On the space \(\omega\). (English) Zbl 0635.46007

Liu, Yu-Qiang

Southeast Asian Bull. Math. 11, No. 1-2, 19-22 (1987).

The author solves the following two problems concerning the space \(\omega\) of all real sequences.
1. Given a real sequence space X, give necessary and sufficient conditions on the infinite matrix A, such that A defines a matrix transformation from \(\omega\) to X.
2. Characterize the orthogonally additive functionals on \(\omega\).
A functional \(f: \omega\to R\) is orthogonally additive if for \(x=(x_ k)\in \omega\) and \(y=(y_ k)\in \omega\) with \(x_ ky_ k=0\), for all k one has \(f(x+y)=f(x)+f(y).\)
Theorem 1: Solution of 1.
A: \(\omega\to X\Leftrightarrow\) A has finitely many nonzero columns and these are the elements of X.
Theorem 2: Solution of 2.
f is orthogonally additive \(\Leftrightarrow\) \(f(x)=\sum^{\infty}_{k=1}g(k,x)\) with
i) \(g(k,0)=0\), for all k
ii) g(k,.) is continuous on R
iii) There are \(L>0\) and \((a_ k)\) with \(\sum_{k}a_ k\) convergent such that: \(| g(k,t)| \leq a_ k\) for all t and \(k\geq L.\)
This last result is a special case of theorem 2 in “Characterization of orthogonally additive operators on sequence spaces” by Chew Tuan Seng [South Est Asian Bull. Math. 11, 39-44 (1987; review below)].

Reviewer: N.de Grande-de Kimpe

MSC:

46A45	Sequence spaces (including Köthe sequence spaces)
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Keywords:

space of all real sequences; matrix transformation

Citations:

Zbl 0635.46008