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EN
Given any sequence a = (an)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n≥1 such that y/a = (yn/an)n≥1 ∈ E. In this paper we deal with the solvability of the (SSIE) of the form l∞ ⊂ Ɛ + F’x where E is a linear space of sequences and F’ is either c0, or l∞ and we solve the (SSIE) c0 ⊂ Ɛ + sx for Ɛ ⊂ (sα)Δ and α ∈ c0. Then we study the (SSIE) c ⊂ Ɛ + s( c) x and the (SSE) Ɛ +s( c ) x = c. Then we apply the previous results to the solvability of the (SSE) of the form (lpr)Δ ) + Fx = F for p ≥ 1 and F is any of the sets c0, c, or l∞. These results extend some of those given in [8] and [9].
2
EN
In this paper we deal with the spectrum of the operator of the first difference A considered as an operator from E to itself where E is one of the sets [...].We apply these results to characterize matrix transformations mapping in E [...] or N. This paper generalizes some results given in [8] and [3].
3
On the [..]- summability and [...]-core
EN
In [6] and [9], the concepts of [..]-core and statistical core of a bounded number sequence x have been introduced and also some inequalities which are ana-logues of Knopp's core theorem have been proved. In this paper, using the concept of [..]-summability introduced by Savas, we characterize the matrices of the classes (...) and determine necessary and sufficient conditions for a matrix B to satisfy [..].
4
Infinite matrices and sigma(A)-core
EN
In [8], the concepts of sigma-core of a bounded number sequence x have been introduced and also some inequalities which are analogues of Knopp's core theorem have been proved. In this paper, using the concept of Vsigma(A)-summabiIity introduced by Savas, we characterize the matrices of the class (Vmu, Vsigma(A))reg and determine necessary and sufficient conditions for a matrix B to satisfy qsigma(A) (Bx) C qmu(x) for all x is an element of m.
EN
In this paper we shall consider the summability with speed. Let [lambda] and m be two monotonically increasing sequences (i.e.speeds) and B be a triangular matrix. In [4] there were found the necessary and sufficient conditions for a matrix M to be transformation of the [lambda]-boundedness field of normal matrix A into the u-boundedness field of B. Now we shall continue the research, started in [4]. More precisely, we shall prove two theorems which give the necessary and sufficient conditions for a matrix M to be transformation of the [lambda]-summability field of [lambda]-reversible matrix A into the u-summability or u-boundedness field of B ((Theorems 1 and 2). Also we shall consider the case, when A is a [lambda]-perfect matrix (Theorem 3). For application we shall consider the special cases when A or A and B booth are Cesaro or Riesz methods. We note that notions [lambda]-reversibility and [lambda]-perfectness were introduced by G.Kangro in [10].
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