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f-conservative matrix sequences. (English) Zbl 0748.40002

Let 𝒜 denote the sequence of real matrices A p =(a nk (p)). For a sequence x=(x k ), (Ax) n p = k a nk (p)x k if it exists for each n,p and Ax=((Ax) n p ) n,p=0 . A sequence x is said to be A-summable to x 0 if lim n (Ax) n p =x 0 uniformly in p. This encompasses the usual summability method A, ordinary convergence and among other methods almost convergence as introduce by G. G. Lorentz [Acta Math. Uppsala 80, 167-190 (1948; Zbl 0031.29501)] Let A=(a n,k ) be an infinite matrix of real numbers a n,k (n,k=0,1,...) and λ,μ be two non-empty subsets of the space s of all real sequences. The matrix A defines a transformation from λ into μ, if for every sequence x=(x k )λ the sequence Ax=((Ax) n ) exists and is in μ where (Ax) n = k a n,k x k . By (λ:μ) all such matrices are denoted. Let f,fs denote the spaces of all almost convergent real sequences and series respectively.

The main purpose of this paper is to determine the necessary and sufficient conditions on the matrix sequence A=(A p ) in order that A is contained in one of the classes (f:f), (f:fs), (fs:f) and (fs:fs).


MSC:
40C05Matrix methods in summability