omitted)

(Equation Omitted)

(Equation Omitted)

converges

(Equation Omitted)

Zq bs; bs~ bv It is known that

(Equation Omitted)

2. Theorems and applications:

Theorem 2.1: Let E, F be sequence spaces such that 0 is a(EO, E) sequentially com lete. If

(Equation Omitted)

is an infinite matrix which is an E-F method, then A' is an 0-0 method.

Proof: Let

(Equation Omitted)

Then

00 E t n Z a nk X k n=0 k=O

(Equation Omitted)

(Equation Omitted)

where

(Equation Omitted)

is the matrix defined by

(Equatio...